ad-a124 841 a modified iolmoooroy-sninov … · 4 ?arariater g~ai~ina idistribution with unknown...

AD-A124 841 A MODIFIED IOLMOOOROY-SNINOV ANDERSON-DARLING AND 12CRAMER-VON RISES TEST F..CU) AIR FORCE INST OF TECH

.... RIOHT-PATTERSON AFB OH SCHOOL OF ENGI. P J VIVIRNO

UNCLASSIFIED DEC 82 AFIT/GOR/R/2D-4 Fla 12/1i N

p..W

VI LIIILu

1..25 1111.4 111 6

MICROCOPY RESOLUTION TEST CHART

NATIONAL BUREAU OF STANDARDS1963-A

-- %7

V1 0

1met

AESCN-DARLINcG, AND CRAMER-VON miIss r, T FORi THE GA:MMA

DXSMIBUTION WITH UNKNOWVINLOCATVON Ai"D SCALE PAflAmETiERS

TiEzSIS

GFIT/GR/MA/ 820 -4 Philip J. VivianloAFIT/V Capt UA

This document has bee pproved -

L 4to public release and sle; itsFE 2 4 093

distributionl is unlimited.

DEPARTMENT OF THE AIR FORCE;OAIR UNIVERSIY (ATC)

AIR FORCE INSTITUTE OF TECHNOLOGY1

Wright- Patterson Air Force Base, Ohio

*too

U,2C LASS I FI ECSECURITY CLASSIFICATION OF THIS PAGE (Whotn Data EFntered*_________________

REPOT DCUMNTATON AGEREAD INSTRUCTIONSREPOT DCUMNTATON AGEBEFORE COMPLETING FORM1. REPORT NUMBER 12 OTACS1N NO. 3. RECIPIENT'S CATALOG NUMBER

AFI T/GOR/tMA/32D-4 5.GV CE9

4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVEREDA Mt:,IFIED !<OLIMiOGOROV-3-MIRNOV, ANDFERSCN,\-DARLING, AND CRAME.-VN NIISES-TEST FR U T~THE GA;IAA DISTR.ESUTI(>J 'NITi U'4t(NO'AiN 6.PEFRIGO.RPRTNMR

LOCATI ON AND SCALE PARAM ETERS7. AUTHOR(&) S. CONTRACT OR GRANT NUMBER(@a)

Philip J. Viviano, Capt., USAF

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT. TASKAREA & WORK UNIT NUMBERS

Air Force Institute of Technolog/AFIT-Ee.!right-Patterson AFi3, Ohio 45433

1I. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

December 198213. NUMBER OF PAGES

14. MONITORING AGENCY NAME & ADDRESS(II different from Controllfig Office) 15. SECURITY CLASS. (of this report)

UrICLASSIFIED15a. DECLASSI FICATION/ DOWNGRADING

SCHEDULE

1S. DISTRIBUTION STATEMENT (of this Report)

Approved for ?,uhlic release; d'istribution unlim~ited.

'7. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, 1 .dfrent from RePort)

* 1S. SUPPLEMENTARY NOTESApi 1R 7p d Iox ibI rle. LAW APR 1907.

11 eder .yl ! Si' TRi'OLAV~f

Eircto Dean for Research and Professional DevelopmlJANi~ 1980 Forenstiutei atTchnology (AW&L

19. KEY WORDS (Continue on reverse side If necesary and identify by block number)m~oified Kolmoorov-Smnirnov 2a m. a Listribution% oditied ,%nderson-narlingj 'onte Carlo Sim~ulation11odif led Crat.er-von rises Critical Value3

* Zry-erical Distribution Function Statistics

20. ABSTRACT (Continue an reverse aide if necessary and Identify by block number)

'lie .,o1:.oorv-3mirnov, Anlerson-rarling, and Crar:'er-von ',,is--sstatistics are used to develop a ne,., test of fit for the thr-3a-

4 ?arariater g~ai~ina Idistribution with unknown location and s~calecarariaters. The ioodness-of-fit tests are valid for sa1 -zplesizes, n =5,10,.,.3C, and shapa paraoteter, K

DO I JAN72 1473 9017014orNOV 411 S OBSOLETE JCASbI~SECURITY CLASSIFICATION OF THIS PAGE (RNien Date Entered)

ULNCLASSI FI L"JSECURITY CLASSIFICATION OF THIS PAGE(Whmu Dora Entered)

A power investigjation of the tests is partormed against tenalternative distributions. The m~ost powerful test againstthe alternative distributions is the Cram~er-von .Iises, fol-lowed closely by the Anderson-Darling; the Xol,.ogorov-Sriirnov is~ the least pow,:rftu1.

A functional relationshi.? between the critical values andshape paraicieter is investigated for each test. The criticalvalues can be expressed as a function of t-,e inverse of thesquare of the shape parameter.

SECUPITv C1. 4SSI~ V, A -1

p--

W A .'.OD1F l D KOL>'!OGOflOV-S ,' IRNOV,

ANDERSON-DARLIN G, AND CRAMER-VOz MI;ES ''ST FOR THE GA,.MA

DISTRIBUTION WITH UNKNOWNLOCATION AND SCALE PARAMETERS

T*iEgSIS

AFIT/GOR/YIA/82D-4 Philip J. Viviano

Capt USAF

L

,.L,

,. .. ..-... ,*--

-. ~~~~~ ~~ ..w-.--~-- . .. .- .. .. .- . ..

AFIT/GOR/MA/8 2D-4

A MODIFIED KOLMOGOROV-SMIRINOV,

ANDERSOM-DARLING, ANO CRAMER-

VON ,,IISES TEST FOR THE GAMmiA

DISTRIBJTION WITi UNKNOWN

LOCATIOnJ AND SCALE PARAMETERS

Presented to the Faculty of the School of Engineering

of the Air Force Institute of Technology

Air University

in Partial Fulfillment of the

" equirements for the Degree of .

.aster of Science , " , *'

. . . . . . . . . . . . . . . . . . . . .. . . . .

by

• ~~pililip) J. Viviano" '"Capt USAF A

Craduate OL erations Research

December 19C2

-LJprovd for pub.lic release; distribution unlimited.

_____ -

S.a c

Pref, fe

Coodness-of-fit tests are developed for the rjamrna dis-

L.r ihkut.iori when the scale arnd location lizrameters are unspec-

i I iL; . iu L lie L-:t i 11ated f ro(i the sa,,iIle data. Tie

',liio irov.' r~jrno, Came-vo miesandArderson-Darling f

.ti.Li~ic r d usod to jevelo,) the tables. Acomprehensive

pu1 .t-v :;tutly is conducted to comapare the Kolmoqorov-Smirnov,

Cr41.cer-vofl ;-1::e3# and Ajidrson-Darlin~j g:oodness-of-fit

test-;. i-.i ij.alysis is performaed to deteri:iine the functional

ul-ionLtLij between tne critical values and the shape

1 .wo k1: 1 ike to thank my advisor, CaPt. Ibriari I.oodruff,

Wb) jui-iiice wa., instrumental to the successful completion

Iii -iidition, I would like to thank 1,r. Albert H. Aoore,

andi Lt. Col. i. ),ies D~unne; their advice and] juidance was very

v£tLo, I w:ould like to thank my wife, lane, who

sulor tud iie th roughtout my Che isrs~ c'

[ineally, I would like to express miy appreciation to mny

claz;.sfiaW: for their help~ and encouragement during the

iiroi~critt io .blof mly thesis.

1thilifJ J. Viviano

Contents

Pagcje

P)refaco. . . . * 0 a . . . . . . . . . i

List oi li /ures . . . . . . . . . 0 V

i I).; tr ac t . . . . . . xi

iiackj round . . . . . . . . 31-iwtirical Distribution Function Statistics .. 4Thie Eolwuo(jorov-)Ifirnov Statistic . . . . . . . 5

The Ariderson-varlingJ Statistic . . . . . . . . 5Tii Cruiiker-von mises Statistic 0 0 . . 0 6Problem Statement ....... . ... 6Objectives . . . . . . . . . . .7

I1I1. ',hA Gaiuwoa Vistribution ............. 8

"tie (awi~ia Density Function. ....... 8

Apilication of the Gamma Density Function . 11

-onte Carlo Sim~ulation Procedure . . . . ... 12Generation of the Three Paramaeter Gainima

tueviates * * * * * * * * * * * * . . . . . . 14

iM-axiiaiun Likelihod stiiLte for theGailia Parameters o 14

C-erivinJ thQ Hiypothesized DistributionFuinction F(x) . 0 . 0 0 . . . . 0 0 0 . 1

Power Comparison . . . . a . . . a . . . . 17Lieteringfl the Critical values . . . . . . * 0 21

4 IV. us~e 01 Taibles . a .* 0 * 0 22

Ex,1.1pl 0 0 . . . .23

Page

V. fiscusision of the Resul~ts . . . . . . . . . . . . . 25

PrULuntatioh of the Kolznogorov-sm~irnov,Cranior-von Fmiscs, and Anderson-oarlinwjTjibIL-s of Critical Values . . . .* .... 25

Coiiut(,.r IPro'jraits validation . . . . . . 0 . 2.

kv\I olt, onshi1) isetw~een Critica~l val.xui;itild Sltia~~ varawetrs.~ ......... .. .. . . . * o 44

VI U! (oii1u; o l;~ d ',co.i~ienrjat ions . . . . . . * . .47

Conclusions . . . . . * * * 0 z,17* * 4

A~~~PXA: 'iables of the Xoliaorjorov-Smirnov Critic-Alvalues; for thie Gamma Distribution o o . 52

All Pti,:ix 01 'iablIe s of the Anderson-IDarlinyj CriLivalLVLalucs for thie Gianiia Distrilbutioli . . . . . 57

AIPHX :Ta bl1e 3 of the Cramer-von ,,ises Criticail*values for the Gammta Distribution . . . . . C

t. 11 II L x ; r a p h.; of the Kolmoyorov-,,;zii rnov Criticalvalues Versus the Gamniia shakpc 6.a~~es . 7

1".~Xi IX E, cr a As of the Anderson-D1"arl1inmj CriticalpValues Versus the Gami ma Shape iP'araineterz * 98

7 11 P'., 1X I. G railhs of the Crai:er-von1 iiSUL Cr it ical'/aliaes% Versus the Gsaima Shalje Parameters 129

ivI" ..

- , ' ' LiSt of __- re

L~ iciurt

la. .,tandard Gamma Density Function with K = .5,2," 3, 4, and 5 . . . . . . . . .

1 .* Staii"ar(. (,ainia Density Function, K = 5L . . . . . 9

2. tandard K.aiima Density Function, ior 1, 2and 9 = 1/3, 1/2, 1, and 2 . . . . . . . . . . 10

""3. C'low Chart . . . . . . .13

4a. Lotnor.wl, Ci = 0, O = 1 . . . . . . .19

,'. Lognormal, W= 0, p= 2 ............. 19

.a. .eibull, I, = 3, 0 = 1, C = 0 . . . . . . . . . . . 19

5b. heibull, E = 2, 0 = 1, C = (1 . . . . . . . . . . . 19

()I . :eta, , = 1, q = 1 . . . . . . . . . . . . . . . . 20

"ib. beta, j) 2, q = 2 . . . . . . . . . . . . . . 20

7. 2hapu vs K-S Critical Values, Level = .01, "F = 5 . Go

;. Shape vs K-11 Critical Values, Level = .Cl, n =1 69

)" . Sida)e vs K-S Critical Values, Level = .0, n = 15 70

it. Shape vs K-S Critical Values, Level = .01, n = 20 71

11.,aj e vs K-S Critical Values, Level = .1, n = 25 72

12. -;iape vs &",-S Critical Values, Level = .1I, n = 3G 73

13. ;haie v-; K-S Critical Values, Level = .05, n = 5 . 74

1.4. S3ha),e vs 1;-S Critical Values, Level = .05, n = 10 75

15. ,;ijape v: K-S Critical Values, Level = .05, n = 15 76

16. S iaj'e vs K-S Critical Values, Level = .05, n = 20 77

17. .ia,,. vs N-; Critical Values, Level = .05, n = 25 70

I v. .%ntiku vs , Critical Values, Level = .015, n = 30 79

V

* t

19. Shape vs K-S Critical Values, Level = .10, n = 5 . 80

20. Shape vs K-S Critical Values, Level = .10, n = 10 81

21. -hape vs K-S Critical Values, Level = .10, n = 15 82

22. S;haee vs K-S Critical Values, Level = .10, n = 20 33

23. :ihape vs 1.-S Critical Values, Level = .10, n = 25 84

24. Sha,,e vs K-S Critical Values, Level .10, n 30 85


26. hape vs K-S Critical Values, Level = .15, n = (' 37

27. St;aipe vs R-S Critical Values, Level = .15, n = 15 8


2; . Shnape vs K-S Critical Values, Level = .15, n = 25 90

3o. :*haie vs K-S Critical Values, Level = .15, n = 30 91

31. hape vs K-S Critical Values, Level = .2(, n = 5 92

32. Shaqe vs K-S Critical Values, Level = .20, n = 10 93

33. Shape Vs K-S Critical Values, Level = .20, n = 15 94

34. f-kiche vs K-S Critical Values, Level = .20, n = 20 95


36. ;IaLe vs K-S Critical Values, Level = .20, n = 30 97

37, 'hape vs A2 Critical Values, Level = .01, n = 5 99

33. -hape vs A2 Critical Values, Level = .l, n = 10 . IOC

37. Ichape vs A2 Critical Values, Level = .01, n = 15 . 101

4C. Sthape vs A2 Critical Values, Level = .01, n = 20 . 102

41. Shae vs AV Critical Values, Level = .01, n = 25 . 103

42. Shaj.e vs A2 Critical Values, Level .(;I, n = 30 . 104

43. Shape vs A2 Critical Values, Level = .05, n = 5 • 105

44. .,lai L vs A2 Critical Vailues, Level = .o5, n = 1 o G

4%

45. Shape vs A Critical Values, Level = .05, n = 15 . 107I 46. ',iiape vs A2 Critical Values, Level = .05, n = 20 . 108

47. Shape vs A Critical Values, Level = .05, n = 25 . 109

4e . '..iiaile vs A2 Critical Values, Level = .1,5, n = 30 . 110

4(j. 1ihace vs A2 Critical Values, Level = .10, n = 5 . 111

50 . IiapL vs A" Critical Values, Level = io , n = 10 . 112

51. At.ape vs A2 Critical Values, Level = .10, n = 15 . 113

12. J;ia Le V, A2 Critical Values, Level = .10 , n = 20 . 114

53. ;hape vf; A2 Critical Values, Level = .10, n = 25 . 115

S4. Shape vs A2 Critical Values, Level = .10, n = 30 . 116

b5. pha[.e vs A2 Critical Values, Level = .15, n = 5 . 117

,se vs A2 Critical Values, Level = .15, n = 10 . 118

1)7. "hoL,, V:i :. Critical Values, Level = .15, n = 15 . 119

5. i),ape v.; A2 Critical Values, Level = .15, ii = 20 . 120

59. ha pe vs A Critical Values, Level = .15, n = 25 . 121

61:. i;hape vs A2 Critical Values, Level = .15, n = 30 . 122

(61. Thape vs A2 Critical Values, Level = .20, n = 5 . 123

62. SIjat,, vs A2 critical Values, Level = .20, n = 10 . 124

i3. Stia vs A2 Critical Values, Level = .211, n = 15 . 125

-4. Shape vs A2 Critical Values, Level = .20, n = 20 . 126

C5. .- ikj)e vs A2 Critical Values, Level = .20, n = 25 . 127

ka. pi ae vs , 2 Critical Values, Level = .20, n = 30 . 12 3

,7. shape vs , Critical Values, Level = .vl, n = 5 . 13,%)

