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Estimation of the parameters of atruncated gamma distributionLaxman M. Hegde a & Ram C. Dahiya ba Department of Mathematics , Marshall University ,Huntington, WV. 25705b Department of Matth.& Stat , Old Dominion University ,Norfolk, VA. 23308Published online: 27 Jun 2007.

To cite this article: Laxman M. Hegde & Ram C. Dahiya (1989) Estimation of the parametersof a truncated gamma distribution, Communications in Statistics - Theory and Methods, 18:2,561-577, DOI: 10.1080/03610928908829919

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COMMUN. STATIST.-THEORY METH., 18(2), 561-577 (1989)

ESTIMATION OF THE PARAMETERS OF A TRUNClATED GAMMA DISTRIBUTION

Laxman M. Hegde and Ram C. Dahiya

Department of Mathematics Department of Math.& Stat. Marshall University Old Dominion University Huntington, WV. 25705 Norfolk, VA. 23508

Key CSords and Phrases: maxz'mum l i k e l i h o o d e s t ima tor ; exponenti a l f ani ly ; modi fz'ed maxi'mum lz*&elz'15ood es t imator; probabi l z t y o f nonexistence

ABSTRACT

This paper deals with the estimation of the parameters of a truncated gamma distribution over ( O , r ) , where r is assumed to be a real number. We obtain a necessary and sufficient condition for the existence of the maximum likelihood estimator(MLE). The probability of nonexistence of MLE is observed to be positive. A simulation study indicates that the modified maximum likelihood estimator and the mixed estimator, which exist with probability one,are to be preferred over MLE. The bias, the mean square error, and the probability of nearness form a basis of our simulation study.

1. INTRODUCTION

The truncated gamma distribution is an important distribution of interest due to its applications in life testing and reliability problems. Broeder(l955),

Copyright O 1989 by Marcel Dekker, Inc.

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562 HEGDE AND D A H I Y A

Chapman(l955), and Gross(l971) are some of the main papers which study estimating the parameters of a truncated gamma distribution. Barndorff-Nielsen(BN) (1978),pages 150-160, has given a set of general conditions for the existence and the uniqueness of the maximum likelihood estimator in a minimal exponential

family. It becomes clear from a review of the literature that the estimation problem in the case of two-parameter truncated gamma distribution needs to be studied in greater depth. Using a few results from BN(1978), we derive a necessary and sufficient condition for the existence of MLE in the truncated gamma family (section 2). In Section 3, we discuss a derivation of the modified maximum likelihood estimator and the mixed estimator. Finally in Section 4, the simulation results are reported.

2. TRUNCATED GAMMA DISTRIBUTION

A truncated gamma population is described by a random variable (Y) with the density function

where O<y <r, *O known, and ( B, a) E R with R = [(B, a) :0<8<~, O<&W 1.

Since the truncation point is assumed to be known, we 7

transform (I), with X = Y/r, el = 1 - - 8'

82 = a 1 ,as

follows :

where a(x; el, e2) = e ( el-1) x+ e210g(x) and (el,e2)= e

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E S T I M A T I O N O F PARAMETERS O F TRUNCATED GAMMA D I S T R I B U T I . O N 563

with e = [ ( el, e2) :-cot el<l, -1< e2<m 1. Note that €4 is a one-one mapping of Q with the inverse mapping given by

7 B = , and a = e2+1.

(1- el) ( 3 )

Let x = (xl, .. . ,xn) be a random sample from (2). Then we can see that the likelihood of x is given by

( el-1) mi+ e2 Clog (xi) e

L(x ; 61, 62) =

1" ( 4 )

[$a@; el, e2)du

We can show that the distribution of (T1 = C Xi, T2 = C log(Xi) ) is of the form

g(tllt2; 01, e2) = h(tlrt2)e eltl+ e2t2-K ( el r 02

( )

function of (tltt2) only. Next we discuss maximum likelihood estimator of ( el, e2) in the family

d = [9(tllt2; 811 9): (611 82) E I ( 6 ) Note that d is a minimal exponential family of order two (See BN(l978) ,page 112, for details). Given a value of (tl,t2), we can see that a maximum likelihood estimate of (el, e2) is a solution of

- 1 - 1 az where tl = -tl , t2 = -t2 and E(X) = - n n nael

where f(x;el, e2) is as defined in (2). We refer to (7) as maximum likelihood equation (m.l.equation) in further discussion.

