asymptotic analysis of estimators for c np ( u , v ) based on quantile estimators

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Asymptotic analysis of estimators for CNp (u, v) based on quantile estimatorsSy-Mien Chen a & Yu-Sheng Hsu aa Department of Mathematics , Fu-Jen Catholic University , Taipei,Taiwan, 242, ROCb Department of Mathematics , National Central University ,Chungli, Taiwan, ROCPublished online: 27 Oct 2010.

To cite this article: Sy-Mien Chen & Yu-Sheng Hsu (2003) Asymptotic analysis of estimators for CNp (u, v) based on quantile estimators, Journal of Nonparametric Statistics, 15:2, 137-150, DOI:10.1080/1048525031000089338

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Nonparametric Statistics, 2003, Vol. 15(2), pp. 137–150

ASYMPTOTIC ANALYSIS OF ESTIMATORS FORCNp(u, v) BASED ON QUANTILE ESTIMATORS

SY-MIEN CHENa,* and YU-SHENG HSUb

aDepartment of Mathematics, Fu-Jen Catholic University, Taipei, Taiwan 242, ROC;bDepartment of Mathematics, National Central University, Chungli, Taiwan, ROC

(Received 28 July 2000; In final form 29 September 2000)

Pearn and Chen (1997) proposed a class of capability indices CNp(u, v) which generalize the process capabilityindices Cp(u, v) (Vannman, 1995) to accommodate cases where the underlying distributions may not be normal.The current indices CNp(u, v) are functions of quantiles which may not be known. In this article, we apply someexisting quantile estimators to estimate the class of indices CNp(u, v). It is shown that the estimators of CNp(u, v)we propose are all asymptotically normally distributed and equivalent and can perform better than the estimatedCp(u, v) in some instances.

Keywords: Quantile estimator; Process capability index; Asymptotic distribution; Specification limits

1 INTRODUCTION

A process capability index is a numerical measure which provides information on whether a

production process is capable of producing items within the specification limits predetermined

by the designer. Recent research and advances made in this subject are neatly summarized in

Sullivan (1984), Kane (1986), Chan et al. (1988a,b), Spiring (1991), Pearn et al. (1992),

Katz and Lovelace (1998), and Vannman (1995), among others. In addition to their typical

applications in industry, process capability indices are particularly useful in chemical

batch processes (Morris and Watson, 1998), rubber-edge production lines (Chen and

Pearn, 1997) and academic fields for studying the classroom performance (Johnson and

Smith, 1997).

Vannman (1995) presented a unified approach to capability indices research by introducing

a class of capability indices Cp(u, v) that generalizes the four basic indices, Cp, Cpk , Cpm and

Cpmk . It has been shown that Cp(u, v) are appropriate for normal processes, but not for non-

normal process data. To deal with this difficulty, Pearn and Chen (1997) and Chen and

Pearn (1997) generalized Cp(u, v) to a class of capability indices CNp(u, v).

In this article, we propose some estimators of the indices CNp(u, v) based on some existing

quantile estimators, and study their large sample properties. It is shown that all estimators of

CNp(u, v) we propose here are consistent, asymptotically normally distributed and equivalent.

* Corresponding author.

ISSN 1048-5252 print; ISSN 1029-0311 online # 2003 Taylor & Francis LtdDOI: 10.1080=1048525031000089338

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In Section 2, the definition of CNp(u, v) is given. In Section 3, the asymptotic distributions of

two new proposed estimators and the estimator proposed by Pearn and Chen (1997) are

derived. Some comparisons of the estimators are made in Section 4. Section 5 provides

the conclusions.

2 PROCESS CAPABILITY INDICES CNp(u, v)

Let X1, . . . , Xn be a sample of measurements from a process which has distribution F with

the lth quantile, xl, mean m, and variance s2. Let LSL and USL be the lower and the upper

specification limits of a product characteristic, respectively. Denote by d ¼ (USL� LSL)=2

the half length of the specification limits, m ¼ (USLþ LSL)=2 the midpoint of the specifica-

tion interval, and T the target value. A class of process capability indices proposed by Pearn

and Chen (1997) is defined as follows:

CNp(u, v) ¼d � ujx0:5 � mj

3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi((x0:99865 � x0:00135)=6)2 þ v(x0:50 � T )2

p ,

where u, v � 0.