"8. :;Aaie vs , Critical Values, Level = .(:l, n = I0 . 131

69. Lt;ha e vs : 2 Critical Values, Level = .(11, n = 15 . 132

7C;. , vs t2 Critical Values, Level - .01, n = 20 . 133

vii

• -.:. ijure

71. Shape vs N2 Critical Values, Level = .01, n = 25 . 134

72. Shape vs N2 critical Values, Level = .01, n = 30 . 135

73. Shape vs ItN2 Critical Values, Level = .05, n = 5 . 136

74. Shape vs V.'2 Critical Values, Level = .0 5, n = 10 . 137

75. Shape vs 12 Critical Values, Level = .05, n = 15 . 13876. ',hape v.; Critical values, Level = n = 20 . 1397-. ;fd;V" =, .5, n =2

77. Shape vs w2 Critical Values, Level = .135, n = 25 . 140

75. Shape vs 1.2 Critical Values, Level = .05, n = 30 . 141

79. Shape vs W2 Critical Values, Level = .10, n = 5 . 142

C,. Shape vs t2 Critical Values, Level = .10, n = 10 . 14381. ZhIape vs . Critical Values, Level = .10, n = 15 . 144

82. mae vs t' 2 Critical Values, Level .10, n = 2 . 145

8'3. :hape vs 1.42 Critical Values, Level = .10, n = 25 146

2!4. rsi jat vs , 2 Critical Values, Level = .10, n = 30 . 147

S85. ,Shape vS W 2 Critical Values, Level = .15, n =5 . 146

i. Sha.,e vs 1 2 Critical Values, Level = .15, n = 10 . 149

P7. Shaj~e vs t2 Critical Values, Level = .15, n = 15 . 150

38. .hajLe vs 1.2 Critical Values, Level = .15, n = 20 . 151

f89. Shape vs W2 Critical Values, Level = .15, n = 25 . 152

818. shap1 e vs I,2 Critical Values, Level = .15, n = 30 . 153

19. tihaj,e vs W2 Critical Values, Level = .20, n = 5 . 1540-" ,. 3ihal'o vs 112 Critical Values, Level =.215, n =3i0 153

94. sfia;e vs "Y2 Critical Values, Level = .20, n = 10 . 155

.)3. Shla 1IA! vs 1,- 1 Critical Values, Level =.20, n = 15 . 156

94. 'i:ape vs '\2 Critical Values, Level = .20, n = 20 . 157

5. Tnap~e vs .,2 Critical Values, Level = .20, n = 25 . 158

96. maie vs IN2 Critical Values, Level = .20, n = 30 . 159

viii

List of Table6

Ta t.,e Lage

I xaiaiple: x, F (X) . . 24

C ri Cante r-von Miises o w...2 27

II Andersofl-i)arling A42 e o o . o e o e 27

IV :\oliiocorov-Saiirnov K-S Thesis Critical Values 28

Aollo:oro-&LlirovK-S Lillieforiscritical Values....... ... . . . . . . .2 3

VI i~ovwer Test f~r the Gainna DistributionAl:Gattma DistriDution, K 4ot

Ai: ,nother DiStributionLee fSignificance = .0~5 3( ... .;

Vii Power Test for the Gaiqma DistributionHo Gamina Distribution, K =1.5

1i itnother DistributionLevel of Significance .05 . . . . . 0 . a 32

Vill Powefr TPest for the Gamma DistributionHO Gamma Distribution, K = 4.01

ii.: Another DistributionLe~vel of significance = .01 o o 34

Ix 1ower 'lest for the Gainia DistributionGamma Disitribution, K = 1.5

I Another DistributionLevel of Significance =., .01 o 36

x Coetticients and R2Values for theh elationships Between the Kolmogorov-"i.irnov Critical Values and the Gai.iinaShape Parameter, 1.03 (.5) 4.V~ . . * . . . . 38

X1 Coefficients and IZ2 Values for theIelationships Between the Anderson-.ir].ingj Critical Values and tlia 3amma

4 'Shape Parameters, lal (o5) 4.V o . . . 40

XITI Cocfficients and k Values for thulelationships between the Cratmer-von

i e.-r .L iL..1 Vcdius aim( tUe GammaShtap~eParameters, 1 .o (o5) 4.0 . . . . . . .42

ix

Pa,Lill

XIII Koluogorov-S:mirnov Shape Parameter = 5 .... 53

XIv Koljiogorov-Smirnov Shape Parameter = 1.0 . . . 53

XV Kolmogorov-Smirnov Shape Parameter = 1.5 . . . 54

iXVI j[ol1ioijorov-Siairnov Shape Parauieter = 2.0 . . . 54

XVII Iolio,jorov-Smirnov shape Parameter = 2.5 . . . 55

xviii i olmojorov-Smirnov Shape Parameter = 3.0 . . . 55

XIX olmogorov-Smirnov Shape Para,:eter = 3.5 . . . 54

X,X Kol1ogorov-Sinirnov Shape Paraiaeter = 4.04 . . . 56

XI Anderson-)arling Shape Parameter = .5 .... 58

Xxii p nderson-Darling Shape Parameter = 1.0 . . . . 58

XxIII Anderson-Darling Shape Parameter = 1.5 . . . . 59

XxIV ihriderson-Darlinj Shape Parameter 2.0 . . . . 59

/ o'lXXV ,\nderson-Darliny Shape Parameter = 2.5 . . . . 60

XXII Aniderson-oarling Shape Parameter = 3.0 . . . . 60

XXVII Anderson-Darling Shape Parameter = 3.5 .... 61

XXVIII Aiderson-Darling Shape Parameter = 4.0 . ... 61

XXIX Cramer-von Mises Shape Parameter = .5 .... 63.. xX Cramer-von Mises Shape Parameter = 1.C .... 43

XxXI Cramer-von Mises Shape Parameter = 1.5 . . . . 64

XXXIl Cramer-von Mises Shape Parameter = 2. .... 64

-.. 1III Crdnhor-von Mises Shape Parameter 2.5 . . . . 65

/ XX iV Cr,.ier-von Mises Shape Parameter = 3.1; . . . . 65

xxxv craimer-von vises Shape Parameter = 3.5 .... 66

AXXVI cru.ier-von Mises Shape Paramcter = 4. 0 . . .

x

A Lst rct

Tihe A nierson-D arl1 miq, Cr a ier-von ~*ises, and(- thc

! olionorov-S.iirnov '7-tetistics are usec' to devulo:' P, now test:

of fit for L~i t!rreo-,:ara.-,eter -yinmc &"istribution wit"I

unknown sihap c;w6 location -arar.eters. "Che critical values

gneratt.c; ,hero obtziine-d, by a i~n~Carlo ; ,roc.2dure. I- or

e a c, v 1uo a t rn ( s;.Ie si 4-e),5--t s a~11u its i*;ere drar.n

frot.,. a jz ta~ulCo w'iiosa shalpe is siec if ied. T I Ie

location an( scale .,roi:.cters are Lasti.-ia-teJ, fromi the data,

an, t;.:- ti 6 rec statistics are calculatc-d )oe, on thc esti-

mated distribution. The simulation was, preformed for sai-. 1Je

sizas n S 5±tl... 3(4- and sha~ 1-arauietprs, : 5 7r

isinyj gammna distributions for siia:,e ccual to 1.5 and

't~,the i~ower of each test is investigat:ed ag~ainst ten

alternative cistributions for savi;'le sizes n S 5, 15, and

30. In general both the \nderson-Darling and the Cramer-von

!,ises. tests are more powerful than the ''o1rogorov-Sinirncv

test. Excevt for the case whiere the altLernative di stri Lu-

Lion is~ lc-Jrr:' al, ttie Crainir-voi. ;1,ises test is the iniost

iowerful test.

'.'he functiona~l reliticnisliip bctween tne critical vaIlues

of the Anderson-Earlin,., Craier-von 'uises, and X<oli,ogorov-

Smirnov is also exaiiinad. 1\ critical value for a ij)4-C

* paraiiieter between 1.5 and -~which is not included in t~ie-

relationship.

xi

5 A ;AUDIED Li) tOLmiGGOkOV-SP IR(NOV, CkAM~ER-

VMi4 m~ISiS ANE, AN[WR5014I-DARLING TEST

ei Ok Tlii (CAM~MA DI STRILA0iION 1-;ITh

LJNRNO1Mq LOCA'iION AND SCALF:

1 . Introuction

Curren~tly the U.S. Air Force is placing~ m~ore and more

N eimipasis on systeim, availability, wiaintainability,' and reli-

ability, both in research and development and in day to day

operations. Of4 particular importance to the Air Force is

the atbility to predict time-to-failure of equipment.

.tudies ini probability and statistics have increased under-

* -standirU ot S-Oie Key probability distributions used in pre-

dictin; tirve-to-f a ilure. Among the most commonly used

coritirnuou . distributions in this area are the beta, gamma,

exPoniriunial, 'eihull, and logjnormal distributions.

utt,21 il, Lhese studies, analysts are confronted with

the probltem of testing agreement between probability theory

and ictu 41 observat ions. In other words, given n observa-

tions of soize variable, say ti ime-to-f ai lure, the problem is

Lo tind out it it can be recjarded as a random variable

fizcvinj a cji'v'GJ probability distrib~ution. The general

ai-p~ruucl to tke solution of this pjrobleia is known as the

yoodie~;-uftutst. In more precise terms let x 'x 2 .x~

be it ran,_Aoti :iatmole. Then a stateiient of the joodness-of-fit

* test i.:

H A: lI(X) v, W~1 x

where ix) is the actual distribution function of x and

11,,(x) is the hypothesized distribution function.

2

a kroun d

Yewo co~imunl1y used 9 oodne ss-o f- f it te st s a re the Ch ii-

-;.,uare ttest aind the Kolrioyorov-Sinirnov test. The Chi-sqjuare

te-It cottilret; observed frequkencies with exi.ected frequencies

ot tHLu JLyottiesiized distribution. It is restricted to lar-jo

1,. 1L;L-..Jrxmt 25 or gjreater (:1:73). The K\oliiocjorov-

..uiirnov (-)tuSL c0Lmparces cui,iulative trequencies between

Ihe ZCLucal ~~l ujiiw - a step function, agdiinst orrespond-

in ..i~~~usin~j tin(- hypothiesized cuildulativu distribution

junction. Oiie I(-S test can be used [or laryc or small

tdifijlLS; ihovvever, it is restricted to distributions which

kj illy sj:peci1ied. Ii. N1. Lilliefors developed a goodness-

ot-l1i t test Cor the norwal (19), and exponent ial d istr ibu-

t iu o2~ ia .ji ich can Ibv used for s~ia 11 samples where thle

~.~r~etersiust be estii mated f rom the sai.iple data. lvhen

~uri~cer5are estimated froii saiipic data, the test is said

to !,L a modified test.

I-ollowini1 Lilliefors' technijue, several other modified

tLe:,Ls have been documented. u. Cortes developed a mrodi fied

:ul ~J(Jv!.,ri~vLest for thle three-paramtner Vveibull and

*joVInlaa diLstributions (5). J. Bush expanded the .joodness-of-

lit te!SLt or ue t'.ibull to include tile i,oditied Crafier-von

S V:; i. ti~d .%riuerson-Iarlinj ( 2) tu.;tsI (30i). The modi-

Lie(, ..uiiujuruv- b3i-oirnuv, Cr~;dier-von *l1.e, nu Anderson-

;,cirlii i tu.-L_5 have also been done for thie uni form, normall ,

La 1.,x~L:., Lcx, oirientiu 1 and Cauviiy rdi~tr iiutions (11). In 1973

*.n, cictier , and iLerticjJrc!.ijneuI two new tcest statistics

3

*called the L andI S statistics. The L and S test statistics

were used to develop a goodness-of-fit test for the two

paraiieer eibll ithunkownparameters (22).

In l1"l , outrouvelis and rKellerraeier introduced a

coodncss-of-fit test based on the empirical chacteristic

function -when the parameters maust be estimated (17). This

tust statistic could be used as an alternative to the F:DFE

statistic if the characteristic function is more easily

ut LeCrmint.:,, than the distribution function.

tEhm irical iistribution Function *Stcatistics

A jererzal class of statistics used for the goodness-of-

lit tLeStS is called empirical distribution function (LDF)

statistics. wistorically EMDi statistics have been used in

cases where the parameters are either known or unknown. In

mrost instances EDF' statistics are easily calculated and are

competitivu in termns of power. This class ot statistics is

base] on a cosmparison between the cumulative distribution

function, F(x), and the empirical cumulative distribution

Eunction ,;,(X) defined as

no. of xSn W = --

n()

'he test procedure is summarized as follows: given a sample

Lrm somum jopulation, the ! DI te!sts, rejo~ct i(,:F(x)= (x

wheni the ditterence between i,,1,(x) and Sn(x) is largje. liere

L.*,(x) iLi tku hylpothusized distribution. In jeneral, EOF

tests are valid when the distribution is fully specified.

* iouwevur, 11ivid and Johnson showed that the aistrioution oi

4

S,'an L. statistic depends only on the functional form of the

distribution and not on the unknown parameters when the

estin: ated parameters are location and scale (6). It is this

principle that permits us to generate valid critical value

tables for the gaim ,a distribution which dePend only on the

shape pjarar:ieter and sample size.

It is imL portant to note that this thesis uses a modi-

fied form of the EDF statistic, because the cumulative

distriuution is not fully specified. An estimated distribu-

tion function is used whose parameters are derived from the

observed sa,.Iple.£2_. 5"_ojorov-S-.iirnov Statistic

.The Kolnogorov-Smirnov (K-S) statistic is defined as

the absolute value of the difference between F(x) and Sn(x)

or,

,'.D = I F(x) S Sn l -~ (2)

In usinj the K-.; statistic for the goodness-of-fit test, we

are intcerested in the greatest absolute difference between

i(x) and I;n(x) (2). Therefore the test statistic is

sP IFl(X)-Sn(X)I (3)

h'e Amderson-Darlinrj Statistic

It i:; known that goodness-of-fit tests which use actual

observation:; without grouping are sensitive to discrepancies

at thu tails of the distribution rather than near the nedian

(1!:2). The Anderson-Darling test statistic overcomes this

problemm by accentuating the values of Sn(X) - F(x) where the

"- test statisLic is desired to have sensitivity. More

5

specifically, tne Anderson-Darling statistic is based on a

weighted average of the squared discrepancy, (i.e. [Sn(x) -

F (x)1 2 weijhted by *(F(x)) or

An2 = nf[Sn(xl-F(xl]2 (4)Fx) dF(x) (4)

where

* (;(x)) = [F(x) " (1-F(x))] 1 . (5)

Using the computational form

nAn -n - . E12j-l)[in F(xj) + in (l-Fnj+l)], (6),. n j=1

the test procedure is as follows:

1) Let Xl<X 2 <_..._Xn be n observations in the sample.

2) Coi~q.ut : An2

3) If An 2 is too large, the hypothesis is to be rejected.