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5 64 HEGDE AND DAHIYA

As discussed laterlit is possible that equation (7) may have no solution for (el, e2) in e. Whenever a solution lies outside 8, MLE of (el, e2) lies on the boundary of 8 with el = 1. Thisfin turn, implies that a maximum likelihood estimate of 5 equals a. Here, we define that MLE exists if and only if there is a

solution for (7) lying in €3. The MLE being infinite with respect to (5,a) and being nonexistent with respect to (el,e2) are used interchangeably. Since the uniqueness of MLE in an exponential family is well known, we discuss here only the conditions for the

existence of a solution to (7) in terms of (tlft2).

Let r = [(el, e2) E ~2 : l ~ ( e ~ , 82) 1 <a I. ~t is easy to check that

r = [ ( el, e2) : - w el<aI -1< e2<a 1. Note that r is known as the natural parameter space and

% = [g(tltt2; '1, 02) : (01, e2) E r I as the full family. Since e is a convex open subset of r, the equation (7) admits a solution in f3 if and only

if (flrf2) E I where

1 = [ (ml,m2) : ml = E(X), m2 = E(log(X) (elr 02) E 01. One may refer BN(1978) for further details. Theorem 1 below determines the set I explicitly. Theorem 1: Let D(ml) = [ ( el, e2) E 0 :E (X) = ml] , O<ml<l. Then for ( el, e2) E D(ml) , we have

Proof: Since the upper inequality in (8) is well known, we need to show only the lower inequality. Let

* * e* be the boundary of e . It is obvious that e* = 8 U 0

1 2 where

* * * * * * e = [el,-1) :-we a], and e = [(I, e 2 ) :-lie <a 1. 1 1 2 2

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ESTIMATION OF PARAMETERS OF TRUNCATED GAMMA DISTRIBUT1.ON 565

* * * Let (e -1) and (1, e ) be any given points in e and

1 I 2 1 *

e respectively. Then the following limiting results 2

hold :

and

e +1 2

(ii) limit * [i x f ( x ; ~ ~ ~ e2)dx (ell 92) +(I1 e2) 2

and

- 1 1 imit * [i log(x) f(x;el1 e2)dx

(911 9) +(I1 e2) 2

= F2 ( el2) , say.

In the results (i) and (ii) above, the integrals

* E(X) and E(log(X)) respectively. Note that Fl(e ) is a 2

* strictly increasing function of e with the range

2

* (0,l) and F2(e ) is a strictly increasing function of 2 * e with the range ( - m , O ) . Hence for a given ml in the 2

* interval (0,1), there exists a unique e such that 2

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566 HEGDE AND DAHIYA

* * F1 ( e2) = ml. In other words, (1, e ) is the boundary 2

point of a(ml). Note that D(ml) does not have any *

boundary point in 9 It is easy to check that 1 '

Furthermore, the following monotonicity result in a(ml) is true (see ~ ~ ( 1 9 7 8 ) ,page 121) .

- - - - where ( el, 62 ) , ( el, e2 ) are in 9 and (ml , m2 ) , (ml, m2 ) are

- - the values of (E (X) , E (log (X) ) ) at ( el, e2) and ( el, e2)

* respectively. Note that e2> e for any ( el, 9) in

2 D(ml). Hence using (9) and (lo), we can check that the lower inequality in (8) follows.

1111

Now due to Theorem 1, we can write the set I as

Theorem 1 gives us a necessary and sufficient condition for the existence of a solution to (7) and the same is stated in Corollary 1 below.

Corollarv 1: Given a (tl, t2) , O<tl<l, the maximum likelihood equation (7) admits a solution in 9 if and

only if (tl,E2) satisfies the inequality 1 - -

1 - - < t2 < log(tl). - (11) t 1

Proof:The proof is a immediate consequence of Theorem 1 by noting that a solution to (7) must lie in

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ESTIMATION OF PARAMETERS OF TRUNCATED GAMMA DISTRIBUTION 567

3)(fl) where - -

qtl) = [(el, e2) E 8: E(X) = tl I, 0<tl<l.

