The asymptotic distribution of the estimated process capability index (C)over-tilde(pk)

(1)

This article was downloaded by: [National Chiao Tung University 國立交通大學] On: 28 April 2014, At: 06:36

Publisher: Taylor & Francis

Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Communications in Statistics - Theory and Methods

Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/lsta20

The asymptotic distribution of the estimated process

capability index C

pk

Sy-Mien Chen a & W. L. Pearn b a

Department of Mathematics , Fu-Jen Catholic University , Taipei, Taiwan , ROC b

Department of Industrial Engineering & Management , National Chiao Tung University , Hsinchu, Taiwan, ROC

Published online: 27 Jun 2007.

To cite this article: Sy-Mien Chen & W. L. Pearn (1997) The asymptotic distribution of the estimated process capability index Cpk , Communications in Statistics - Theory and Methods, 26:10, 2489-2497, DOI: 10.1080/03610929708832061 To link to this article: http://dx.doi.org/10.1080/03610929708832061

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no

representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any

form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions

(2)

THE ASYMPTOTIC DISTFUBUTION OF THE ESTIMATED PROCESS CAPABILITY INDEX

Cpk

Sy-Mien Chen Department of Mathematics

Fu- Jen Catholic University, Taipei, Taiwan, ROC

W. L. Pearn

Department of Industrial Engineering & Management National Chiao Tung University, Hsinchu, Taiwan, ROC

Keywords and Phrases: process capability index; asymptotic distribution; process mean; process standard deviation.

ABSTRACT

Bissell (1990) proposed an estimator

ck

for the process capability index Cpk assuming that P ( p

2

m) = 0, or 1, where p is the process mean, and m is the midpoint between the upper and lower specification limits. Pearn and Chen (1996) considered a new estimator

CPk,

which relaxes Bissell's assumption on the process mean. The evaluation of

cPk

only requires the knowledge of P ( p

2

m) = p, where 0

<

p

5

1. The new estimator

Gk

is unbiased, and the variance is smaller than that of Bissell's.

In this paper, we investigated the asymptotic properties of the estimator c P k under general conditions. We derived the limiting distribution of

Cpk

for arbitrary population assuming the fourth moment exists. The asymptotic distribution provides some insight into the properties of

Cpk

which may not be evident from its original definition.

Copyright O 1997 by Marcel Dekker, Inc.

(3)

2490 CHEN AND PEARN

1. INTRODUCTION

Process capability indices, whose purpose is to provide a numerical mea- sure on whether a production process is capable of producing items satisfyng the quality requirements preset by the designer, have received substantial at- t,ention in the quality control and statistical literature. The two most widely used capability indices are C, = , and Cpk = 3a ' 30 },

where U S L is the upper specification limit, LSL is the lower specification

limit, p is the process mean. and o is the process standard deviation. While the C, index reflects only the magnitude of the process variation, the Cpk index

takes into account the process variation as well as the location of the process mean relative to the specification limits.

For processes with two-sided specification limits, the process yield can be calculated as F ( U S L ) - F ( L S L ) , where F ( . ) is the cumulative distribution function of the process characteristic. If the process is normal, then the process yield can be expressed as :

where

a(.)

is the cunlulative function of the standard normal distribution. If the process is perfectly centered, then the process yield can be expressed alter- natively as 2@(3CPk) - 1 = 2@(3C,) - 1. For example, Cpk = 1.00 corresponds to process yield 99.73%. If the process is non-norlnal (symmetric or asym- metric), then these indices provide only approximate measures on the process performance.

For Cpk, if we assume P ( p

1

m) = 0, or 1, where m = (USL+ L S L ) / 2 is

the mid-point between the upper and lower specification limits, then the Bissell (1990) estimator

ck

is defined a s follows:

ck

= (USL - x ) / 3 S if p 2 rn;

otherwise.