The Cramfur-von Mises statistic

The Cramiier-von mises statistic is a special case of the

A 2 with ['(x)] = 1 and is written asiiSn2 = M (X) - F(x)] 2 dx. (7)

This test procedure is the same as outlined for the

Anderson-Darling goodness-of-fit test (26). The computa-

tiurnal form used in this case would be

nV[i x ) = 21 (8).. 'n2 12n j=l 2n

!i L, ~~robl1em,___/ ;ta tement

Few joodness-of-fit tests are available to perform on

sample data when the parameters of the distribution are not

known. As mentioned earlier, Lilliefors developed a test

" for thu unspecified exponential and normal. Also, bush

6

........ ........."..".......".... ... --..... .. ,

. -. generated a set of critical values for the Weibull with

unspecified scale and location parameters (3). There still

exists the need to develop a valid goodness-of-fit test for

the janiia density function when the scale and location

paraieturs are unknown.

The purlpuse of this research is to develop a goodness-

of -fit test for the 3 parameter gamma when the scale and

location parameters must be estimated from the sample data.

This involves generating a table of critical values based on

the sample size and the shape parameter. The accuracy of

the critical values must be sufficient enough so that data,

sai. pled from other populations are rejected.

.,I c tives

This thesis has the following objectives:

1) iTo generate and document the Anderson-Darling, Cramer-

von ,.ises, and the Kolmogorov-Sirnov rejection tables

[or time three paramqter gaitima distribution where the

.calu and location parameters are unknown.

2) To conduct a power comparison between the Anderson-

iarlij., Cramer-von Mises, and Kolmogorov-Smirnov

goodness-ot-fit tests.

3) To investigate the possibilities of a functional

4| relationship between the shape parameter and the

critical vdlues in objective one.

7

7

-. ., - .. .,. ,,, ,-.. ..- - ..- . --. -

: I• The , Di;tribution

i'lhe jan.,u density function is usefuil in reliability and

,: d, intainability theory. It also has ap.plications in the

natural .;ciences. If the randoi. variable x is jwuia distri-

buted tinun the probability density furction taKes the form:

f(x) = (( <(ex (X)), (9)r(K) e

", , > 0; x> c > .

" .'iece are three parameters which Lipecify the gamma. 0 is

thu scale -;araiieter; K is the shape parahieter, and c, the

lucation pardieter.

Thu cort ;lexity and versatility of the jamia distribu-

tioo can ou observed by examininyj the graphs of the distri-

5ution for vairious; values of the shape and scale paraneters.

k itj re 1, show:i jrapjhs of the standardized gawma (i.e., c-0,

=I) tor and 5 (7:370i). isVhen <=1, the gamma is

the exponential. Also, it is interestinj to note that when

is l;. thmi one, the jamma closely resetitbles the exponen-

t i, .1 i~ Ar i but io||.

,..Lu, 1 ijure lb shows that for larje K (K=5f), the

a r0!;C!.Ilcl..s the norral distribution. eigure 2 illus-

I. rd tii intIluIcr e of on tir -jr dh of tnc jaAi.ma. here

w(. UL -=. n =2 aria s~etci the cjraj~hs fot 0 =1/3, 1/2, 1

a, ;.' (7:371).i.

. .. .

.3

.07

1.0

[Ic; 1. raphs of the standard a)amioa density function for(aK .5,2,3,4 5 and (b) t,=50

.2. eti

VI G 2. cralliis of the standard qainina density function forc 0 (, K =2 and O= 1/3, 112, 1, and 2.

"A ),lication ot the Gamma Vensit Function

The 'Jalilbaa distribution is otten used in reliability

theory to represent the distribution of the time between

lailures of a system. Assume, for example, that a system is

*;ade up ot r coiiponents, all of which must fail for the

-, - system to fail. furthermore, assume that the time to fail-

ure x i of each component is independent and exponentially

distLriuteS. Then the time to system failure Y =X 1 + X2 +

+ Xr is jamma distributed (7:369).

In clueueing theory, the random variable T follows a

juo!lia distribution, where T = X + X 2 + ... X is the total

ti.,e to -service K customers assuming that the time of

service of each customer is independent and exponentially

* distributed.

Tije twio cases described can be modeled as a special

c 0Z 01 theL! jamma known as the Erlanj distribution and

uxi.re:;sud as:

Kf (t) t ) .tK-I e-_ t' t>0. (I(;)

A random variable having a ygannia distribution has also

t,een used to represent or measure the occurrence of physical

"liino meii . [-or examiple, Slack and ,rubein (1955)

"(d,:eonstrat:d that the mean value x of radioactivity (alpharic~le per Linute) witrin a sample of Pennsylvania shale

followe,j a ..,waistrihution (7:37(0).

4"

L1

I + . . . I • ; ,i

Tfhis chapter presents the Monte Carlo simulation proce-

dure used in jenerating the critical value tables for the

rioJ i fied :"oliiojorov-Smirnov, Crarier-von ivises, and Anderson-

;i r I iii~j t uiL:;. The L)rocL'duru 1:S outi inc~ L.;in~J the flowi

citir t in I- iju re 3. Secondly, an outl ine of the power coin-

pari1 on aimonlj the thrue tgoodness-of-f it tebts is given.

Thirdly, a discussion of the analysis of the functional

relationship between the shape parameter and critical values

is presenteo for each of the test statistics.

i-ionLtC Carlo Simnulation Procedure

The followingj procedure is used to yenerate the criti-

cal value tables for the modified goodness-of-fit tests. As

weuntioned earlier Figure 3 presents these steps in flow

chart formiat.

1) For a fixed sample size n and tixed shape paramieter ;

n standard randood gamma deviates are generated using a

computur subroutine. The standard gammna deviates are

converted to random deviates with location parameter C

I,;~ and scale parameter 0= 1.

2) The ni randomr deviates are ordered, x ( 1 )' X( 2 )1

X(n

43) Th'le ordlered random deviates are used to

eLtiiiate the maximuai likelihood scale and

location parameters.

a 12

ST~AR T

Generate Random Step 1.;ammca leviates

Order iandom Step 2Ca6%i; a ;Jeviates

E'stimate Location Step 3and Scale Parameter

Determaine Hypothesized Step 4,:- Distribution Function F(x)

o 0 Calculate A2 , W2 or Step 5: I1*.-,S Statistic

.eteriaine the 80th, Step 6:-.-3 5th, 90Jth, 95th

aaij v.bth iercentilu

LW

FIG 3. Flow chart

, 1 3

4) The eLtimated scale and location parameters and fixed

.Shape parameter are used to deterj:iine the hypothesized

distribution function F(x).

5) The test statistic is calculated using equations three,

six, and eight, for the modified 1"olriojorov-Smirnov,

Anderson-Darling, and Cramer-von iiises tests respec-

tively.

6) tepA s one through five are repeated 5000 times.-

7) 'i'he value of each statistic are ordered in ascending

or, er and the 80th, 85th, 90th, 95th and 99th percen-

tiles are used as the critical values of the test.

IteneratiorL of the Three Parameter Gamma Deviates

ior the gamma distribution function, there is no closed

forsj,, for which we could obtain an inverse; however algo-

ritL, is are available which can be used to generate random

garluaa dleviates. The IMiSL subroutine GGjvtAR is used to gener-

a e standard ganma deviates in this thesis. These standard

duviate; are converted to deviates having location C = 10

and scale 0= 1. This is done by using tne transformation

z = " x + C ()

where x represents a standard random deviate. This trans-

oriat ion i nade to avoid a problem with the parameter

L!;ti;ating routine. Further discussion on this matter is

prosented iik tne following section.

,"a.<1imu;d lLikulihood Lstimates for the Gaiimia 1,arameters

ihe procedure used to calculate the maximum likelihood

es ti.ctes Lor jamma parameters was developed by Harter and

14

Moore (12). Their analysis involves the derivation of the

maximum likelihood estimators and includes an iterative

method for solving the simultaneous equations. To derive

the 1aaximum likelihood e'.uations we begin with the jamma

density function with location parabieter C>_, scale param-

uter 9 , anu sha,e i:arameter K:i:-.[(xc, Ohl) = 1 "A-c k- u x _f- _c

.f- x , -x, (12)

KTlie l iilinood function of the order statistics xl, X2,..xn

ot a l of size n is

L 1 )n lt 'k-1 exp x=1 (13)

.e wisih to Iind the values of 9, K, and C which riaximize L.

Irv Thi-; is done by taking the natural 1',garithm of L, and

sutting tlhe ,aLtial derivatives with respect to the three

pa,iiraeters ceIual to zero and solving the three simultaneous

equations. Tfe partial derivatives are shown here:

::: i ,L n, k x i- c.'r-L . "k + - (14)

i=l 09

c)in L n. -nln9 + ln(xi-c)_ n "A-K) 1 (15)

-. i=l

"-1-nT = (l ) (xi-c)-' + n i,* (1-K) + n1(

IL s3tould be noted that because ol a limitation of the

hart-r an(,+ ";oore subroutine, it is not possihle to estimate

) . tilL iaramiet ers when tne -gamma deviates are ,jenera'Led with

location C = 0:. In the event that gamma deviates are

15(

generated with C o ~, it is poss~ible to obtain negative

estimates for the parameters. The subroutine imaps these

negative estimates onto zero. Thus, C' ind *will not retain

the invariant property which is needed for these tests to be

V'11idj. In addition, the iterative techniqjue used in the

iarter and [foore subroutine does not work for the special

case whien the shape paraimiter is set to one. because the

caiais aim exponential ditribtion when the shape param-

etur is e. 1 ual to one, we can use the tm,'&ximum likelihood

estim~ators of the location and scale pjarai~ieters for the

ux1 orintil. ierefore, setting K eqjual to one and solving

uutions 13 anid 1'&: we obtain

C = (17)

anu

e=A Xi * - 1) (18)

-erivi the iyothesized Distribution lFunction F (x)

Th~e jidaximuia likelihood estimnates for the location and

scal I raiiw-rs and the fie hpeprntr deter ineI

Wx. Obtainincj a numerical value f or Fix) req4uires an

inteLjrai calculation; this calculatioii was done using the

1;i.';L suhroutine MiDCAM (13). ivDGAM calculates time probabil-

iLy tliat a railoni variable x from a standard 'jamma distribu-

to(i.c., C banid O= 1) is less tnan or equal to x. 'I'o

t r"An;iori t.-ic -!Viates, yi, trom a joneral ized gamiira di stri-

buit jog, i nt-o !tandari yamiiia deviates we use

4 aC

The details ol this transforimation are provided in Cortes

• (5: 17).

Poa.r C ci,;. l [ ition

von uIsen, d t.nderson-Dar]inj tests -ire compared for ten

alternative distrioutions. Sauiiples of sizes equal to five,

15, ,i nc 25 are drawn from tte followirt selected

.1i st r i h, t ions :

1) G , shiape equals 1.5

2) Gauunia, shape equals 2.5

3) Gawwa, snape equals 4.0

4) .cibull, shape equals 2.0

') ,ci,uIu , share equals 3.1'

C.) [jorial (10{,1)

7) beta (j = i, q 2)

P) b!eta (j, = 2, q = 2)%.I 9) Lo-jnorcial (, I , w 0 )

lo() Lojnormal ()- 2, w 0)

Thezcs, distributions are tested according to:

- .;: The samjle variates follow a gamiia distribution

havinj shape parameter K.

.. ,, The samaple variates follow some other distribution.

'lie 1,owt-r i nvustijation was conducted under two null hypoth-

"z-eb, one tor shape parameter, : 1.5, the othur for shape

L. ..cL.r, r: = 4.r0 . h rac n ot.: devi,tes for the above,iLurnative distributions were enerated using IMSL

sulroutiins.

17

-- - .... . -

Tihe lojnor,4al density tunction is written as

1- Iin. x-w ] 2x x ( [), (>O (20 )t (x) ... L. ex, p- I2.~Z),((

1= otherwise.

I id j i., ill u,Lz,.t in [ijuy es 4a, aiv! lb for the ,arameters

p W -j. JiVn above.

, n.ull .cnsity unction is

x ): X S) I ex1 , ]K , 9 >( C. x (21), exi 0

= otherwise.

and i,; shown in liyures 5a and 5b.

'ihleta density function is expressed as

f~) £~)xpll-x),i-, l:<X<l (22)

= 0, otherwise.

arn,, iLt jra i.1 for the two cases of interes;t, is presented in

I i,,Ur C )d and Cb.

, (r each oL the Saimple Si;es 11eritioned above, five

thou~A~ni ~~uAL l;CL ets were generated tor the alternative

ui~triuUciu . The location and scale !arametert are calcu-

-l ,ed rmer the null hyp othesis and tlie three test statis-

Lic, ]:-.;, A 2 I and N2 arv evaluated. ilhe value of these

stiLizLi-'S aL% cum|pared to tne critical values derived in

S tIhi t;i,:ii:.. IL the value of the statLitic is greater than

the critic ,l value, the iiull hipothesis is rejected. The

11

Ad..

Fl (G 4 a Lo~jnorinal FIG 4b. Lognormalw 0. p=1 w0. p=2.

I- 12i 5a. Ueibul1, K =3. 5T 9b. Iveibu11, K =2.11.

1. C=0. ei 0.

19

7,4

I C3 (I. EA 1eta, FIG 5b. beta,p=1. q=1. p =2. q =2

total number of rejections are counted. The power is the

total number of rejections divided by the number of trials,

Determininj the Critical Values

t. repjeatiny steps one throujh five 5[0 tiles as shown

in thu flow chart in Figure 7, 50 .,[ vAlucs for K-S, A 2 , and

1 2 trt calculated. These critical values are ordered and

the 83jth, 85ti, 90th, 95th, and 99th percentile are used as

Lhe criticaL values of tne tests.

''hlu voiraLditer pro~jrams used in this thiesis are Presented

in iqA enrdix C.

21

IV. Use of Tables

In this cha[pter a set of steps is jiven which is used

to jperfurk~ii z;joodness-of-fit test by appjlyingJ any of thle

LlhrLu c te:.ts eveloped in tiiis thesis. Als~o, an example

i1Lustr-tiCiutAj Andersun-!,-:rlinj LesL is presented.

Thu.. lolow~in~j steps are use(d to j erforii a goodness-of-

L tet:

1)i)eteri~iine tile shalpe paraireter, 1K, and the desired level

2) 1 rw~i thu d-ata to be tusted, calculate tile i~iax imum

I iK e Iihood est ima to rs f or the location and so-ale

3) L'roit the ai,;rapriate- table, select the critical value,

dcorrespondin(I to a ,the sample size n, and shiaje

4) JUi~-j the !w.x1ilauj~ likelihood estif-1-ators, determine tile

es3timated hypothesized distribution, arid use equation

tlirveu, six, or eiyjht to calculate the !oliaocorov-

.nirrnov, Anderson-Darling, or Crari-r-von Nises test

,itatis;tic respectively.

It) [Like value obtained in step fuur is jreater than tile,

critical value tound in ster) three, then reject tile

kL1.o~~sicd is-tribution. IL it is siiialler than thle

critical value, ttlen thu hly~otihesizudI distribution can

nlot, &LCuCteJ.

22

x ai't k' C

The time between failures of a particular subsystenm of

a radar system is believed to be distributed according to a

.jaitia distribution with shae parameter equal to 3.0. A

test .cimineer recorded the following tii;es between failures

o0 that sub!;ystem: 11.1, I(0.6, 10.4, 13.0, 1 1.3, 10.5,

1t.C, II:.9, i0.(, 10.8 days.

,. rioditeid AnderSon-Darling test at a .05 level of

Ss ijri i i'alice is performed usint4 the critic.uI values in this

triesi;. The proulem can be stated as a test of hypothusis,

tlat is,

i U: The distribution is gamma (shape = 3.0).

iA: The sawple comes from another distribution.