When the process has symmetric tolerances (i.e., T ¼ m) and the underlying distribution is

normal, the new indices reduce to the indices Cp(u, v) that generalize the four basic indices

Cp, Cpk , Cpm and Cpmk .

3 ASYMPTOTIC PROPERTIES OF ESTIMATORS OF CNp(u, v)

3.1 Estimator Based on Empirical Quantiles

For a distribution function F, the lth quantile of F is defined as xl ¼ inf {x: F(x) � l},

0 < l < 1. Let xxl denote the empirical lth quantile of the sampling distribution function

Fn. Then xxl is the empirical quantile estimator of the population quantile xl. For 0 < l < 1,

assume that xl is the unique solution x of F(x�) � l � F(x). Then the sequence of empirical

quantile estimators xxl will converge to xl almost surely (Serfling, 1980, p. 77).

Consider the following estimator of CNp(u, v) which is based on the empirical quantile

estimators:

CCNp(u, v) ¼d � ujxx0:5 � mj

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi((xx0:99865 � xx0:00135)=6)2 þ v(xx0:5 � T )2

q :

In order to find the limiting distribution for CCNp(u, v), we need the following lemma.

LEMMA 1 Suppose that in a neighborhood of each quantile, x0:00135, x0:5, x0:99865,

F possesses a positive continuous density f. Then

(a)ffiffiffin

p[(xx0:5, ((xx0:99865 � xx0:00135)=6)2) � (x0:5, (x0:99865 � x0:00135=6)2)] � AN ((0, 0), S),

where

S ¼s1 s2

s2 s3

� �,

138 SY-MIEN CHEN AND YU-SHENG HSU

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and

s1 ¼(0:5)2

f 2(x0:5),

s2 ¼x0:99865 � x0:00135

18

(0:5)(0:00135)

f (x0:5) f (x0:99865)�

(0:00135)(0:5)

f (x0:00135) f (x0:5)

� �,

s3 ¼x0:99865 � x0:00135

18

� �2"

0:00135(0:99865)

f 2(x0:00135)þ

0:99865(0:00135)

f 2(x0:99865)

�2(0:00135)2

f (x0:00135) f (x0:99865)

#:

(b) For a normal process N (m, s2),ffiffiffin

p(xx0:5, ((xx0:99865 � xx0:00135)=6)2) � AN ((m, s2), SN ),

where

SN ¼

p2s2 0

04(0:00135)(0:9973)pe9

9s4

0B@

1CA:

Proof

(a) By Theorem B on page 80 of Serfling (1982),

(xx0:00135, xx0:5, xx0:99865) � N (x0:00135, x0:5, x0:99865),1

nS�

� �,

where S� ¼ (sij)3�3, s11 ¼ 0:00135(0:99865)=f 2(x0:00135), s22 ¼ (0:5)2=f 2(x0:5), s33 ¼

0:99865(0:00135)=f 2(x0:99865), s12 ¼ s21 ¼ 0:00135(0:5)=f (x0:00135)f (x0:5), s23 ¼ s32 ¼

0:5(0:00135)=f (x0:5)f (x0:99865), s13 ¼ s31 ¼ (0:00135)2=f (x0:00135)f (x0:99865).

Let (A1(a, b, c), A2(a, b, c)) ¼ (b, ((c� a)=6)2),

D� ¼

qA1(a, b, c)

qaqA1(a, b, c)

qbqA1(a, b, c)

qcqA2(a, b, c)

qaqA2(a, b, c)

qbqA2(a, b, c)

qc

0BB@

1CCA

evaluated at (a, b, c) ¼ (x0:00135, x0:5, x0:99865), and S ¼ D�S�D�.