Ck

=

(Xn

-LSL)/3S, where 1, = Xi/n and S = { C !=AXt -

(4)

X n ) 2 / ( n - l ) ) l / ' are conventional estimators of p and u , respectively, derived

from a sample X I , X 2 ,

. . . ,

X,. Pearn and Chen (1996) considered a Bayesian-

-

like estimator Cpk to relax Bissell's assumption on the process mean. The

evaluation of the estimator

ck

only requires the knowledge of P ( p

2

m) = p,

0

5

p

5

1, which may be obtained from historical information of a stable

process. If P ( p 2 m) = 0, or 1, then the estimator

ck

reduces to Bissell's estimator. The estimator is defined as

ck

= {d -

( X n

- m ) I A ( p ) ) / 3 S ,

where I A ( p ) = 1 if p E A , I A ( p ) = -1 if p

6

A, A = { p ( p

2

m ) and

d = ( U S L - L S L ) / 2 .

Pearn and Chen (1996) showed that by multiplying the well-known cor-

rection factor bf to the estimator

ck,

where bf = { 2 / ( n - 1))1/21?((n -

1 ) / 2 ) { F ( ( n - 2 ) / 2 ) } - * , an unbiased estimator

ePk

=

bfck

can be obtained. Pearn and Chen (1996) also showed that on the assumption of normality, the

distribution of the estimator 3 ( n ) ' f 2 C k is t,-'(S), a non-central t with n - 1

degrees of freedom and non-centrality parameter S = 3(n)'/'CPk. In this paper, we investigate the asymptotic distribution of c P k under general con-

ditions. We derived the limiting distribution of

epk

for arbitrary population assuming the fourth central moment exists. The asymptotic distribution provides some insight into the properties of G k which may not be apparent from

its original definition. Consequently, some approximate statistical testing on whether a process is capable can be performed.

2. ASYMPTOTIC DISTRTBUTION OF

Gk

Let X I , . . .

,

X , be a random sample of measurements from a process

whch has distribution G with mean p and positive variance u 2 . Let /z =

'2 - C ;= ( X ,

-x,"

) 2

' : ' = l x ' = X n , a n d o

-

= S;-,

,

be the sample mean and sample variance, respectively.

(5)

2492 CHEN AND PEARN

Define d =

v ,

and m =

v .

Then, the process capability index Cpk is defined as

When both the mean p and the variance a' of the measurements are un-

known, an estimator proposed by Pearn and Chen (1996) is given by:

Lemma 1: Define pk = E(X - p)k as the k-th central moment of G. If

p1 exists, then a s n + m ,

where

[Proof]: See page 72 of Serfling (1980).

Lemma 2: Let { u ( n ) ) be a sequence of m-component random vectors

and b a fixed vector such that f i [ u ( n ) - b] has a limiting distribution N(0, T)

as 71 + m. Let f ( u ) be a vector-valued function of u such that each component f,(u) has a nonzero differential at u = b, and let Wlu=b be the (i, j)-

th component of

ab.

Then, f i { f ( u ( n ) )

-

f ( b ) ) has a limiting distribution

N(O, Q;,TQ,,).

(6)

[Proof]: See Theorem 4.2.3 in T.W. Anderson (1984). ( u-m d

I

(1

-

TI.

3+7q

,

if

P

>

m.

du,

v ) = u-m

.

1

I

[ I +

7 1

3m , i f p < m . Since then Cpk = g(p, u 2 ) / b f . Case 1: When p

>

m, u

>

m. Since u - m d U - m d . b f g(u,v)=[l--I.--== = [I-

-1.

- d 3Jv/b+ d 3 & '

is a real valued function and is differentiable for all u E ( L S L , U S L ) , v

>

0 ,

with

(7)

CHEN AND PEARN

and

Define

then Dl

#

(0,O).

Hence, by Lemma 1 and Lemma 2,

where

Case 2: When p

<

m. u

<

m.

Since

is a real valued function and is differentiable for all u E ( L S L , U S L ) , v

>

0,

and

Define

then D2

#

(0.0).