Lirst, tie level of significance a, and shape parameter K,

have been ,deteriained to be .05 and 3.o0 respectively.

er;UI', Lh tiiaxlmuiI likelihood estiinators calculated usiny

the Htarter ard ',oore subroutine are C = 9.319 znd 9 = .U2(.

reuxt, th. critical value from Table XXVI is .8415. The

hiypothesized distribution is completely deterwined by the

tiXUL.' snape paramieter and the estimated location and scale

.aramuters; these values are presented in Table I. The

value oi: the Anderson-Darling statistic is A2 = 1.7342.

,incc 1.7342 is (jruater than .""415, the null hypothesis is

roj-ected. Therefore, the conclusion is that the sampjle of

tiiiu betweoun failures comes from soie othier distrLC 'ion.

23

'±AL LL I

i t x L(x)

1 10.4 .1532 10.5 .1633 i1.6 .2034 10.6 .2115 10.8 .2696 10 . 9 .3 07 11.1 .3846 11.3 .4329 13.0 .822

l0 18.6 .999

42

| 24

V. Discussion oL the i<esults-

Tkiis chapter presents the results obtained with respect to

the objectives stated in chapter 1. These objectives were to

develop modified Kolm~oorov-Smirnov, iAnderson-uarling, and

Cr,-Iite[-VOn i-isus tes~ts for the qartm&wz ans,' coiiipare their

o w ers. Also included was an investigation of the relation-

si ji Luctwien tie cri tical values of each test and he

pa raw.et !r s. Included with these results is a report on the

val1 'latioii of the coni,,uter progjrams used in this thesis.

ire :;Ltc.Lrion of ttue I~ln~rvSnroCramer-von Mises,

andi Aneron-jrin Tables of Critical Values

The tables of critical values for tthe modified K-S, A2 ,

aa * 2 tust , are presented in Append ices A, 6, and C,

'-Alil theu Slape paraitetur is fixed, both the K-S3 and A2

critic.,il values are decreasing as the sai~iple size increases.

Tile rate- UL dlecrease is siialler as n increases; this is an

ir~dic'atiori that the critical values appear to be convergingj

for larcju siml szs The Cram-,er-von :iises critical

values, ol 'lhe other hand, are increasing with respect to

thte saii, 1 le size. Again, the rate of increase is smaller for

LaIr~1er valties of n, indicating that the critical values are

convor jiIn tor ]care sam-,ple sizes.

ILt shouldib notedJ that becauIsO the critical values are

Je.r ivc>i LIiruj~jli Monte Carlo imul at ions, the values are iiot

error frue and toat ttic,aimount of error decreasus as the

nui,,&ur o1 Lriu.1s increases (25). The 5000, repetitions used

25

77 7

in this thesis was a practical compromise based on computer

time required and accuracy desired.

L£- Luter L) h Validation

he computer trograis are verified by generating crit-

ical values for the exponential distribution with unknown

Wean. 'hi.; is done by generatiny gamma deviates with shape

[arameter equal to one, fixing the location parameter at

sume arbitrary value, and estimating only the scale

parameter

'I'le critical values are calculated for sample sizes n =

5, lo, 2U, and 30. The critical values from the Anderson-

Darlinj and Cramer-von Piises statistics are modified using

expressions (23) and (24) derived by Stephens (27):

~A 2 (1 + 1 5 _ 5)(23)n n2

'IN2 1i + ._-1 ). (24)n

*T1i e comiuted critical values are compared to those

calculated by Stephens (27) for significance levels .15,

S'hiie critical values calculated for the Kolmogorov-Smirnov

statistic for the exponential are compared ditectly to those

c(erivuld by Lilli',fors (2 ). The critical values which are

derived from the programs in this thesis are presented in

Ti,le IV and can be corompared to Lilliefors results in Table

IV .

''ic critical values for the Kolmogorov-Swirnov statis-

tic cumipared very well to thi. Lilliefors values. The

26

crainer-von miesW

2(1 + .16/n) Stephen'S1-x ICr itical

n= r=l(j n=20 f iz30 values

S .42.149 .148 .1523 .149.177 .7 .174 .17,r .171

.5 .22C .21 .221 .221 .224

.9 .341 .332 .335 .325 .337

i\nd(rson-Darlirvlj A~

I~ ( + 1 .5/n - 5/n-) St e Ph en s1-x:- Critical

nn= I n=20 n=3.I Values

.41.992 .5 61.922

(5 1 ."17 1. 3 81 1.37G 1.372 1.341

0 2.7t; 2.219 2.077 1.974 1.957

27

TABLE IV

i Kolmogorov-Snirnov-__Thesis Critical Values

1- n=5 n= 1 n=20 n=30

.85 .378 .276 .19.) .166.401 .295 .214 .178

5 .447 .328 .235 .193

.99 .531 .384 .27t; .232

TA uL L V

Kolmogorov-Smirnov K-SLilliefors Critical Values

I- n= 5 n=lf) n=20i n=30

k.5 .382 .277 .199 .164.9'r .4C6 .295 .212 .174.95 .442 .325 .234 .192

.99 .5o4 .3803 .278 .226

28

jreatest deviation occurs for n = 5 at significan'ce level

.0l. The Cramer-von Mises values generated are very close

to Stephens values with the greatest deviation being 3.6%

for n = 3(o and significance level at .0I. There was also a

ood matcl between the Anderson-Darlin,. values, with ilost

:.eviations 1,etween 3% ant 4%; however the 'jreatest deviation

i 13.4, tor n i0 and signiticance level .il.

v'ower .Irivest i.jat ion

,m power coij. arison is made between the iKolmogorov-

Sinirriov, i\nder son-Darling, and Cramer-von Plises goodness-of-

fit t*.x ts cieveloped in this thesis. The gamma distribution

4ith sha~ie parameters equal to 1.5 and 4.0 were both used

against the alternative distributions listed in chapter III.

5).,a l sizes five, 15, and 25 were used in the power studies

at botil an a-level of .0,5 and .01.

T'ah1,es VI through XIII show the results of the power

co,;j.ar i:tons tor a-levels .05 and .(I. 'hen the null hypotl-

esis is truu, the power meets the claimed level of signifi-

canc,_ to the second decimal place in most cases. For all

test-i tiie power is low for sample sizes equal to five. In

tact, in ;,iost cases for n = 5 the power is nearly equal to

the s -jnificance of the test, indicating that the goodness-

of-tit test has no practical use for very snall sample

. z e- s.

Iii nuerly all cases, the power:; of the Cramer-von Mises

and/or ,nuersor,-Darlinq are qreater than Kolmogorov-Smirnov

test!;. Fased on this study, the lattLr test would not be

29

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434

used as lo 9j as tihe first two are available. An examination

.1of Tables VI through XIII reveals that for both levels of

sijnificance, the Cramer-von Mises test is more powerful

*}. than the Anderson-Darling test. The only exception to the

previous stitement occurs when the alternative distribution

is lo -noraal.

K' 'Y'he following observations are macde concerning the

power of both the Cramer-von Mises and Anderson-Darling

t~tSt when samlle sizes are either 15 or 25:

1) It the null hypothesis is a gamna with shalpe equal

to ].5, the i,ower is high acjainst all alternative distribu-

tio ns exceopt 1or another yamma. The tests are especially

hijni a:ainst the lojnormal distribution.

2) Nhen the null hypothesis is the yamoa with shape

ejual to four, the power is hi9h ajainst the lognormal and

normal distributions. The power is not qu te so high

ajairist the beta. Agjainst the other yaminas and the VWeibul]

witit shipze ekual to two, the power is low, even when the

sa , 1,le size is 25.

F<lation shi between Critical Values and Shape Parameters

An investigjation of the relationship between the shape

L, drz,,uter and the Kolmogorov-Smirnov, Anderson-Darl-ing, and

(Cra;,,r-von mises critical values is summarized in this sec-

tion. Cor each test statistic, the shape parameter versus

critical values are plotted. These grallis are presented in

-A-Lpp.iiix P for the K-S, Appendix E for A 2 , and Appendix F

- for th '.' critical values. The 9rajihs of all the test

K:,. 44

statistics appear to exhibit a common and consistent behav-

ior. This consistent behavior observed from the graphical

roi)rusentations can be suLinaed up as follows:

1) For all test statistics, the relationship of crit-

ical values as a function of shape is always decreasing for

shai-u -jreater than one; this decrease' appears to be an

inverse relationship.

2) T'he graphs of the Anderson-Darling critical values

al. .ys show an increase as the shape increases from .5 to

3) ThIe loliltogorov-Smirnov and Cramer-von Mises crit-

ical values increase or decrease, as the shape varies from

.5 to l. I, ciependinq on the sc.:iile size.

/ regression analysis is perforoed to determine the

furwntional relationship between the critical values and

shape Iaratmetcrs, as suggested by the yzaiphs. The study

inludus values for the shape between 1.5 and 4.0). Shape

it r4:,eters leiss than 1.5 are not considered in the regres-

,:sion c.,,alysis because a different estimating technicue was

U-,vu to calculate the critical values for the gamiaa when

I1hua,- iS u ual to one. In addition, more information is

nredud about the behavior of the function between .5 and 1.5

* for all test statistics.

"el exj.ru-;sion which best re ,resents the relationship

lor ll te:,t statistics is

C = aV, + a 1 ( I / K

S, tti. AUI,.bCK whic measured the amount of variation

a4

ill all case'i. This expression can bie used to find the

critical value corresponding to shar-e parameters between 1.5

and 4.0, riot found in the tables in hppend ices A, E3, and C.

'±heretore, a test of hypothesis can be 1,erformed for the

null hypothesis beiny, Lor examnplu, a cjami.-a distribution

haviiivj sflape equal to 2.75.

The vi.lue of i 2 and coefficients a0 and a1 are re-

cordod in Tables XIV, XV, and XVI, for the 1K-S, A2 , and.2

criLiCal value!.; respectively.

I4

VI. Conclusions and Recoiienda tions

Conrc I us ion s

based on results obtained in tnis thiesis, the following

conclusions are noted:

1) 11tic Kolmorjorov-S.1mirnov, Anderson-Darlinj and

Crairier-von :izscritical values for the three-parameter

.aCI~IIaa a~re valid. The power study revealed that when the

null Iiyj.othesis is true all three tests achieve the claimed

level of sit.nificance.

2) The power comparison study based on the ten alter-

native distributions listed in Chapter 3 shows that in

jent,-ral thie powers in decreasingj order are W2 , A2 , and K-S.

'Yh ie A2, however is more powerful against the lognormal

distribution. All three tests dc!nonstrated low power for

sari:Ae sizes eqlual to five, indicating a coodness-of-fit

t--st; involviroj a saw~ple size of f ive using the tabled

critical values would not be practical.

icos..ei')tion.

The following recommendat ions. are sur'>jested for further

investiqjation:

1) uuvelo7) a more efficient technique to calculate

theO wa~x1irtuIi likelihood estimators for the parameters of a

(Jani.ij distribution.

2) Investigate a functional relationship between the

* saz~j~Ie sze and critical values sC) that gjoodness-of-fit

tu~ts can be done for sample sizes other than those

psented in this thesis.

47

- 3) Ixtend the goodness-of-fit test to include

parameters butween zero and one.

4) Lxamine the feasibility of developing a goodness-

of-fit test for distributions whose parameters are unknown,

based on Lhe characteristic function.

-4

i48

1.Anderson, T. and D. Darlinj. "A Test of Goodness oflit", Journal of Amderican statistical Association, 49:765-7G9 (1954).

2.ArI3traditer, B. lkelialbility Niathenatics. New York:mvcGrw-ill1 book Company; 1971.

3. irush, J. "A modified Cramier-von misci~s and Anderson-

Force [)ase, Ohio, 1931.

4. Capcn, P. A Practical Approach to Reliabilit.Lonooni: Busines s Ciooks, Ltd., '1972.

.Curt--es, ."A modified Koliaoorov-Smnirnov 'rest for the(ai~ia and loeibull Distribution with Unknown Locationand 5cale Parameters", Unpublished ml,' Thesis. Air;:orce Institute of Technology, V.right-Patterson AF'I,01hio, 19J0o.

6. David, V'. and N. .Johnson. "The Probability IntegralTiransformation When Parameters are Estimated from the

~;ja1c, Limetra, 35j: 182-11930 (1948).

7. kler~ian, C., L. Glesery and I. 01kmn. A Guide to i~elia-bilit- Theory and .L2 1 Ii cat ions. New York: Holt,iMnehart, and lhinston, Inc., 1973.

6. Lhil lot, 3. and C. Singh. Engineering 1,elability NowToce iLO, and Applictions. Now York.- John hiley arnd

. ELron, 13. "bootstrap Methods: IAnother Look at thlei~tkkn~e" The Annals ot Statistics, 1-26(17)

10' ~;iLbons, J. 4onarametric Statistical Inference. iwewYork: ucGraw-1ill book Company, 1971.

11. ,rucn, J. and Y. I, jazy. "oeflMdfe-~

6ouolness-uf-Fi t Tests", Journal of the A,;er ican*Stzitistical Association, 71:- 2T-2 9 (fL))**

12. Harter, Ii. and A. ,oore. jImaxim~um Likelihood kEstima-Lion of. Parameters of Gamma and k.eihul1 Populations.t r o v Com~pl ete a nd f rom rcensored Samples",Tu'chno;etrics!, 7: 6;39-643 (19 5).

13. Interniational Mathematical and Statistic, Lihrary7 --rnc -ianuTToco Houston: LmS;L, 19331.7

49

-. - "- ° : -- -S -- .... . " " .-. --.. . .

14. JohrISon, N. and S. Iotz. Continuous UnivariateDistributions. Vol. 2. Boston: llouton-Ft,7 iTT'(T Tn Co.,197).

15. Johnston, J. "A Modified Double Monte Carlo Technique% to Approxirziate Reliability Confidence Limits of Systems

with Comiponents Characterized by the Weibull Distribu-tiof)". Unpublished iy1s thesis. Air Force Institute of'Technology, bright-Patterson AFB, Ohio, 1980.

11. tilahal1, L'. "On the Choice of Plotting Positions onvrobaility Paper", journal of the Aerican statisticalAssociation, 55: 546-56J (196U).

17. Koutrouvulis, I. A. and J. Kelleriiieier. "A Goodness-of-kit Test Based on the Einperical Charteristic Func-

tion ihere Paraiheters Miust be Esti.nated", Journal ofthe zuyal Statistical Society, Ser, 43: 173-176 (19BT

Ig. Law, A. M., and W. 1. i'elton. Simulation Modeling andAnalysis. New York: McGraw-flill uook Company, 1982.

19. iilliefors, Hi. "On the Koliviojorov-Sinirnov Test foruiorimality with Mear. and Variance Unknown", Journal of

the Aiaerican Statistical Association, 62: 199-402

2:. ."On the JXolmo.jovor-!'hirnov Test for the Expo-nential Distribution with Mean Unknown", Journal of theAierican Statistical Association, 6_4: 387-399 (1979).

21. Littell, it., J. McClave and 1,:. Offen. "Goodness-of-FitTests [ot the Two-Parameter 'Weibull Distribution",Com.,,unications in Statistics, B8 (3): 257-269 (1979).

22. .;ann, :4., E. Scheuer, and K. Fertig. "A New Goodness-of-Fit Test for the '1wo-Parameter Weibull or Extreme-Value Distribution with Unknown Parameters",

1.CowIunicat ions in Statistics, 2(5): 383-4u0 (1973).