Then

D� ¼

0 1 0

x0:99865 � x0:00135

180

x0:99865 � x0:00135

18

0@

1A,

S ¼

s22

x0:99865 � x0:00135

18(s23 � s12)

x0:99865 � x0:00135

18(s23 � s12)

x0:99865 � x0:00135

18

� �2

(s11 þ s33 � 2s13)

0BBB@

1CCCA:

In view of (xx0:5, ((xx0:99865 � xx0:00135)=6)2) ¼ (A1(xx0:00135, xx0:5, xx0:99865), A2(xx0:00135, xx0:5,

xx0:99865)), the required result follows by Theorem A on page 122 of Serfling (1982).

ASYMPTOTIC ANALYSIS OF ESTIMATORS FOR CNp(u, v) 139

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(b) For a normal process with mean m and variance s2, the desired conclusion follows from (a)

and the following fact: x0:00135 ¼ m� 3s, x0:5 ¼ m, x0:99865 ¼ mþ 3s; s11 ¼ 2(0:00135)

(0:99865) p e9 s2 ¼ s33; s22 ¼ 2(0:5)2 p s2; s12 ¼ s21 ¼ 2(0:00135) (0:5) pe4:5 s2 ¼

s23 ¼ s32; s13 ¼ s31 ¼ 2:(0:00135)2pe9s2 � f (x0:00135) ¼ (1=ffiffiffiffiffiffiffiffi2ps

p)e�4:5 ¼ f (x0:99865),

f (x0:5) ¼ 1=ffiffiffiffiffiffi2p

ps: j

We are now ready for the asymptotic normality of CCNp(u, v).

THEOREM 1 Suppose that in a neighborhood of each quantile, x0:00135, x0:5, x0:99865,

F possesses a positive continuous density f. Define

G ¼ G(x0:00135, x0:5, x0:99865, v, T )

¼x0:99865 � x0:00135

6

� �2

þv(x0:5 � T )2

( )1=2

,

H ¼ H(x0:5, d, m, u) ¼ 1 �ujm� x0:5j

d,

and

sgn(m� a) ¼�1, if m < a:þ1, if m > a:

�

Then

ffiffiffin

p(CCNp(u, v) � CNp(u, v)) !

LN (0, DSD0), if x0:5 6¼ m,

�1

3GjW1j �

d

3G2W2, if x0:5 ¼ m,

8<:

where D ¼ (sgn(m� x0:5)(u=3G) � dv(x0:5 � T )H=3G3, � dH=6G3), !L

means conver-

gence in law, (W1, W2) � N ((0, 0), USU0), S is defined as in Lemma 1(a), and

U0 ¼

uv(x0:5 � T )

G

01

2G

0B@

1CA

Proof Define

g(a, b) ¼ 1 �uja� mj

d

� �d

3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibþ v(a� T )2

p ,

for a 2 (LSL, USL), then

CNp(u, v) ¼ g x0:5,x0:99865 � x0:00135

6

� �2 !

,

CCNp(u, v) ¼ g xx0:5,xx0:99865 � xx0:00135

6

!20@

1A,


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and

ffiffiffin

p(CCNp(u, v) � CNp(u, v)) ¼

ffiffiffin

pg xx0:5,

xx0:99865 � xx0:00135

6

!20@

1A

24

�g x0:5,x0:99865 � x0:00135

6

� �2 !#

:

Case I When LSL < x0:5 < m, LSL < a < m,

g(a, b) ¼ 1 �u(m� a)

d

� �d


p :

Define

D ¼qg(a, b)

qa,qg(a, b)

qb

� ��(a,b)¼(x0:5,((x0:99865�x0:00135)=6)2)

,

then

D ¼ sgn(m� x0:5)u

3G�dv(x0:5 � T )H

3G3, �

dH

6G3

� �

Since CCNp(u, v) ¼ g(xx0:5, ((xx0:99865 � xx0:00135)=6)2), by Theorem A on page 122 of Serfling

(1980) and Lemma 1(a), we have

ffiffiffin

p(CCNp(u, v) � CNp(u, v))!

LN (0,DSD0):

Case II When m < x0:5 < USL, m < a < USL,

g(a, b) ¼ 1 �u( � mþ a)

d

� �d


p :

The proof is done by the same technique used in Case I.