1111

As by-product results, we may note that the following remarks hold: i) The m.l.equation in the full family

[g(tl, t2 ; el, e2) : ( el, e2) E ] admits a solut.ion with probability one.

ii) The m.l.equation in the subfamily 0 0

[g (tl, t2 ; 81, e2) : e l , e >-1 known] admits a solution 2

if and only if

0 Note that e2 = 0 refers to a truncated negative

exponential family. iii) The m.l.equation in the subfamily

with probability one. Next in Theorem 2, we give a result characterizing the

set S where

Theorem 2: The following inequalities charaleterize the set S.

- - - n- 1 -a < t2 5 log(tl), for O<tl 5 -- n ' (12)

1 - n-1 - -log (nf l- (n-1) ) < t2 < log(fl) , for -- <tl < 1. n n

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568 HEGDE AND DAHIYA

Proof: Since the right inequality, c2 409 , is well known, we only prove the left inequality in (12). Let

1 Note that ,clog(xi),being a concave function of

(X~,...~X~), attains its absolute minimum on the set

of extreme points of Hence the left inequality in (12) follows by noting the following:

k-1 - i) For -tl +, k = 1,2, . . . n, the set of extreme

n

points of x(tl) consi'sts of all the distinct n!

pemutaions ( of (X~,...~X~~...~X~) (k-1) ! l! (n-k) !

where xj = 1 for j = l121...k-ll

- = ntl-(k-1) for j = k, = 0 for j = k+ll .. .n.

ii) The set of extreme points of x(T1) contains at

least one zero for 0c;~22, and does not contain zero n

- n-1 for tl> -.

n

1111

Now the set S can explicitly be expressed as S = SIU S2 where

Let C denote the closure of convex hull of S. It is easy to see that

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ESTIMATION OF PARAMETERS OF TRUNCATED GAMMA DISTRIBUTION 569

FIG. 1 The Set I Superimposed on the Set S

BN(1978),page 151, considers C instead of S, for discussing the existence of a solution to the maximum likelihood equations in a minimal exponential. family. Note that S+C as n+m. However for a given n,

(Cl,t2) E S with probability one for this family. Furthermore, we can show that I c S. Hence the maximum likelihood equation (7) does not yield a solution in 9

whenever (tlrt2) E (S-I) .The Figure 1 below gives us a graphical presentation of the previous discussion.

The following comments may be noted for Figure 1. i) The set I is superimposed on the set S and hence

- the ml-axis is the same as the tl-axis and the m2-axis

is the same as the t2-axis.

ii) The solution to the maximum likelihood equation

(7) exists if and only if (il,f2) E I. That is, the equation (7) admits a solution in 9 if and only if

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570 HEGDE AND DAHIYA

(flrf2) lies in the shaded area between the curves - - 1 - t2 = log(tl) and t2 = 1- -.

1

3. MODIFIED ESTIMATORS

From the discussion in Section 2, it is obvious that the maximum likelihood estimator(MLE) becomes nonexistent. Blumenthal and Marcus(1975) encounter a similar problem with reference to estimating the total sample size based on truncated samples from a negative

exponential distribution (e2 = 0) and rectify this problem by introducing a class of estimators called Bayes modal estimators or modified maximum likelihood estimators(MMLE). Mittal and Dahiya (1987) studied a mixed estimator(MXE), a mixure of MLE and MMLE, in the case of a doubly truncated normal distribution. Here, we present a brief sketch of the concepts and derivation of these estimators.

3.1. The Modified Maximum Likelihood Estimator(MMLE)

Let x = (xlr...,xn) be i.i.d. observations with p. d. f . hl (x; W) where w is a scalar parameter. Let L(x; W) = ll hl(xi; W) denote the likelihood function of x. Then MMLE of w is derived by maximizing the - modified likelihood function L* (x; W) = L(x; w)p( w) where p(w) is a properly chosen prior density(weight function) for W. Under certain regularity conditions, the MMLE of w is a solution of

In the truncated gamma family (6), we choose a prior distribution p ( el, e2) as

a ( el-1) +blog (1- el) P(elr e2) " e I (13

where O<a<m, -l<b<m, are the prior parameters. Hence

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ESTIMATION OF PARAMETERS OF TRUNCATED GAMMA DISTRIBUTION 571

given a (tl , t2) , we can verify that MMLE of ( el, e2) is a solution of

E(log(X)) = t2 where E (X) and E (log (X) ) are as defined in ('7) . Next in Theorem 3, we show that the equation (14) admits a solution in 0 with probability one provided :b>O.