23. Li'Lorhiall, W. and R. Schaeffer. viatihematical Statis-., tic.-, .ith /.L ications. North -cituate, Ma: Duxbury

K 24. V2to, i . and . Leu "Weibull Distributions for Continu-

ous-Carcinogensis Experiments', biometrics, 29: 457-470(1973).

25. 2iioo,,aAi, M. Probability Reliabilityi An EnjineerIn i

/\2 j roach. New York: McTraw-hill book Co., 1968.

p.-

50

r7

2C. Stefletnc', .I. "The Anderson-Uarlinj Statistic". Unpub-1. ihied Technical Report, No. 39, for the U.S. Armyihestcarclt Off ice, Stanford Univer.-ity, Stanford,California, 1979.

27. -:"DF Statistics for Goo('ness-of-Vit and Soi.ieCowldjir isonS", Journaal of the f\jier ican Statistical

28. c~ton Tabl 531.7. (1974).in iednjAjrxm~ePec

La--JL Point!s of thle Statistics D, V, 'v , U , and A inI'jnitu Samples of N Observa tions", ILionnetrika Tablesf or !-titisticians, Vol. II, Li., CanbrillWeTT1:92, ip.359.

15

PPENDtIX A

:dj*,Of the K~om9arov-1znirnov Critical

Vcilucs for the Gaiana Distribution

I5

TABLE XIII

Kolmogorov-Smi rnovShape Parameter = .5

'avi)le Level of Sitjnificance

n 2 o 15 .u o5 .01

.3521 .3681 .3913 .4330 .52.17

10 .2637 .2798 .3010 .3331 .3862

15 .2249 .2365 .2514 .2804 .3317

2 0 .1967 .2077 .2225 .2457 .2941

25 .1759 .1870 .2006 .2227 .2623

3f, .1635 .173f .]A44 .2(139 .2449

TABLE' XIV

Kolmogorov-Si rnov8 hape Parameter = 1.0

Sampie Level of Significance,-. ~size,•-

n .20 .15 .10 .05 .01

5 .3701 .3848 .3994 .4433 .5269

BI .2651 .2788 .2958 .3267 .393J

15 .2184 .2306 .2451 .2709 .320J

.1923 .2038 .2177 .2392 .2774

25 .15,93 .1788 .191) .2113 .2520

30 .1561 •1650 .1759 .1958 .2297

53

I

TAb L E XV

Kolmogorov-Smi rnovS8hape Parameter = 1.5

aiu le Level of Significanc;i ze.

n .20 .15 . ;,., .,'

.3333 3505 373: .4151 .4725

,i .2496 .2624 .2797 .3045 .3597

2 '6C . 21&i7 2336 . 2554 .2995

.i41 .1935 .2058 .2244 .2658

21, .1624 .1713 .1 12 .1979 .2320

3'. .1507 .1579 .1677 .1 39 .2222

TA13LE XVI

Nolmogorov-SmirnovShape Parameter = 2.1'

S aml P1 eLevel of SignificanceSize

n .20 .15 .10 .05 .01

5 .3255 .3423 .3635 .3927 .4482

l;; .2432 .2567 .2730 .296) .3377

15 ..2885 .2105 .2235 .2464 .2845

.1748 .1838 .1976 .2157 .2480

.1591 .1671 .1773 .1943 .2264

3*1463 .1537 .164s .1793 .2099

54

ITAbLE XVII

Wohno roy -SmirnovShape Parameter = 2.5

.Laile Level of Significance

n .20 .15 .1G, .. 01

5 .319 .3366 .3.81 .3898 .4399

37 . 037 .2483 .234 .2884 .3378

15 .1963 .2067 .2187 .2414 .2814

.1718 .18k3 .1911 .212 .2457

25 . 1552 .1636 .1738 . 1890 .2209

3,,.1417 .1483 .1584 .1730 .2023

TABLE XVIII

golI moorov-Smi rnovShape Parameter = 3.0,

Saiile Level ot Signiticance-~ize

n . .15 .111 .05 .01

.3179 .3342 .3524 .3799 .4365

'2338 2449 .2595 "2829 .3282

.1939 .2041 .2172 .2352 .2751

2.1710 .1792 .189[; .2067 .2395

25 .1520 .160o .1695 .1U58 .2130

4 3(, .1411 .1478 .1573 .1711 .2026

55

5 ITAIL: XIX

Kolmoyorov-Siairnov______ Shape Parameter = 3.5

a, pLevel of Significancen S~.ize,-

LI •2 .1-5 .lif' A _Al

5 .3161 .3317 .3508 .3769 .4314

lI .2310 .2421 .2561 .2812 .3234

15 .1923 .2001 .2127 .2326 .2724

..1695 .1775 .1884 .2059 .2403

25 .1511 .1588 .1689 .1829 .2127

310 .1382 .1452 .1543 .1684 .1992

TAiLLLE XX

Koliaogorov-Smi rnovLihape Parameter = 4.0

barjlu Level of SigniticanceSize

n1 .20 .15 .1. .0 1

5 .3131 .3289 .3471 .3731 .4266

lb .2308 .2416 .2567 .2792 .3267

15 .1904 .1992 .2116 .2312 .2666

2c 1, 7 .1748 . 1ir,3 .201',2 .2312

., .1505 .1570 .1466 3 .2137

.1381 .1450 .1544 .1690 .1970

56

a.

qi APPENDIIX Bi

T A hles of thu Anderson-Dariinj Critical

Valuej for the Gainina Distribution

57

- 'TABLE XXI

Anderson-Dar IinjShape Parameter = .5

Level of Significance

.20 .15 .0 .0:5 .01

5 1. 0724 1.1679 1.3135 1.62332 2.5899

-0 .9224 1.0295 1.1919 1.4375 2.2323

P, .8797 .9824 1.1258 1. 4 083 2.2116I.

2C; .87 04 .9759 1.1352 1.4831 2.1989

25 .8367 .9472 1.0963 1.3910 2.0626

3.) .8332 .9428 1.1107 1.4316 2.3048

TAbLE XXII

Anderson-Darl inL'hape Parameter = 1.11

',',dlI},ICLevel of Significanceizen o.2o .15 1( lu .5 .0 l

2.0880 2.3079 2.5926 3. 0710 4.3921

.1.501 5 1.6604 1.8807 2.2980 3.496

15 1.2827 1.4396 1.6493 2.0360 2.9562

1. 1.2124 1.351, 1.54CO 1.9o38 2.7789

1.',)831 1.2147 1.31'54 1.7118 2.5678

3;' 1 0558 1.1750 1.3756 1 6C94 2.4482

58

TAiLL XXIII

Anderson-Dar 1 ingShape Paraldeter = 1.5

Wap1I V Level of Signiiicance

r, .2i) .15 .iL .0:5 .0 1

5 .69C8 .7577 .6491 1.(!603 1. 430C,

1 kI .7104 .7d 32 .9017 1. 0120 1.5597

15 .763 .7940 .9(107 1. 1607 1.5828

2k: .7145 .79(;3 .9120 1. C',9 23 1.5715

2 .7278 .8j53 .9170 1.1332 1.5569

3! .7296 .8105 .92 G 1.1189 1.6133

TABLE XXIV

Aav-er:on-Darl inrgShape Parameter = 2.0

1',a, cLevel of Ligniticance.o .15 Ic .1 1, 5.1 1 .05 .0)1

5 .6106 .6666 .7417 .8673 1.1757

10 .6265 .6940 .787w .924;1 1.2260

4 ]! .6330 .6927 .7880 .941;5 1.3451

6, .6422 .710] .8 151 5)2 1.3437

4 .5429 .7002 .7984 .967o 1.3956

3(, .,4 b .7167 .U213 .9976 1. 4,)4 ,

4 59)

TAbiLE XXV

Anderson-DarlingShape Parameter = 2.5

Level of ,'8ignificancei '>;i ze ,__ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _

n .20 .15 .10 5 .0l

! .5700 .6261 .6964 .8324 1. 197(,

,1 •.5f",9 I G 5 C,4 .74019 .8740) 1.2080

15 .5915 .6556 .7330, .8693 1.2046

2, 1.9 .6699 .75 4 .9029 1.2276

25 .5991 .65918 .7462 .9023 1.2328

"30) .51311 .6573 .7394 .91J25 1.2614

TABLE XXVI

Anderson-DarlingShape Parameter = 3.11

Level of Significance

.20 .15 .10 .0'J5 .I

5 .5549 .5962 .6730 .7V13 1.0634

Ii .5687 .6276 .7 056 .8415 1.1702

15 .5757 .6376 .7251 a615 1.1651

2() .5749 .636w .716" .8755 1.1499

. 25 .5708 6364 .7225 .C80 3 1.1714

K 3,) .5352 .6490 .7355 .8 28 1. 2601

K- .

"46

TABLE XXVII

Anderson-Darl in

Shae parawc-ter = 3.5

acl , Level of Significance

n .21, .15 .1c .05 .01

5 .5472 .5967 .6588 .7748 1.0173

il .5522 .6024 .6730 .8 u72 1.1318

15 .5577 .6209 .6942 .8081 1.115V

2o .5686 .6259 .7061 .8432 1.1503

25 .5603 .6190 .6960 .8418 1.159{1

3; .563 .6130 .6973 .8298 1.1755

TABLE XXVIII

iAnderson-Darl ing-hape Paraw eter = 4.11

, :a .1p Level of Significance

n .20 .15 .1) .05 .01

.5293 . .,464 .7535 .9715

* i! .5475 .5996 .672 .8 1.1020

15 .5435 .6 ()6 .681( .8154 1.1036

2 G .5565 .6060 .3915 .82(44 1. 1021

25 .5557 .6124 .6998 .8425 1.2094

43; .5577 .6193 .P)7 o .8239 1.161 Ii.

APPENhDIX C

'Pal-les of the Craer-vof l is3es Critical

V~le~;for the Gammna Distriboution

66

.- - - 4 k L : - ' L ¢ /-_.

, :- --,- *: -i -: - . -. : - , - - -- -- -

TAbLE XXIX

Cramer-von Mises

Shape Parameter = .5

i r;at1e Level of S;ijnificance, ;ize

n .20 .15 .111 .05 .01

5 .1217 .1363 .1576 .1928 .3200

0 .1327 .1530 .1795 .2259 .3569

15 .1379 .1596 .1863 .2340 .366o

20 .1425 .1625 .1918 .2526 .3743

25 .1453 .1654 .1951 .2514 .3769

3o .1440 .1649 .1986 .256E, .4170

TABLE XXX

Cramer-von MisesShape Parameter = 1.0

Sarijple Level of Sigjnificance8 ize

.20 .15 .10 .05 .01

5 .1397 .1561 .1809 .2314 .3571

.1341 .1511 .1775 .2210 .3573

15 .1310 .1490 .1738 .2112 .332 .'

90, .1316 .1484 .1748 .2220 .3327

25 .1299 .1487 .1742 .2250 .3239

31 .1319 .1501 .1757 .2228 .3415

63

TABLE XXXI

Cramer-von kIisesSha1 e Parameter = 1.5

-°titpltl Level of S;ignificance-i ze,V, 2 .15 .10 .05 .i

) .1092 .1210 .1390 .1741 .2363

1(, .1132 .1268 .1471 .1015 .2671

15 .1133 .1287 .1541 .1846 .2740

20 .1199 .1353 .1551 .1917 .29,2

S.1157 .1307 .1503 .1847 .2791

3.) .1171 .1324 .152 .1870 .2872

riBLE XXXII

Cramer-von misesShape Parameter = 2.0

p a e Level of SignificanceS i ze

n .20 .15 .10 .115 .01

5 .10I7 .1118 .1261 .1A81 .2096

.1 i45 .1160 .1336 .1620 .222F3

15 .104C .1154 .1325 .1627 .2409

2C; .1055 .1169 .1355 . I([86 .2404

25 .1051 .1174 .135r .1695 .2481

,..36) .1077 .1220 .1415 •1739 .25901

64

'l'TAtOl" XXAIII

Cramler-von i A1'L s.(;hiat.e Iyar ajeter = 2.5

i ;al, pLe I.evel of ,ijnificance

.2o .15 .1 .15 .01

5 .095,3 . D153 .1187 .1432 .19 (;j

. . l1j3 .1263 1522 .2194

109013 .1100 .1261 .1531 .2217

2 . 1) 91 .1097 .1256 .1531 .2207

-.21979 •1099 .1277 .1547 .22.19

-7 3i .1091 .1255 .1543 .2237

TABLIE XXXIV

Cramer-von Njises;hape Parameter = 3.r,

,e Level of Significance

n .20 .15 .i .05 .01

9 .927 •1011 .1145 •1367 .898

o 4 •09 .1042 .12.0 .1441 .2052

15 .0l954 .1075 .1234 .1501 .2085

S'942 .1057 .1214 .i514 .2,1(1

25 .(,()35 .1050 .12-24 1524 .2137

3.9,957 1'b1 .1247 .1521 22 6

I| 65

TABLE XXXV

Cramer-von Mises,hape Parameter = 3.b

;a!,,iple Level of ,;inificance

n .21 .15 .10 0 5 0 1

5 .,919 .1009 .1127 .1341 .1827

10 .0914 .1006 .1149 .1396 .2007

15 .0925 .1032 .1172 .1418 .2002

21 .0939 .Iu51 .1203 .1473 .2112

25 .0920 ."1134 .1139 .1447 •2115

3; .926 .1024 .117(; .1416 .2056

TAHLL XXXVI

Cramer-von Nises

Shape Parameter = 4.0

::J:;aplu Level of Significance

•21 .15 •10 .0 5 il1

.0839 .0968 .11 V9 .1295 .1763

12 .8906 . 102 .1150 .1391. .1974

15 .915 .1F12 •1152 •1396 .1967

2'. .0908 . 115 .1151 .1424 .1959

2, .916 1021 .1182 .1433 .2110

0- - 3. .092 .10 19 .1178 .1431 .205c

r..,

AIP

~.L otthe Korl ~o-i.irnovo ofroCritical Values versus the

LEVEL=.Ol N=5

Lli

co

ci

0

Th. co 1.00 2.00 3.00 4.00SHFIPE PARAMET[ER

FIG; 7. iie. vs K- Critical Valucs, Level =.'.I1, n S

46

SHAPE VS CRITICAL VALUES

:. OnririA

c'J_ LEVEL=.O1 N:-IO

D

0

)

w

(r)

-J -

-C)

v-

0

0.00 1.00 2.00 3.00 4.00SHAPE PARAMETER

"I'; b. ",,iU vi r-8 Critical Values, Lev l .t , n = 01

T9- I

5 SHAPE VS CRITICAL VALUES

GAMMA

r- LEVEL=.Ol N=15

CD• CO.

(.f) c-

1J-o

co:

r-J

.. -J- cici

H-

coN

)(04u -

0. 1.00 2.00 3.00 4.00SHAPE PRRAME 4ER

" , v; V-: Critical Vaue, Level , n

7 1)4 7

SHAPE VS CRITICRL VALUES

GAMMA

LEVEL=.0I W420

C)

LuI

CE

C:

I-

NN

0

'b. oo 1.00 2.00 3.0 40SHPPE PRRRMETER

1;jI. : ii vsi K-S Critical Values, Level =.021, n 21

71


GAMMA

LEVEL=. 0 N=25eli

-4

cli

U)

ci:

Li

CJ

C) 0

00 1.00 2.00 3.00 4 .00SHRPF PRMETER

L i, 11. I..;,e vs K-S Critical Values, Level =.ln =25

U 72'


GAMMA

LEVEL=.O1 N=30

C-.

'0)F-4.