Case III If x0:5 ¼ m, then

ffiffiffin

p(CCNp(u, v) � CNp(u, v)) ¼

ffiffiffin

p(

1 �ujxx0:5 � mj

d

" #

�d

3


q

�d

3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi((x0:99865 � x0:00135)=6)2 þ v(x0:5 � T )2

p):


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Let

V1n ¼1

3


q ,

V2n ¼d


p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi((xx0:99865 � xx0:00135)=6)2 þ v(xx0:5 � T )2

q ,

W1n ¼ffiffiffin

p(u(xx0:5 � m)),

W2n ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix0:99865 � x0:00135

6

� �2

þv(x0:5 � T )2

s

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixx0:99865 � xx0:00135

6

!2

þv(xx0:5 � T )2

vuut :

Then ffiffiffin

p(CCNp(u, v) � CNp(u, v)) ¼ �jV1nW1nj � V2nW2n:

By page 24 of Serfling (1980), we have

(V1n, V2n)!P 1


p

�d

3[((x0:99865 � x0:00135)=6)2 þ v(x0:5 � T )2]

��

1

3G,

d

3G2

� �:

Let

(k1(t, s), k2(t, s)) ¼ u(t � m), � G þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisþ v(t � T )2

q� �:

then

(W1n, W2n) ¼ k1 x0:5,x0:99865 � x0:00135

6

� �2 !

, k2 x0:5,x0:99865 � x0:00135

6

� �2 ! !

:

Define

U ¼

qk1(t, s)

qtqk1(t, s)

qsqk2(t, s)

qtqk2(t, s)

qs

0BB@

1CCA,

evaluated at (t, s) ¼ (x0:5, ((x0:99865 � x0:00135)=6)2).

Then

U0 ¼

uv(x0:5 � T )

G

01

2G

0B@

1CA:


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Hence by Theorem A on page 122 of Serfling (1980) and Lemma 1(a),

(W1n, W2n) !L

(W1, W2),

where (W1, W2) � N ((0, 0), USU0). Defining V1 ¼ 1=3G, V2 ¼ d=3G2, by Slutsky’s

theorem,

(V1nW1n, V2nW2n) !L

(V1W1, V2W2):

Letting r(h, l) ¼ �jhj � l, by theorem on page 24 in Serfling (1980),

r(V1nW1n, V2nW2n) !Lr(V1W1, V2W2):

Recall

r(V1nW1n, V2nW2n) ¼ffiffiffin

p(CCNp(u, v) � CNp(u, v))

and

r(V1W1, V2W2) ¼ �1

3GjW1j �

d

3G2W2:

Then we have

ffiffiffin


L�

1

3GjW1j �

d

3G2W2:

Consequently,

ffiffiffin


LN (0, DSD0), if x0:5 6¼ m,

�1

3GjW1j �

d

3G2W2, if x0:5 ¼ m,

8<:

and the proof is completed. j

3.2 Estimator Based on Kernel Quantile Estimator

Let F�1(l) be the lth quantile of a distribution function F, l 2 (0, 1). Assume F is smooth

near the lth quantile F�1(l), and F possesses a positive continuous density f . In Falk (1984),

under suitable conditions on F, an > 0 and k: R ! R, a kernel type estimator of the lth

quantile is defined by lln(Fn) :¼ a�1n

Ð 1

0F�1n (x)k((l� x)=an) dx. By Falk (1985), the kernel

quantile estimator and the empirical quantile estimator have the same limiting distribution.

Denote xl as the lth quantile F�1(l), and ~xxl as the kernel quantile estimator lln(Fn) of

F�1(l). By the theorem on page 429 in Falk (1985) and the same argument as for the

proof of Theorem 1, the following result holds.

THEOREM 2 Under the assumptions given above in this subsection, define

~CCNp(u, v) ¼d � uj~xx0:5 � mj

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi((~xx0:99865 �

~xx0:00135)=6)2 þ v(~xx0:5 � T )2

q ,


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Then

ffiffiffin

p( ~CCNp(u, v) � CNp(u, v)) !

LN (0, DSD0), if x0:5 6¼ m,

�1

3GjW1j �

d

3G2W2, if x0:5 ¼ m

8<:

where W1, W2, D, S and G are defined in Theorem 1.