Theorem 3: Given a (tlrt2), O<tl<l, the equation (14) admits a solution with probability one for any b>O. Proof: It is clear that a solution to (14) must lie in

- - e(t2) = [(el, e2) E e : E(log(X)) = t2 I , t2 < 0.

Using the arguments of Theorem 1, we can show that for

( el, e2) E e(t2) , the following inequality holds: -

1 et2< E ( X ) < -. 1-t,

* Let (1, e ) be a boundary point of a(t2) . N0t.e that

2 * -

e2> e for any ( el, e2) E e(t2) . Hence the following 2

limiting results, in fZ(t2) , can be verified:

- a b a

ii) limit [ E ( X ) - n - 1 = et2- - n( el-1) n '

(ell 92) '(-a, a) Before concluding the proof, we may note the following:

i) E(X) is strictly increasing in &(it2) , -

ii) et2<t1 <1 for a fixed i2 < O.

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572 HEGDE AND DAHIYA

The previous analysis holds for any a>O, and b>O. However the question of choosing the optimum values of a and b is not answered here. Blumenthal(l982) has given a detailed analysis of choosing the optimum parameters, involved in a prior density, by minimizing the asymptotic bias. Also Mittal and Dahiya (1987) have shown, using Blumenthal(l982), a derivation of the optimum prior,parameters in the case of a doubly truncated normal distribution. Following a similar

1 analysis,we find that a=-and b = 1 are the optimum

2 values in our case and the same are used in the

simulation study.

3.2 The Mixed Estimator(MXE):

A mixed estimator, as the name suggests, is derived as a mixure of several estimators. Given a random sample x from a density hl(x;w), let U1(x) and U2(z) denote any two estimators of W. Then the MXE Urn@) is defined as follows:

Um(x) = U1(x) if x E A, = U2(y) if x E A', (complement of A)

where A is a proper subset of the sample space with positive probability measure for every W. We may call the set A as a mixing criterion. We derive a MXE of

( el, e2) as a mixure of MLE (U1 (11) ) and MMLE (U2 (y) ) with the mixing criterion(A) as

Note that Um(y) exists with probability one with the mixing criterion(A) as above.

4 . SIMULATION RESULTS

Now we discuss the simulated behaviour of MXE and MMLE for the ( el, e2) -parameterization. Since MLE

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ESTIMATION OF PARAMETERS OF TRUNCATED GAMMA DISTRIBUTION 57 3

becomes nonexistent(1ies outside e)for certain samples,it is not considered for simulation. The truncation interval is fixed to be (0,l) for all the cases of parameter values studied. The bias(1ength and direction), the mean square error, and the probability of nearness form a basis of our simulation study and these concepts are discussed below.

* * Let T = (T1,T2) and T* = (T T ) be any two

1' 2 * *

estimators of ( el, e2) . Let (t , t2i) and (t ,t ) be l i lfi 2i

* * the observed values of (T1,T2) and (T T ) 1' 2

respectively for the ith sample, i = 1, . . . , M where M is the number of simulated samples. Then we define the simulated quantities as follows. i) Let B = (bl, b2) denote the bias vector of T. Then

the length Ib 11 of B is defined as

where bl =

ii) The direction angle D I R of B is

-1 bL D I R = COS [C

Note that the quadrant in which D I R lies is determined based on the signs of bl and b2 together. For example, if bl<O, and b2>0,then D I R lies in the second quadrant of a coordinate plane. iii) The mean square error of T (MSE(T)) is defined

as

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5 74 HEGDE AND D A H I Y A

iv) The efficiency of T* w. r. t. T (EFF(T*,T) ) is

defined as

EFF(T*,T) = MSE (T) MSE (T*) '

v) The probability of nearness of T* w.r. t. T

(P (T*, T) ) is defined as

where m is the number of times where

r

the event E occurs

vi) The probability of nonexistence of MLE (Po) is defined as

where no is the number of times the sample statistics

(tl,t2) belongs to A', A' being the complement of the mixing criterion defined for MXE in the Subsection 3.2.