)

WC.)".001.02.03.04 0

-JP PRMEE

I Ao si rtclvle , QC 0 ,n 3

U7


GRMMFI

LEVEL.05 N=S

1)

__j

CC

I-

CV)

Th. oo 1.00 2.00 3.00 4.00SHRPE FRRRMETER

* LI~ 13 . halpe V-. K-S Critical Vj 1uU.S, LoVi 1 .05, n=5

74

SHAPE VS CRITICAL VAILUES

Cn

* LEVEL-.as N=10Co

0)C\1

*0)

C)

Luo)

-J

C-4

0)

b.oo 1.00 2.00 3.00 4.00SHAPE PARAMETER

11,, 14. chj': vs L.-S Critical Values, L~evel =.5,n =1

75

- ~ ~ 1 77 X .~.

SHAPE VS CRITICAL VPLUES

0:)

LEVEL=.OS N=15

CDr-

C)

C'

CELei

p..

03

ctO 1oI. 00 2.00 3.00 4.00

SHRPE PRRRMETER

FIGj lt. Ai1ai1 o vs K-S Critical Values., Level = ,n =l

76


GAMlMA

00

*LEVEL=.05 N=20

0

C)

UI)

J

cC

S.-

U

030

C3

0

c'*b 00 1.00 2.00 3.00 4.00SHRPE PRAMETER

~IUU-. 'haout. v.. IV-S Critical values, Loe1 .05, n 20

77

SHARPE VS CRI.TICA~L VALUES

LEVEL=.05 N=25

04

N

CD

LI)

-J.

-J OCc

(D

'0.00 1.00 2'.003 00 40

SHPPE PRRRtIETER

LIG 17. ;tji.EA juV! K-S Critic-,Il Values, Level =.015, n=25

78

SHAPE VS CRITICAL VARLUES

GAMMA

-4

LEVELz.O5 N=30C14

0

~C%j

-j

Cc)

OD

(D

0.00 1.00 2.00 3.00 4.00SHRPE PRRRMETER

:,1( l6b. t vs i,-,s critiCa1 Values, Level =.0~5, n 3

79


.-. G A M M A

LEVEL=. 10 N=5

U)LLJ C)

;-:)

:Li

Co

C)

-" •oo 1.00 2.00 3.00 4.00SHRPE PRRRMETER

ti( l '. W , v' ;-' Critical Values, Level .18, n =

...

_ .S , . . . , . . " . . ' . _ _ - . , . ; . .. . . ...


LEVEL=.1O N=10

C0

-j.cCJ

CL

CD

CJ

Cet 00 1.00 2.00 3.00 4.00

SHAPE PRRRMETER4

[I GJ v: 1..-Sh Cr it ical Values, LeVe 1 1~ n

81j

0-0124 941 A MODIFIED KOLROGOROY-SMIRNOY ANDERSON-DARLINO AND 2/2CRANER-YON NISES TEST F.. CU) AIR FORCE INST OF TECHWRIGHT-PATTERSON AF8 OH SCHOOL OF ENGI.. P J7 VIVIANO

UNCLASSIFIED DEC 82 AFIT/OR/NR/82D-4 F/G ±2/i N

lomEohmohmohismhhhhhhhhhhhEhhhhhhhhhhhhE

v.A .

L~L6-12.

.. k


LEVEL=. 10 N=15

U)LLJIcJ

CE

-Jw

Coi

0b 0 1.00 2.00 3.00 4.00SHRFE PRRRMETE<

Vc 21. fiapu vs K-5 Critical Values, Leveil =.10l, n =15

82


GA MM A

cliCr)

YAC%j

004

:)

Dco

-. o .020 .040SHPEPRMTE

-J sK5CitclvleLee 1,n 2

L83


GAMMA

0*LEVEL=. 10 N=25

0

N1

N

-- 4

Co

-oo 1.00 2.00 3.00 4.00SHRPE- PlRfM E T E R

* i.23. 'Iia,,, vs U-F Critical ValueS, Level .1(!, n 25

* 84


GAMMA

LEVEL=.]0 N=30

.- °

U)

_..

-

0)

13.00 1.00 2.00 3.00 4.00SHRPE PRAMETER

FI(G 2 4. 'A4Lc vs I-'Cri tical VatlUCS, Lovo1 -lilt ~ 301

85

.". ..- - - - - - -

.'iA*s' -- - . - -- - ..-7.

SHAPE VS CRITICAL VAlLUES

GRMMR

a)

LEVEL=. 16 N=5

cf)

C,

Li0)

D

10 0


GAMM iA

OCxCo

LEVEL=.15 N=10

COJ

-S-

o-J

aC

0I

'b o 10 .0 .0- 40SHPEPicRMEE

FI (.!U p - riia aus ee

'-47

SHAPE VS CRITICAL VRLUES

GAMMllA

LEVEL=.IS N=15

Cr)

LIJ

CE)

af

~)

oh~o I . 00 2.00 3'.00 4.00SHAPE PRAMETER

F iIG 27. '.-Aiape Vs K-IS Critical values, Level =.15, n =15

El8


GAMA(0

LEVEL=.15 Nz2OC%1

C"-

ID

Li)

:Dca)

uat:

00 1.00 2.00 3.00 4.00SHRPE PHRFMETER

2L U v 1:- Critical VIducL;, Level =.15, 11 20

89

SHRPE VS, CRITICRL VALUES

LEVEL=.15 N=26

U-4

LU

-j

'01.01.0203.04 0

SHPEPAAMTE

w 9 tkCevsi- rtca aus evl .1,n 2

c,90


LEVEL=. 15 N=30

C0

NA

'-4

CD

U)

tL 0

C)

-b0 10 20 3.0040SHPUPRMEE

F1G 10 Salev -sciiclvlus evl .5,n 3


GAMMA

S LEVEL=.2O N=5

CfD

3 C)

W)

cCr

c-) 0:Cf)

CJ

0 0oo 1.0 2.00O 3 .00 4.0 0SH-RPE PRRRMETER

h~IL 31. 1J1,ie vs K-S Critical w~ilues., tcvol =.2(;, 5

92


C - LFVF[=.?l N=10

D

X

C

-4

U

L)

b.00 1 .00 2 .00 3 .00 4.00

~I. SHAPE PRRMETER

FIGi 32. v6 ih-s critical Values, Level =.20~, n= I

9 3


'2 GAMMA

LEVEL=.20 N=16

'CD

:)

ci

-j.

OD

Th.oo 1.00 2.00 3.00 4'.00SHRPE PRRPMETER

IG33. :-,ni Aju vs it-s critical Values, Level =.2o~, n =15.

94

SHAPE VS CRITICRL VAILUES

LEVEL=.20 N=20

'-4

Ld0i

-

-4

Luc

-J

L) .01.0 0 3.040SHP PRMEE

16 3 .- v; c iia a UQ ,Iee 2 ,n 2

-95

_ - . .... 7T77 77~7T


GAMMA

*0

D

=)to

(D

U,

0

-b. oo 1.00 2.00 3.00 4.00SHAP'E PAIRAMETER

* i HU 3r.~ip v:-. K-S Critical Vu~ ee ~

9; A


GAMMA

LO,

LEVEL=.2O N=30

u -)

D

'-o

m)

-b o10 .0 .040SHPjPRMEE

vs-JCiialVius ovl .1,n 3

cJ97

APPENDIX L~

I..aijis of the AridersoiDaln

Critical Values Verse.s the

Cainna Shape iParaiaer.;


GAMMA

C0Co

LEVEL=.D1 N=5u')

CI)

uJ

CCC:)>0j

I-"Cb

Wkw

Co

Co

'0.00 1 .00 2.00 3.00 4.00SHRIDE PRRRMETER

42kIG.1 7. ;hli v.; A2 Critical ValUes, LeVel =.(!I, n =5

4 99


GA MMRi

C)

0)

-4c

LU

CD

CD

4-

FIG38.~Iaf' v~h 2 Critical values;, Level = ~,n lo1(

1oci

No-.


GAMMA

cJ

LEVEL'. O N=15

C)

0

U

>0

C-- to

0

| '-4°

,U).

0

--

b.0oo 1.00 2.00 3.00 4.00SHAPE PRRHMETER

Ll

'I C; 3'. -v; CW Critical Valut:s, LaveI .t'I, n = n ')

101


:1 : GAMMA

LEVEL=.Oi N"20

C,I °JCC

-

)

OD

c. . 0 1.0 200 300 40

LJ2

0 2

.- .O0 1.0O0 2.0O0 3.0O0 4.O0

:% SHRPE PRRMIIETER

... ' " LIG 4.;. IhL , vs A2 (ritical Valaes, Level = .(1I, n = 20

i. .. ' . .- ,, . .. ' -. ,' ." . -. - " . ' ",-- ..' .' - - '- - . . ., . ". . . -- . . -'_ -. _ • " " . " - -' .' .- " " -


GAMMA

0-4

LEVEL=.O1 N=25

0,a,

0iL °-J

CI

0

04

0q I II

boo 1.00 2.00 3.00 4.00

SHRPE PRRRMETER

I,; 41. 2ha, vs r Critical Values, l.evel = .ol, n =

103

U SHAPE VS CRITICAIL VALUES

LEVEL.0I N=30

C3)

LUJ

c-a

C0

cr_

c1 00 1.00 2'.00 3.00 4.00SHRPE PRRRMETER

I'I G £2 . :',hope vS A2 Critical Values, Level o .0, n =311

104


GAMMA

. LEVEL=.05 N=5

C)

0

C

(-j

cc

C)

>0

_.

I-~

-o

0

00O0 1'.O0 2.00 3 . 0 4 . 0

SHRPE PARRMETER

kI(; 43. L-'-we v A Critical Values, Level .05, n 5

10• q 1 5


GAtMMA

LEVEL=.05 N=10

(cm

-LJ

CC>0

to

C)

'b.o .0 .03.040

SHP PRMEE

FI .,aev . rtia0a~s ee (5

- - - - - - -- - - - - - - - - ..74.


LEVEL=.05 N=15

N

LuJ

-E

>0

":

C)

I-

0

w

cb o 1.00 2.00 3.00 4.00SHRPE PRRRMEIER

iii~~~~ *). .ii v z-critical Value~s, Le VO1 il-, n 15

107


GAMMAip.

LEVEL=.O5 N=20

C%

-4"

0(0LUJ

-:CE>0 CD

b°'1

cc

D

O:c%

'O 0 1.00 2.00 3.00 4.00SHAPE PARAMETER

, "l'l,; 4%. ."l,;[e v~s A2 Critical Vailues , Level = .(.,5, n =2v:

10

%-"

." -.- '-.-, ' ".' - . ''2 ' . " _ -: - , ,. ,. , . - . ' . , . " -0- ' -

U SHAPE VS CRITICAL VALUES

GAMMA

LEVEL=.O5 N=25

C)

U

c-

>0 c

I-

14 0

0

Tho .0 . 2.00 3.00 4.00SHRPE PRRRIIETER

FIG 47. Iat vs A2 Critical Values, Lcvkil =.0~5, n =25

10~9


GAMMA

LEVEL=.05 N=30

C3J

CD

Lli

:D>0

Tw-i-

QCJ

cb 00 1.00 2.00 3 .00 4.00SHRPE PRRRMETER

.7 1 ( 'i. J11JOL' V:,; A2 Critical Val~ue.-;, Level = .;-5, n=

lk)

07

LEVEL=. 10 N=5

0

IC

LJ

CC

C-)

0

0

9j. oo 1.00 2.00 3.00 4.00SHRPE PRRRMETER

VIC' 4'. :h A2Criticali ValuCS, Levul =.(,n 5

SH-RPE VS CRITICAL VRLUES

GAMMA

LEVEL=.10 N=10

cli

0

(D-

-J

CE

CJ

C)

I-

0

Th 00 1.00 2.00 3.00 4.00SHAPE PARAMETER

*L P.i; ::11aijc VS A Critical Values, Level =.1c, n =1

112


p ,q ?" 'GAMMA

LEVEL=. 1O N=15

(DC)

cc>- C

0>0

(3

I-o

,.00 1.00 2.00 3.00 4.00SHRPE PRRRMETER

VI G 111. '~i. VS A2 Critical Values, Level = 1o, n = 15

113


GA MMAP

LEVEL.10 N=20

bLJ

>0 C

0."J

-9

0O,

CD-

Th. oo 1.00 2.00 3.00 4 .00SHAIPE PARAMETER

FIG 52. .,ha,-v vs A2 Critical Values, Level *lt, n =201

114

SHAWPE VS CRITICAL VALUES

LEVEL=. 10 N=25

c:)cm-

-LJ

a:C:00

1-4

C

'.0 1 .00 2.00 3.00 4'.00

SHAPE PARR~METER

[It, 53. :t1 1' VS A: 2 ritical. values, Leve~l .l:n

SHRPE VS CRITICRL VRLUES

K GAMMA

C"D

LEVEL=.10 N=30

- -4

0.2

0

CJ

U

-J

CI:C00

G:

F-

L.." C:-

"0,

CD

I II

00 O0 1.00 2.00 3.00 4.00SHRPE PRRHMETER

H IC 4 11 a ie vs *2 Critical Values, Level = .10, n = 30

r11

SHAPE VS CRITICAL VRLUEIS

GAM MR

* 0

C - LEVEL=.15 N=5

0

-C)D

-:)

Li

U-:

-00o.0 .0 3.0 40SHrE PRcMTE

02

11

S SHAPE VS CRJTICRL VRLUES

GRMMA

* LEVEL=.15 N=10

C3

::D

CE

LU

~-J2F a I rtclVlus :vl .b


GRMMA

0CO

LEVEL=. 15 N=15

U-

LUJ

CJ

Il-

0

0

OD

Th.oo 1.00 2.00 3.00 4.00SHRPE PARAMETER

5i s7 'tiapc v-, i\ Critical Values, Level1 .15, n~ 15

119


LEVEL=.' IS N=20

0

C~

>0

12


GAM MA

tEDLEVEL=.15 N=25

C.,J

LLJ

-

)

-4

'C)

M

'b. oo 1.00 2.00 3.00 4.00SHRFE PRRRMETER

!IC IL' VS A2 Critical Values, Level .15, n 25

121


" GAMMA

Q

LEVEL=.15 N=30

:C:

0

U0

(I)

_J

>0

0

.-

I I I

9" O0 1 .00 2.00 3.00 4.00

SHAPE PARAMETER

. I'I.; <i;. "i,,, vs A2 Critical Values, Luvel = .15, n = 3(

122


GAMMA

LEVEL=.20 N=G

0

CJ

LUJ

CE:

-o:C'.

C).

0

CV 00O 1.00 2.00 3.00 4.00F SHAPE PRIRRMETER

i1~1.~v., A~ CritiCal Value:;, Lcvl =.20, n 5

123

SH-APE V3 CRITICAL VALUES

GAMMAC0

LEVEL=.2O N=10

Lu

t-4

c

C)

~~124


C)

LEVEL=.20 N=15

~0

CJCM-

LUJMJ

CC:>0C

C:

0

Lo

9b. 0 1.00 2.00 3.00 4.00SHAPE PARAMETER

6B (3 Woa*,0 V6~ A- Critical Valucs, Level .20, n =15

125


GAMMA

CDLEVEL=.20 N=20

-4

CJ

-J-:

a:)U;2b

C)

'b o 1.00 2.00 3.00 4.00SHRPE PRRRMEITER

FIjG C4. Sjha; 2 V I; A2 Critical Values, Level .20, n =20

4 126

SAPE VS CRITICAL VRLUES

CR MM R

C0

LEVEL=.20 N=25

-

-

0

CI)'LO

b o10 .0 .040SHP-PRMEE

>0' . ri ia a u sLv l .0

012


GA MM A

* LEVEL:.20 N=30

0

UC-

C)

L)

F-

CD

0

Th. oo 1.00 2.00 3.00 4.00SHRPE PRRPIMEIER

I'lG 6f,( Shwje vs A2 critical Values, Ltovel =.201, n =31

1 23

APPI'''L)Ix k,

Gr~ Li ts o f the Cranter-von t-tise~

Critical Valueii Vor,.,es Ltie

Czmhia shap~e Plaraiiieters

120

,. .- 4 - -.. -.