3.3 Estimator Proposed by Pearn and Chen

For unknown quantiles, Pearn and Chen (1997) applied the method in Clement (1989) to find

the quantile estimators for the generalizations CNp(u, v), and obtained a superstructure for the

estimators as follows:

�CCNp(u, v) ¼d � uj�xx0:50 � mj

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi((�xx0:99865 �

�xx0:00135)=6)2 þ v(�xx0:50 � T )2

q ,

where

�xx0:00135 ¼ X([R1]) þ (R1 � [R1])(X([R1]þ1) � X([R1])),

�xx0:5 ¼ X([R2]) þ (R2 � [R2])(X([R2]þ1) � X([R2])),

�xx0:99865 ¼ X([R3]) þ (R3 � [R3])(X([R3]þ1) � X([R3]));

R1 ¼ 0:00135nþ 0:99865;R2 ¼ 0:5nþ 0:5;R3 ¼ 0:99865nþ 0:00135:

In this setting, the notation [R] is defined as the greatest integer less than or equal to the

number R, and X(i) is the ith order statistic of X1, X2, . . . , Xn.

The limiting distribution of �CCNp(u, v) is given below.

THEOREM 3 Let 0 < l1 < l2 < l3 < 1. Suppose that in a neighborhood of each quantile,

xl1, xl2

, xl3, F possesses a positive continuous density f. Then

ffiffiffin

p( �CCNp(u, v) � CNp(u, v)) !

LN (0, DSD0), if x0:5 6¼ m,

�1

3GjW1j �

d

3G2W2, if x0:5 ¼ m,

8<:

where W1, W2, D, S, and G are defined as Section 3.1.

Proof Let Rn ¼ nlþ (l � l), then 0 � Rn � [Rn] � 1.

By the Bahadur representation (Serfling, 1980, p. 93),ffiffiffin

p{X([Rn]þ1) � X([Rn])} ¼ Op(n�1=5):

It follows that

ffiffiffin

p(Rn � [Rn]){X([Rn]þ1) � X([Rn])} ¼ op(1):

Therefore,ffiffiffin

p{�xxl � xxl} ¼ op(1), for all l 2 (0, 1). Consequently,ffiffiffin

p( �CCNp(u, v) � CNp(u, v)) ¼

ffiffiffin

p(CCNp(u, v) � CNp(u, v)) þ op(1),


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and the asserted result follows by Theorem 1 and Slutsky’s theorem. j

By Theorems 1, 2 and 3, the three estimators of CNp(u, v) – the kernel based estimator~CCNp(u, v), the empirical based estimator CCNp(u, v), and the estimator �CCNp(u, v) proposed by

Pearn and Chen (1997) – are consistent, have the same limiting distributions, and are

asymptotically equivalent.

The above theoretical results are important, but it should be noted that a large sample size

may be required to achieve the normality. See Table I for simulation results illustrating this.

4 COMPARISONS BETWEEN ESTIMATORS OF CNp(u, v) AND Cp(u, v)

In this section, we make comparisons first under normality and then under some nonnormal

distributions. When the process is normal, the class of indices CNp(u, v) is equivalent to

the class of indices Cp(u, v). It is of interest to compare CCNp(u, v) with CCp(u, v) proposed

by Chen (1997) and Chen (1997) in terms of the asymptotic distributions. [It is clear that

the theorem 3.2 in Chen, 1997 is equivalent to the combination of Theorems 1 and 2 in

Chen, 1997, taking into account that in Chen, 1997, p. 153, the author left out a factor

1=s in the expression for Cov(Z1, Z2)]. First, we need the following Lemma.