We carried out the actual simulation using Fortran interactively with IMSL subroutines. The simulated values of

for MMLE and MXE, are based upon M = 500 random samples of sizes n = 10,20,40. The samples are drawn using IMSL routine GGAMR and all the non-linear equations are solved using IMSL routines ZSPOW and DCADRE. Next we give concluding remarks of the simulation results presented in Table I.

Comments on Table I: We see that Ib 11 of MMLE < Ib 11 of MXE with the exception of a few cases when el = 0. By

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ESTIMATION OF PARAMETERS OF TRUNCATED GAMMA DISTRIBUTION 575

W d r l N w P 0 m L n . . . w N d

cow -Y m d w d e P . . . N d O

m m o m a 0 . . .

d

m m m I D w w d d d

m m rl d -3 0 w e N . . 0 0 0

m d P 0 0 0 o m m . . . m m r l d

P c o W

m m 0 . . . r lmCJ rl

* C O P m co o -3 m r( . . . N m N

o o e I. w m . . . LnPP mmI. m m r l

C O P P r n w ~ -I'm d . . . 0 0 0

m -3 P d 0 CV d 0 N

a s .

c o m e N

m co m m m m

m - 3 m comm -11- w w w d d d dr ld d d d

N 0 0

0

0 0 - m o o c o m r c o d o c r ldd --l 0 0 0 . O O C . . 4 . . . m . . .

I I I1 II

m d W rl a d a 0 CD 0

0001 0001 O O C r lNe d d m -

0 0 rl P P P d r ld

a, 0 0 0 C P P P +' dr ld

m -3 4J

C 0 o m 0 a, I m (I) 11 m w m 13

I k CI 4 0 0 a

a, ... h

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5 76 HEGDE AND DAHIYA

studying DIR, it being mostly in the second quadrant, we find that both MMLE and MXE tend to underestimate el and overestimate e2. By comparing the MSE(MMLE) with the MSE (MXE) , we see that l<EFF(MMLE,MXE) . That is, MSE(MMLE)<MSE(MXE). By looking at the probability of nearness, it becomes obvious that P(MMLE,MXE)>O.S. In other words, MMLE tends to scatter near the true value more often as compared to MXE. As a final comment on Table I, we note that the probability of nonexistence of MLE(Po) is observed to be as high as 17% when the sample sizes(n) are small and the truncation probability(qo) is about 10%.

BIBLIOGRAPHY

Barndorff-Nielsen O.(1978).Information and Exponential Families in Statistical Theory,John Wiley.

Blumenthal,S.(1981). Stochastic Expansions for point estimation from complete,censored,and truncated samples. Technical report, Department of Mathematics, University of Illinois, Urbana-Champaign.

Blumenthal,S., and Marcus,R. (1975). Estimating population size with exponential failure. Journal of the American Statistical Association. 70, 913-922.

Broeder,G.G. den (1955). On parameter estimation for truncated Pearson Type I11 distributions. Annals of Mathematical Statistics. 26, 659-663.

Chapman, D.G.(1956). Estimating the parameters of truncated gamma distribution. Annals of Mathematical Statistics. 27, 487-506.

Gross,A.J. (1971). Monotonicity properties of the moments of truncated gamma and Weibull density functions. Technometrics,l3, 851-857.

IMSL Library: Reference Manual(1982). International Mathematical and Statistical Libraries,Inc. Houston,TX.

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ESTIMATION OF PARAMETERS OF TRUNCATED GAMMA DISTRIBUTION 577

Mitta1,M.M. and Dahiya,R.C.(1987). Estimating the Parameters of a doubly truncated normal distribution. Communications in Statistics, Simula, 16(1), 141-159.

Received S e p t e m b e h 1 9 8 8 .

R ecommended by S u b h a h m a n i a m Kochehlahota, Univehnity a 6 Manito ba, CANADA.

Reieteed Anonymounlg.

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