*SHAPE VS CRITICAL VALUES

GRMMR0

: LEVEL=.Ol N=5r0

C;

C"-

Li

23

-Jccl%

C)J

0

Th. oo 1.00 2.00 3.00 4'.00SH-RFE PRRRMETER

II(7. vi;~C criticalu values, L.ovel .01, n

130

... *

' .c -'.-.'-- -t

- SHAPE VS CRITICAL VRLUES

GARMMA

' LEVEL=.O1 N=1O

C.)

Co~

Li:D

>Co

,JJ

i

CE)

(If C'J

Cbo0 1.00 2.00 3.0 400SHRPE PRRRMEIER

*'u rGE 5. iha,,c vi. h" critical Values, Level =.(')I, n li

131

aa

SHAIPE VS CRITICAL VRLUES

GAMMA

LEVEL=.O1 N=15

C)

N

LUJ

.~j

I-

(J5

U

c-

9.00 1.00 2.00 3.00 4.00SHAPE PRAMETER

i; ~m mz~. ~ 2CriLic'1 Values, Lcvel=.1,n=1

* 132

SHAPE VS CRITICHL VALUES

GA MM A

* LEVEL=.O1 N=20

C7)

C,

-

to

'00 .020 .040

SHPEc:;MTE

cc2j G 7; ,hiov rtia au-),Lv ln 2

I13


GAMM1A

LEVEL=.D1 N=25

0

C)

LLu

-JO

C)

C

C3JC I

'0.00 1.00 2.00 3.00 4.00SHRPE PRRRMETER

K ~~ ~ L v tiG N1 ,d~ CriLiCal Values;, Level = .i1,

134

7 SHAPE VS CRITICAL VALUES

GRMrIR

LEVEL=.OI N=30

0

Q

C;

(no

-J

c:Cr)

cin

I-

I...

D

4 1w

Th 00 1.00 2.00 3.00 4.00SHRPE PRRRMETER

ii72. !jaj;c w;~ Critical Values, Lovel .0ll, n 3

135


GAMMA

C:)- LEVEL=.05 N=6

C)

Q

LU

LiCE

CE

N

1.L2.0 '0O 40

ULJ

0


GAMMA

_ LEVEL=.05 N=1O~0

)oLUI

-

ic;-a:

--

(-5

•I III

-00 1.00 2.00 3.00 4.00SHRPE PRRRMETER

'1 -4. .%'2 Critical VlIues, Level = .05, n IA)

137

- . . --!,---

SH-RPE VS CRIIICRL VRLUES

ORMMR

C~j CEVEL=.05 N=150

CJCLi-i

CE

C-

F-

co

Too 1.00 2.00 3.00 4.00SHAPE PRAMETER

tFic 71,. VL 1 -.2 Critical1 Value s, LCV(el =L.5, n =15

138


GAMMA

(D

C;- LEVEL=.05 N=20

0

CJ

LuJ

-JccE:

>0

.13


*LEVEL=.05 N=25

N

C

o

i

-Jc

Lj )

14


GAMMA-

to

0* LEVEL=.f35 N=30

0

)

L)

CJ

co>0

'o N10 .0 .040SHP PRcEE

C.J2

FI 7( hvCs.IDrtclVle , ee 0,n 3

-14


GA MM A

CMj* LEVEL.1O N=5

C).

-9

-LJ

cc

I-

C'.'

00 1.00 2.00 3.00 4.00SHRPE PRRRMEIER

lAG ~ ~ V 7 U 2~ ~ Critical Values, Level .10, n 5

142


LEVEL=.1O N=10

C)

LuJ

CE

0

c4

cb 0 1.00 2.00 3.00 4.00SHRPE PRRRMETER

12I IG '8 .1p v ; W Critical values, Level .ln=

143


GAMMR

0CJL .1 N 1

C

-i

Li

>-(

F-

$- It

9xoo0 1.00 2.00 3 .00 4.00* SHRFE PARRMETER

Flu !I. Shatu v.. . Critical Values, Levul h 1t, n 15

144


GAMMA

C* LEVEL=.1O N=20

0

0

CE

LU

-

C:

(3

'0.00 1.00 2.00 3.00 4.00SHRPE PRRRMETER

F IG 1!2 ;iicm ~,,~critical Values, Level Iol~, n =20

145


K GAMMA

N LEVEL.IO N=25

CJ

M0

-4c

C(f)UC3

0

C 1.0 0 2.03.0 0SHP PRMEE

(J21:.I-p s Ciia VleLvl 1" n 2


GAMMA

C i

0

u-I

c-

>LO

-E

0

9oJ.00 1 .00 2 .00 3 .00 4.00SHRPE PRRRMEITER

4

iv;.- ~. ~ V. Critical ValueS, LtuVel =.(,n 31

74

SHRPE VS CRITICRL VPLUES

GRMMR

0,U,

LEVEL=. 15 N=5

0r)

U)

LLCD

)C)

01


GR MM A

0

* LEVEL=.15 N=10

CD

0

)K

U)Lij C

0

C)

oh o 1.00 2.00 3.00 4.00* SHRPE PRRRMETER

i~; r. ~ vs Critical values, Level =.15, n=

g 149


GAMMA-

! Co_ LEVEL =.15 N=15

Ln

U)

-is

CE,:. --

U

0

0

& t

-b. oo 1.00 2.00 3.00 4.00SHRPE PRRAMETER

i IG ,f7. l vs tN2 Critical Values, Level .15, n = 15

151)

U.' SHAPE VS CRITICRL VRLUES

GAIMMR

LEVEL:.15 N=20

c15LLU

CE

-j

cr

)

1151


GAMMA

cm LEVEL=.15 N=25

C-

C-

LLJ

CC

CE

-o

)

00

C1 1.00 2 .00 3.00 4.00SHRPE PRRRMETER

ii. ~.v. ~ V ** 2Critical Val1ues, Lovel =.15, n =25

152


GAMMAi °i

I CJ LEVEL=.IS N=30

Lo

CE

CE

C)

C)

1.00 2.00 3.00 4.00SHAPE PARAMETER

,

FIG V .wp V 2 Critical Values, Level = .15, n =3

153


GAMMA

C0

LEVEL=.2O N=G

0

'0

Dc'.C,

CE

0

'b o 1.00 2.00 3.00 4 .00SHRPE PRRRMETER

F'1G 9, 1. S1hape vi; N2 Critical Vailues, Level =.202, n =5

* 154


LEVEL=.20 N=10

OD

C),

-4

(I)

LU

cb.0 .0 0 .040

-J. PRMEE

I' . .cr t c l V lu s e e .1, n 1

-15

SHAPE VS CRITICRL VRLUES

GAMMA

0)

LEVEL=.2O N=15

CO,

LLJ~~

n-4

CE:

-

C-.j

00. 0 0 1.00 2.00 3 .00 4.00SHRPE PRRRMETER

1, IG1 93. iiape v.i critical ValuteS, Level .20, n =15

156

. SHAPE VS CRITICRL VRLUES

GAMMA

C,•) _LEVEL=.20 N=20

C-

.4-

4')

b.4

•~- . -.

crp-.,,-

C)

-4

C,

'"1 00 1.00 2.00 3.00 4.00SHRPE PHRRMETER

FIG ".. L-, vs '.2 Critical Values, Level = .201, n = 2'

157

. . ~ ~ ~ ~ Li . ..--.. ..-- . .- / .. :

SHAPE VS CRITICRL VRLUES

ORMMA

Li,p•-4 LEVEL=.20 N=25

w--..

C31

LU)

°. 4.

cr

-Ja:

4 C)

c -j5-' 0 I I

.00 10 2.00 3.00 4.00SHRPE PRRRMETER

I'

- . iIC V. , ,.,, Vi 2 Critical Vi.luei, Levul = .20, li 25

158

SHRPE VS CRITICAL VALUES

GAMMA

LEVEL=.20 N=30

U4

4

'Oo..

cr

)

',-J.

-D

-159

*g 0

cb~oo 1.oo 2.00 3.00 4'.00

SHRF'E P RIJE TER

:'-" ~I, ,)* .tla~ vuj t.* Critical Values, Level' .?. , n =3

159

'N D I X G

~PROGRAM AD

C

C *THIS PROGRAM GENERATES rHE 4OOIFIED A-D STATISTICSC o5000 RIPS *

C THE TABLES GENERATED ARE VALID FOR THE GAMMA DISTRIBUTION a

C ft = SAMPLE SIZE : 5%(5)930C IsSSz 0 IF SCALE PAKAMETER (THETA) IS KNOWN aC *SS1 = 1 IF THETA IS TO HE ESTIqATEDOC oSS2 = 0 IF SHAPE PARAMETER (K) IS KNOWNC *SS2 = 1 IF K IS TO UE ESTIMATED aC eSS3 = 0 IF LOCATION (C) IS KNOWNC *SS3 I IF C IS TO BE ESTINATEDC eCl = INITIAL ESTIMATE OF C (OR KNOWN VALUE) aC eTl z INITIAL ESTIMATE OF THETA (OR KNOwN VALUE)C *A= INITIAL ESTIMATE OF ALPHA (OR KNOWN VALUE)

COMNON/VALUE/P(100)COMMON/RAY/T(100)COMMON/MANA/NSSI*SS2.SS3,NwCer tI o1NRDOUbLE PRECISION OSEEDoToCloTLeAIDIMENSION FX(6bO)AA(50001XK(002)oY¥T(5002) 6(50)INTEGER REP9PPOSEED=150 00.000MR-O

NONE :0NZERO=OREPzbO02NDS=REP-2NUMZREP-2

CCC &CALCULATES 5000 PLOTTING POSITIDNL. ON INTERVAL (0911 At (I-053/NC sTtwESE ARE USED FOR INTERPOLATIONS ON PERCENTILESC

VYI REP):I.00 405 L=2oREP-l

VTY(L):( (L-1l-.51/NOS

405 CONTINUEC 6END OF PLOTTING POSITIONS R3UTINE

READ AO#,SSS2. SS .53*CT rl*AtPRINTavSSl#SS2pSS39ClvTloA1PRINT*PRINTPh~INTOPR.NTt(2XA.F5 .1et'SHAPEz:9AlPkINT9(2X9A)9 so ----------- 8PRINT*

C *SAMPLE SIZE ASSIGNED HERE00 100 PP:15025N:PPM:N

C *LOOP FOR MONTE CARLO SIMULATION STARTS HERE00 99 KK:=15000CALL GGAMR(OSEEOAlNeGP)

161

*~0 119 coKri 9N

CALL VSRTA(PNl00 3 11=19N

3 CONTINUECALL GAMMA(tCSJ rSJASJ)

C *CALCULATES LST[MATEO FIX) FOR~ EACH SAMPLE POINT00 335 LzlohM=IPIL)-CSJI/TSJXI=ASJCALL MDGANME9I9PRO8gIER)FXtL I=PROBIF(FX(L).EQ.Go)T1IENFX IL 5-F N LP*. . 01NZERU=#ZERO.1END IFIFIFX(L).EQ.L.)THENFX4L 3:FX(LI-.OO01NONdE NONE.1END if

333 CONTINUEMA 0:0.XN=N00 500 1=19N11:1MAD=WAD.12.eXI-l.).(LOGIFK(I )3*L06(l.-FX(N*1-I5)I

500 CONTINUEMAD=I-MAD/xkd-XkAAtKK):UAO

99 CONTINUECALL VSRTAIAA,5000100 400 L:1,REP-2

400 CONTINUECALL ENDPTIXoVVREPgNUN)

CC *PAINTS, PERCENTILESC

PRINTl(2XA,12)lv9FOR N = 0PPPRINT9(2X9A)990 ----------PR INTs00 410 J=8099595DO 420 11=1.REPI=REP*1- IIIFEVYtl ) LT. (J/100.O3 ITHEN

ZZ=-SLOPE*XX(I I.VVII)PRlNTt(2X9A, 12jA#F9oW9'

19THE*,JoIT PERtCENTILE IS',24(J/100. )-ZZ)/iSLOPEPRINT*GO TO 410ENO If

420 CONTINUE410 CONTINUE

10G2

00 430 AK=19REPK=REPOI-AKIFITTIC) .LT. .99)TIEN

60 TO 999END IF

430 CONT INUE

Liz- SLOP a EXt K) ( K)PR INT* (2 XA, f9.4)

IOTHE 99TH PERCENTILE IS2,24.99-ZZI/SLOPEPRtIN T*PRINraPR INT&PRINT*

100 CONTINUEPR INTO (2X9F9o* 94v~44XF.) qCSJqTSqJvASJElio

CCCC 0SUBROUTINE TO EVALUATE ENDPOImrsCCC

SUBROUTINE ENDPT(XXYYREPvNUHIINTEGER REPDIMENSION XX4REP),TY(REP)SLOPE=(VY(2)-VY(3) )/CKX(2)-XXIS)a3zlY(2)-SLOPE*XX(2)Viz-el SLOPEIF(VI.LT*Q*)THEN

VI:O.END If

SLOPE:CYIINUMI-VV(NUM*ib)I(XX(NUN)-XE(WtUN,1))B1i:VT(NUH)-SLOPEBeXINUM)V2=(I.-01)/SLOPEXX4REP)=V2RETURN

163

PROGRAM PIOGGI 74/i, OPT"- FTN 5*1*52g

PROGRAM 110G61

C THIS PROGRAM GE1dCRATES A POW7q STUDY BETWErk THE FOLLOdINGv'C KS.CRAMII_ VON-MIStS, A4O7_RS0N-4ALINGv AND CI-SQUARE STAfC 5000 REP"C THIS POWER STUDY IS VALID FOR rTH GAMMA DISTRIBUTIONC N =SAMPLE SIZE z25 TC SSE:O IF SCALE PARAMETER TH-T IS KNOWNC SS1:1 IF THETA IS TO BE ESTIMATED 4C SS2=O IF SHAPE (9) IS KNOUVdC SS2=1 IF IKI I-S TO BE ESTIMATE)C SS3=0 IF LOCATION (C) IS KNOWNdC -SS3zl IF C IS TO BE ESTIMATEDC CI=INIFIAL ESTIMATE OF C (OR KVOMN V4LUE)'C ' TI:INITIAL ESTIMATE OF THETA ()A K14OWN VALUE)C AIzINITIAL ESTIMATE OF 9 (OR KIDUN VALUEDC s* 1U *t 4. k~ a 0. 6 ,~ a &

COMMON/VALUE /P(100 ICDMHON/RATIT(1001COMMON/MANA/)..SSI.SS2,SS3,MC1,TlA1.mqDIMENSIONe FX(601.@FIX (60),

2AAWdCVM(SOOOI .AAhA(50001,AAKS(j000)DOUBLE PRECISIOV OSEED*T#:1,rIAIINTEGER PPOSEED:25000. ODDHR=0RWCVM:O0.RWKS=O.

NONE :0710 CONTINUE.