LEMMA 2 Let X1, X2, . . . , Xn be a sample from the normal process with mean m and

variance s2. Denote d ¼ m� T , G1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þ vd2

p. Then,

(a) the results in Theorem 1 follows with G ¼ G1,

VNp ¼ DSD0 ¼ps2

2sgn(m� m)

u

3G1

�dvdH3G3

1

� �2

þdH

6G31

� �24(0:00135)(0:9973)pe9s4

9

� �,

and

SWNp¼ USU0

¼

u2s2 p2

2uvds2 p4G1

2uvds2 p4G1

4v2d2s2 p8G2

1

þ 2s4 (0:00135)(0:9973)pe9

18G21

0BB@

1CCA:

TABLE I The Quantiles in the Tails of the Asymptotic Distribution and the Empirical Quantiles Derived by aSimulation from N(3.05, 1) with Different Sample Sizes when LSL¼ 0, USL¼ 6 and Target¼ 3.

n 0.01 0.05 0.1 0.5 0.9 0.95 0.99

Empirical 100 �0.4949 �0.1159 �0.1006 0.8907 1.9194 2.2033 2.7876500 �1.2012 �0.6871 �0.3929 0.6324 1.7082 1.9817 2.5756

1000 �1.5029 �0.9244 �0.5883 0.5018 1.6941 2.0158 2.630215,000 �1.8946 �1.2821 �0.9384 0.2531 1.5347 1.8833 2.546010,000 �2.0525 �1.3765 �1.0092 0.2147 1.4611 1.8282 2.474870,000 �2.2227 �1.5299 �1.1488 0.0925 1.3725 1.7127 2.384190,000 �2.2524 �1.5582 �1.2179 0.0854 3.3358 1.6792 2.3340

Asymptotic �2.3263 �1.6448 �1.2815 0 1.2815 1.6448 2.3263


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(b)

ffiffiffin

p(CCp(u, v) � Cp(u, v)) !

LN (0, Vp), if m 6¼ m,

�1

3G1

jW3j �d

6G31

W4, if m ¼ m,

8<:

where Vp ¼ s2[sgn(m� m)u=3G1 � dvdH=3G31]2 þ (dH=6G3

1)22s4,

(W3, W4) � N ((0, 0), SWp),

and

SWp¼

u2s2 2uvds2

2uvds2 4v2d2s2 þ 2s4

� �:

Proof

(a) By Theorem 1 and the fact that x0:5 ¼ m for N (m, s2) the result follows.

(b) This is shown by using Theorem 3.2 in Chen (1997). j

We are now ready to make a first comparison between CCNp(u, v) and CCp(u, v).

THEOREM 4 For a normal process N (m, s2), the limiting distribution of CCNp(u, v) has more

variation than the limiting distribution of CCp(u, v).

Proof

Case I m 6¼ m.

The result follows by Lemma 2 and the fact that VNp � Vp is positive.

Case II m ¼ m.

Let r denote the correlation coefficient of W1 and W2. Then Theorem 1 implies that

Cov(jW1j, W2) ¼ EjW1jW2 ¼ E{E[jW1jW2jW1]}

¼ E{jW1jE(W2jW1)} ¼ E jW1jr

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiVar W2

Var W1

rW1

� �

¼ r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiVar W2

Var W1

rE(jW1jW1) ¼ 0:

Since W1 � N (0, u2s2p=2), we obtain

VarjW1j ¼ EW 21 � (EjW1j)

2 ¼ Var W1 �2

pVar W1 ¼ 1 �

2

p

� �Var W1,

and

Var �1

3G1

jW1j �d

3G21

W2

� �¼

1

9G21

1 �2

p

� �u2s2p

2

þd2

9G41

4v2d2s2p8G2

1

þ2(0:00135)(0:9973)pe9s4

18G21

� �:


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Analogously, by Lemma 2

Var �1

3G1

jW3j �d

6G31

W4

� �¼

1

9G21

1 �2

p

� �u2s2

þd2

36G61

(4v2d2s2 þ 2s4):

Clearly, Var( � (1=3G1)jW1j � (d=3G21)W2) > Var( � (1=3G1)jW3j � (d=6G3

1)W4), as was to

be proved. j

The above results could have been foreseen, since CCp(u, v) is the maximum likelihood

estimator of Cp(u, v). However, if the process is non-normal, then the natural estimator is

not necessarily better, as the following examples show.

Example 1 Let X1, X2, . . . , Xn be a sample from the exponential population with parameter

y, namely

f (x) ¼1

ye�x=y, y > 0, x > 0:

Assume y 6¼ m. By Theorem 1,

ffiffiffin

p(CCNp(u, v) � CNp(u, v))!