(8)

Hence, where Since p ( f i [ b f

(qk

- Cpk)]

5 t )

= p ( f i [ b f ( G k - C,n)l

5

tip

>

m)

.

P ( p

2

rn)

+

p(&[bf(qk - C,k)l

5

tip

<

m)

.

P ( p

<

m), then

& [ b f ( C k - Cpk)]

5

N(0, o&)

.

P ( p

L

m)

+

N(0, u;,,)

.

P ( p

<

m).

Where

b; d

0 , k i = ?-{I+ =(1

-

7 ) [ 2 p 3

+

$ ( 3

- 02)(1 - y ) ] ) ,

2

3

d

opk2 = 9 { I

-

w ( l -k 7 ) [ 2 p 3 i-

?(B

2 oL - u2)(1

+

y)]}.

Thus, the asymptotic distribution of

fi[cPk

- bfCpk)] is a mixture of normal distributions provided that the weights P ( p

>

m) or P ( p

<

m) is known.

Theorem 2: If p4 exists, and p E (LSL, USL), then f i [ ~ , k

-

C,k]

5

N(0, a$,) . P ( p

>

m)

+

N(0, u&) . P ( p

<

m).

Where a$, and

aikl

are defined a s in Theorem 1.

[proof]: Since

fi[Gk

- Cpk] =

f i [ b f q k

- c p k ] = f i [ b f

(ck

- Cpk)]

+

n-03

fi[bf~Cpk-Cpk)]=&[bf(ck-~pk)]+fi(bf-l).[Cpk-cpk],andbf -+ 1.

then the result follows from Theorem 1 and Slutsky's Theorem (Lohe (1978)).

(9)

2496 CHEN AND PEARN

3. CONCLUSIONS

Bissell (1990) proposed an estimator

ck

to calculate Cpk value assuming that P ( p

2

m) = 0 or 1. Pearn and Chen (1996) considered a new estimator e p k , which relaxes Bissell's assumption on the process mean. The evalution of cpl; only requires the knowledge of P ( p 2 m) = p, where 0

5

p

5

1. The new estimator cpl; is unbiased, and the variance is smaller than that of Bissell's estimator. In this paper, we investigated the asymptotic properties of the estimator

Gl;.

We showed that the limiting distribution of

Cpk

for arbitrary population is a contamination of normal distributions provided that the fourth rnomeiit exists.

Thus. if the knowledge of P ( p

2 rn)

= p, and P ( p

<

m) = 1 - p is given, then the asymptotic distribution of the estimator

cPk

is a contamination of two nornlal distributions. The estimator treats the process as a mixture of two manufacturing processes . Such situation occurs when the raw materials or components come from two different suppliers, or the machines, equipments have two different conditions, or there are two different groups of workmanship involved in the process.

ACKNOWLEDGMENTS

The authors are grateful to the editor's and referees' suggestions that improve this presentation.

BIBLIOGRAPHY

Anderson, T. W. (1984). A n Introduction to Multivariate Statzstical Analysis,

second edition. John Wiley, New York.

Bissell. A. F. (1990). .'How Reliable is Your Capability Index?," Applied

Statistics, 39, 331-340.

(10)

Chan, L., Xiong, Z. and Zhang, D. (1990). "On the Asymptotic Distributions of Some Process Capability Indices," Communicatzon in Statistics- Theory and Methods, 19(1). 11-18.

Kotz, S. and Johnson, N. L. (1993). Process Capability Indices, Chapman &

Hall.

Lokve, M. (1978). Probability Theory 11, 4th edition, Springer-Verlag, Berlin and New York.

Pearn, W. L., Kotz, S. and Johnson, N. L. (1992). "Distribution and In- ferential Properties of Process Capability Indices," Journal of Quality

Technology, 24. 216-231.

Pearn, W. L. and Chen, K. S. (1996). "A Bayesian-Like Estimator Of Cpk,"

Communication in Statistics

-

Simulations and Computation, 25(2), 321-

329.

Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. John Wiley and Sons, New York, 1-125.

Received October, 1996; Revised May, 1997.