READ eSSI*SS529SS3*CI ,TlAIPRINT' .SSESS2.SS3*C1,TI*AlPRINTPRIN~tkPRINTPRINT*42XvAvF5.I1 ' .SHAPE~' ,I1PRINiT9(2X9Al* to--------------------PRINT-

kPPI

M=NdDO 99 KK=195OOOCALL 6GNLG(OSEED.N*D.,1.,3#DO 719 IKZ1,NP(IKI=1. P4LKI*10.

4 119 CONTINU.CALL VSRTA(PoNlD0 3 XJ=ION7(IiI:P( IjI

3 CONT INUECALL GAMMA4CSJTSJvASJi)DO AS Lz1,N

U:(T(Ll-CSJl/TSJ

PROGRAM P10561 74/74 OPrag FTM 5.1*52B

XX:ASJCALL MOGAM(U.XX*PROBIbIR)FX(LI=PROBFIX( Li :F I(LbIF (FIXtLO.iQ.O. ITHENFIX(L=F IXtL)*.oO01

END IFIF(FIX(L I .o.. to I THENFIX(LlzFIX(L)-.0001NONE=NtON.* 1

TOP: 0.BOTz0.DO 500 I119

RL =IF(RL/XN-FIX(Jb *GT. TOP$ TOPzRLIKN-FIX(I)

UCVM=UCWH.(FX(Il-(2* Xl-?.olf(2* XNIS 2

500 CONTINUE

IFSBOT *GT. OIFlOIF=BOTIKS=DIF

AAKS (KICIOFIF(WKS &GT* *27)!lRKSRiISe1.MCWM=WCVP41. i(12.,XNIAAMCVM(KK i=WCVMIF(MCvM .GT. *1391)RWCVNRICVIO1*WAD=4-UA D/XNI-XNAAWAO(KK I:UAOIFIMAD *GT*.803IP RWiAD:RWADe1

99 CONTINUE'CALL VSRTA(AAKS,50000CALL VSRTA(AAWCVM950001CALL VSRIAAuA3#5O0OiPRINr.PRINT- 9*NZEROZ ',NZEROPRINT*#INON-= *vNONEPRI[NTPRINT. ,'SANPLE SIZE z 09PPPRINTPRINTPRINTb is*TOTAL RrJECTION 2 FOR I-Sz ',RWKS/5000

* PRINT'PRINT-91TOrAL REJECTION I FOR JCVR= foRWCVM/500APRINTPRIN to 99TOTAL RZrJEC TION I FOR d43= *,RWA3/5004PR INTPRINTPRlkT'(2XF9.4tAXF'i*iAKF9.AI'CSJTSJASJ

*~ ENO

CCC *SUBROUTINE GAMMA CALCULATES MLE PARAMETERSCC

SUBROUTINE GAMMA(CS.,,TSJASJ)COMMON /kAY/T(100)COMMNI/NtA/t4,SS1,)2.SS3,M.C1.TlA,MNiDOUBLE PRECISION TvCTtIErAALPlA9OLT9DLA9 ALOLC.CETHvEftENELNMDOUBLE PRECISIONM EMrkEI D2TOrD2A.OA,02CDCENSGAMGMAGAMT ,BMAIDOUBLE PRECISION GMA12,OEXP.DABSDLOGSLDGD6I ,06[2.SRS1.DGANDOUBLE PRECISION DGANqIELCS~,TSJASJoC1 ,T1,A1,OSEEODIMENSION C(110Q),TWETA(jI00),ALPHA(11Q0)DIMENSION OLT(t0) ,OLA(505 ,AL(501,OLC(5O),CE(50)T4(5O5Ji:20

THETAC 15 T1ALPHAE 1)=AU

9 EN=NEM=M

86 ELhM=0.0EMR=MRMRP=MR .1

al NMNNH.DO 88 I:NMN

as ELNN=ELNN*DLOGCEIDIF'CMR)669899109

109 DO 110 I11"

110 ELhM:ELMM-OLOGCEIS89 DO 63 J191100

IF (J-11 66,112,111III 5j:jj-l

If IJ-JI)6,139,139139 IF (J/J"-.JJ/JH)6,6,111I11 .12zJ-2

J33J-3IFCSSI1 1 15119,118

11R O2T=THETA(JJ5-2.0O.THETA(J2,.T1ErALJ3IOT:THETA(JJ1-rTETAJ2IF 102T)135*119,135

135 NT=DABSCDT/O2T)60 TO 120

119 NT=999999120 IF (SS2)122,1229121121 02A=ALPNA(JJJ-2.D0eALPHA(J2).ALPNALJ3)

DA=ALPtIAC4J)-ALPHA(j.25:F 402A)1369,12291316

136 hAzOABSIOA/02A)GO TO 123

4122 NA:999999123 IF (SS3)12591259124124 D2C=C(JJ1-2.iUO*.2)CJ31

OCzC iJj -C (J2)IF (CLJJ55.D-5T(15)140v~I252

1

SUBROUTINE GAMNA 74114 OPT=i FTN 5.1.520

140 IF (C(JJ)-5eD-5J125vI259l11

137 hCzOABS(OC/02C1GO TO 126

125 ft.Cz,1*j991126 ftS=2*MIN0(tdTvNANCI

IFtNS5fi68,12

138 ENS=NSIF CSSI)12791279128

127 THETA(%§)=THETACJJ)60 TO 129

128 TICA(J)zTHtTA(JJ)*(OT..25D0e(--iS.1.DOQID2TBE4STHETA(J) :DNAXI(THfTAC.J),1.D-4)

125 IF CSS2)1.3091309131130 ALPHACJI:ALPHA(JJfl

GO TO 132131 ALPHA(J) =ALPHA(JJ3.(OA..25DO .(EM4S..O)a02A3.[NS

ALPHA(J) :OMAXl(ALPgjA4JJ,1.O-4I132 If (SS3)1.3,1339134133 C(J)=C(JJ3

6O TO 112134 C(J)zC~jjfl(DC..25D00.ENS.1.D03aO2C)eENS

C(J)=ONAXI £C(J).0*DQ)C(J)=DON1(J)9TC 1 )IF (1.O0-EMR)*C(J)-T(l))112,&6

6 THETA(.fl=:THETA J)~i. IF iSS1113913,7

7 SI:0.0000 8 I=MRPgN

a S1=S1*TCt),-C(JJbIF IN-N.IIR)6697S,74

73 TNETA(J)=S1/4EN&ALPHA(JJf)Jso TO 13

74 6MA=GAM(ALPA(.iJ))KS Z0DO 108 K=1950KKZK-IGNAI:GAMIC(T (m)-C(JJ)bITHETA(J,ALPHA(J~J)SNMA2GAN(T(iP)-C(JJ)/rlEr?.(j.ALPHA(JJIIOLT(K)=-ENALPH1A(JJI/THETA(J).Sl/(THETA(Jle.2).(EN-EN).(T(M)-C(jjb

1 3e.ALPM4A(JJ) o(EXP((C(JJ)-T(Pq) )/jETA(J))I(THETA(J)..(AL0,1A(JJ)*..u20)3.GNA-GH4A1.JEH.ALPHAJJ/THErAJ-EMe*(TrMP-C(JJO)a.AL'tiA.J

A 6EP(CJ-~oR)/HTCJ/(p~~~e(LHAJI10)GA2

Th4dKI=TN( VA(J)If COLI(K))11 .15102

101 KSZKS.-1IF (CSPK)10591039105

102 KS=KS~lZF(KS-KJ 105,104i,1O5

103 ThCTA(J)z.5D0&TNt(K)46 OTO lob

104 THETA(J)=I.50.TeI(K1GO TO lob

105 IF fOLT(K)*DLT(KK)Jl1913v16108 KKK-

60 TO 105

167

SUBROUTINE GAANA 14/14 OPIZ0 FYN 5.1*524

107 TiEA(J)=TH(K).OLT(c).(THIK)-rH(KK )I(OLTCKK)-W.TEK)$IF £DABS(rHErA(J3-Tgl(KII-tD-4)13.13.1a5

106 CONTINUE13 ALPHAJI=ALPhA(JJ)14 If £SS2)44#44v1515 SL=0.DO

DO 16 ImMRPN16 SL:SL*OLOG(TCI)-C(JJbI

KS=ODO 43 K=1950IK KK-1GMA: GARICAL PHA CJI bIF £N-N.D4R)66.30921

21 GNAI=GANIC CTCMb-CIJJb)/TiETAI(J),ALPHACJ))6NA12=GANI C(TCMqP)-CC(JJl)/TI4ErA(J) ,ALP"A(.J35

30 DG=DGANtALPHACJJ)76 IFCN-PN.MR)66,71,v3271 DLACK1:-EN.DLOG(THETA(J)).SL-E4*G.O/G"A

60 TO IS32 DGZ:OGAMII(T(N)-cCJJ))/TIIETA(J),ALPHA(JJ)

0612:DGANI CCII NRP)-C(JJ))/TIETA(J ,ALPHA(J))31 DLACK)=-EN.OLOGCTI1EIACJ)).SL-EN.O6/GNA.CEN-EN).(D-DGIlI/G6NA-GNAII

1.ENR*DLOG(1I4ETACJ) 5.ENR*DG12/6qAI278 ALtKI=ALP1A(Jl

IF CDLAIK))35,944,4039 KS=KS-1

IF IKS*KblO.4lglIO40 KS=KS*140 IF (KS-K)70942*70

41 ALPHA(J)=.5O~obALlK)GO TO 43

42 ALPHACJ)=1.50.AL(KI* 60 TO 43

to IF (OLACK)&DLA(KK)172,44l1it KK:KK-1

60 TO 7072 ALPHA(J)=ALCKI.OLACK).(AL(K)-ALK) I(DLACKKI-OLAIK)I

7. IF IOABS(ALPhACJ)-ALCK)3-1.0-43444,4343 CONTINUE44 C(JI:CCJJfl85 IF CSS31112,L1294545 IF 41*D0-ALPIIA(J))799143vI43143 IF (SS1*SS2)b1,57qI91s IF £N.-N)66.bJv4646 GNA=GAN ALPIIAIJI)63 KS=0

00 56 K=1954KK:K-1SR :0 * D00 69 I:NRPIN

6 9 SftSR.D.0(TfI[-CJI))IF (N-N*NR)66,80981

so DLC(K):C1.DO-ALPhAC(J)O*SR.ENItNiETA(JI60 TO 82

$1. 6NAZ:GAMIC(UCNM)-CCJI)/ThETA(J)@ALPiACJ)GHA12=GAHICfV(MRP)-C(J))/TIETAJ)ALPIACJ)IDLCCK3:C 1.DO-ALPNIA(5) 1SR*CCn-rnq)/THETACJ).(EN-EN3.ItTIN5-C(J))s.(

SUBROUTINE GHMA 14174 OPT=O FYN s.t*3a,

IALPIIA(J)-l.DO EX((TM-CJ)/HTAJ)/(THETA(J)eeALPH4A(J'(I2A-GNAIb)-ECN&aT(MRP)-C(J))ae(AL~bNA(JI-1.00).OEXP(-(T(NAPI-C~j3)/TNq3ETA(J)3IOCrHETA(J5..ALPIA(J)e6qAI23

82 CE(K)=C(J)51 IF IDLC(K)1909112#91

* 90 KS=K,"-IF (KS*K)54,52954

152 IF(C(J)CLoC () -1D)121,6

53 CONTINUE(K*50(()C(60 TO 112

112 IF (DLCKI oLC(X))fT91295

113 O 15 I1

66 IF (DBC(J )-C(1.GI*-4-TIb16 111129

56 IFCONINUE,560 T1O 112

S1ZS1.)Ttl3-(J

L2 IF(NMRA 66139

IF (NA JI*AeO ((T(N)-C(JII/rI4(JAPIA(

16 If tN-N)668 O0,9

108 SZC(I)C)

GAJAALPHA(J )IF (-*)660,60~f

6 IF (DAMS(C(-C(J))1.D-A(J~1,61 .6361 F A=G ~AMI(RI-C(J))TA~HETA.oIALPHACJII

628 I ELDENaS(A*LHAIJ)-PHAJJ))-L.DO(TE)4,)*ALHAJ,630b

IF6 REUNMN)610999 EL=LND E)(DO(H-MI)DO(N

5-NeL"C~DLG~EAJ)EtkDO(1A2

100I (SJ= .i

FUNCTION GAN TIq OS FYN 5*1+524

CDOUBLE PRECISION FUNCTION. GAHM

* DOUBLE PRECISION G#Z#DLOGOEXP*1

I IF (Z-9*001292,32 GzG-DLOG(Z)

GO TO I3 GAM=G(Z-.500)aDLO(Z)-Z..5OO'3, 06(2. 0a3.Il I5926535897 iBOI*1.0OI

6AM0(EXP(GAM)RETURNEND

FUNCTION 064H 74/74 OPTZS FIN 5.1*52i

DOUOLE PRECISION FUNCTION OBA4CI)DOUBLE PRECISION D6.L,1,OLOSBAN

I IF tZ-9*DO)2t2e3

2 0:061.001

so T0 13 06AN:O6.(Z-.5OO)IZ.0L06(z1-I .DO-1*1.D/(I2oDO*l..2I.1.O0/(120.080?*4

I )-3.00/(252.00eZ*'6).1.32/(200.00aa8 -I.00l(132.00.Za. 1012 *691.O0/( 32760.DO.ZO*12)-l.0I(12.0Q.Ze'14)VGAN:OGANO6AH(TIRETURN

FUNCTION OGAMI 740/74 OPT:0YN51*2

DOUBLE PRECISION FUNCTION OBAMI(WZ)DOUBLE PRECISION uV#UZqSUqELLDIMENSION Ut5O)#V(50)

W(1b=W&Z/Z&&2SUzU(11-Vt1JDO 1 1:2950LL:1-1ELL=LLU(C):(..ULLROUELL&(ZELL-t.D 3)I(Z.ELLI

I SU=Z;U* UI L I-V IL IDGAMIZSUAE TUANEND

17 Cl

UFUNeTION GAHI i7/1y oprI0 FYN Sol*S24

DOUBLE PRECISION FUNCTION GANICM.ZIDOUBLE PRECISION U9UgZvSU.ELLDIMENSION Uttla)

SU:U91 5DO 1 L=2950LL:L-1ELL: LLUIL):(-U(LL)/ELLS eU.(Z.ELL-t.03 5I(Z*ELLI

I SU=SU*U(L)GAII :SURtETURNE ND

171

Vi ta

,'hilij. .John Vi vi ano wj~ iborn on 1,~Ve~fl~r1)8i

:; ntLou i -- iiji-our i * lie jrad ua ted f roc, Sa int P'aul lii.jih

Lcu&ioi iii :.ij iland, Illinois in 19C.". ine rect-ived a baclic-

lot'-s dtjree in Applied ivathernatics froi Southern Illinois

uiiiv%.rsit in 1;cvardsville in 1971. I~e attended Officers

r~4i i~n~ ', Cool c.,r1 was coululi iissionL-d ai; a U.1,14 a LIIiccr in

11)77. ii- ,ore ~ a plannlinj anal -:,t for tue Deputy for

~k~C )j.LV~tk'~un;,lieCtronic -Sytem6u 1.,ivisioii# Hac~rSolu 1ir

Forc iw~e. ~ then entered the : chul it LErinering, Air

Ijorce Im~t itiwLe of: Technology in Iune 19(C 1.

I-Lrmianeflt AddIress: 126 Collinsville Ave.

Collin:-ville, Il. 62234

172

FILMED

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