LN (0, DSeND

0),

where

SeN ¼

y2

(1 � (0:00135=0:99865))

� ln (0:99865=0:00135)

18y3

(1 � (0:00135=0:99865))

� ln (0:99865=0:00135)

18y3

((0:99865=0:00135) � (0:00135=0:99865))

�( ln (0:99865=0:00135))3

182y4

0BBBBBBB@

1CCCCCCCA

or equivalently

SeN ¼y2 0:3665204y3

0:3665204y3 99:643802y4

!:

(The calculation based on the formula l ¼ F(xl) ¼ 1 � e�xl=y). By Theorem 3.2 in Chen

(1997), ffiffiffin


LN (0, D1SepD

01),

where

Sep ¼y2 2y3

2y3 8y4

!:

Therefore,

Sep � SeN ¼0 1:6334796y3

1:6334796y3�91:643802y4

!:


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Since the matrix Sep � SeN is neither positive definite nor negative definite, neither of the

estimators is always asymptotically better than the other.

Example 2 Let X1, X2, . . . , Xn be a sample from the uniform population U (a, b), and

(aþ b)=2 6¼ m. Then by Theorem 1,

ffiffiffin


LN (0, DSuND

0),

where

SuN ¼

(b� a)2

40

02(0:9973)3(0:00135)(b� a)4

182

0BB@

1CCA:

By Theorem 3.2 in Chen (1997),ffiffiffin


LN (0, DSupD

0),

where

Sup ¼b1 0

0 b2

� �,

and

b1 ¼(b� a)2

12,

b2 ¼b5 � a5

5(b� a)�

(aþ b)(b4 � a4)

2(b� a)

þ 6aþ b

2

� �2b3 � a3

3(b� a)�

3(aþ b)4

16�

(b� a)4

144:

Therefore,

Sup � SuN ¼b3 0

0 b4

� �,

where

b3 ¼�(b� a)2

6,

b4 ¼b5 � a5

5(b� a)�

(b4 � a4)(aþ b)

2(b� a)

þ 6aþ b

2

� �2b3 � a3

3(b� a)�

3(aþ b)4

16

�1

144þ

0:00135(0:9973)3

162

� �(b� a)4:

Since the matrix Sup � SuN is neither positive definite nor negative definite, neither of the

estimators is always asymptotically better than the other.


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Example 3 Let X1, X2, . . . , Xn, be a sample from the uniform population U (a, b), and

(aþ b)=2 ¼ m. When (u, v) ¼ (0, 0),

U0 ¼

0 0

01

2G

!,

and

Sup � SuN ¼b5 0

0 b6

� �

If b6 > 0 (e.g., a ¼ �y, b� y), then U(Sup � SuN )U0 is positive definite, and the limiting

distribution of CCp(u, v) has more variation than the limiting distribution of CCNp(u, v).

5 CONCLUSIONS

In this article, we first proposed two estimators of CCNp(u, v) by applying quantile estimators in

Serfling (1980) and Falk (1985). Then we found the asymptotic distributions of these two

estimators as well as the estimator proposed by Pearn and Chen (1997). The results show

that all three estimators are consistent, asymptotically normally distributed and equivalent.

When the process is normal, the class of indices CCNp(u, v) is reduced to the class of indices

Cp(u, v). It is expected that in this case, the MLE CCp(u, v) proposed by Vannman (1995) is

asymptotically better than CCNp(u, v), ~CCNp

(u, v) and �CCNp(u, v). However, when the underlying

distribution is nonnormal, for instance, exp (y) and U (a, b), the natural estimator CCp(u, v)

need not be better and, in fact, may be inferior to CCNp(u, v), ~CCNp

(u, v) and �CCNp(u, v).

Acknowledgements

The authors appreciate two referees’ and editor’s comments from which the presentation of

this article is greatly improved. The research of the first author was partially supported by

Grant NSC 88-2118-M030-001, National Science Council, Taiwan, Republic of China.

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Chan, L. K., Cheng, S. W. and Spring, F. A. (l988b). New measure of process capability indices: Cpm. Journal ofQuality Technology, 20, 162–175.

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asymptotic analysis of estimators for c np ( u , v ) based on quantile estimators

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