對稱型分位數管制圖

對稱型分位數管制圖 研究成果報告(精簡版) 計 畫 類 別 ： 個別型 計 畫 編 號 ： NSC 97-2119-M-009-007- 執 行 期 間 ： 97 年 08 月 01 日至 98 年 10 月 31 日 執 行 單 位 ： 國立交通大學統計學研究所 計 畫 主 持 人 ： 陳鄰安 計畫參與人員： 碩士班研究生-兼任助理人員：游雅芳 碩士班研究生-兼任助理人員：侯智飛 博士班研究生-兼任助理人員：謝宛茹 報 告 附 件 ： 國外研究心得報告 處 理 方 式 ： 本計畫可公開查詢 中 華 民 國 98 年 12 月 03 日

Reporter: Lin-An Chen (Institute of Statistics, National Chiao Tung University)

Project number: NSC 97-2119-M-009-007 Report type: Brief Version

Excution period: 2008/08/01 - 2009/10/31

Contents:

5. Results and Discussion 6. References 7. Self Evaluation Typeset by A M S-T E X

1. Introduction

In statistical process control, one understanding is that it is not enough to control just the distribution mean, nor it is enough to control just the distribution standard deviation. Repco (1986), Van Nuland (1992), Chao and Cheng (1996) and Spiring and Cheng (1998) developed combination control charts to control the normal mean and standard deviation simultane-ously. A good review of the existing works for control chart of this problem can be found in Cheng and Thaga (2006). These works basically proposed tests dealing with hypothesis of the normal parameters as

H0 :  = 0 = 0

in a simultaneous surveillance of location and scale parameters. It is known that the performance of normal-based control charts is seriously degraded if the underlying distribution is dierent from normal while manufacturing process with non-normal characteristic variable is, however, very common (see, for examples, Cheng and Thaga (2006), Schiling and Nelson (1976) and Kanji and Arif (2000 )).

2. Research Purpose

With full investigation of parametric control chart studies, our research interest is constructing the nonparametric control charts since it makes no assumptions concerning the type of controlling variable.

Some nonparametric control charts are suggested. For exam-ples, Janacek and Meikle (1997) considered the median chart, Liu and Tang (1996) considered the boostrap control chart and Grimshaw and Alt (1997) considered using quantile function to construct the control chart. Very few of the existed non-parametric control charts that are designed to simultaneously control distribution parameters where the quantile based con-trol chart by Grimshaw and Alt (1997) is an exception.

With considering nonparametric control chart, it is interest-ing to study if there is an alternative quantile technique that its produced quantile control chart may gain benit of bet-ter eciency in some sense than the empirical quantiles based control chart of Grimshaw and Alt (1997). In an attempt to improve the eciency of the trimmed mean for estimating the location parameter, Kim (1992) and Chen and Chiang (1996) introduced the symmetric quantile to construct an alternative trimmed mean. Chen and Chiang (1996) observed that this trimmed mean can has asymptotic variances very close to the Cramer-Rao lower bounds for several distributions, including heavy tail ones. Can this interesting result be carried over to the construction of control chart. Our aim in this research is to construct an alternative control chart by symmetric quantiles and show that it does gain better eciency than the classical version constructed by empirical quantiles.

4. Methodologies

consider the population quantile vector Q(1:::k) = 0 B B @ F;1( 1) F;1( 2) ... F;1( k) 1 C C A

for monitoring that can be estimated by the corresponding em-pirical quantile vector

Qe(1:::k) = 0 B B @ F;1 n (1) F;1 n (2) ... F;1 n (k) 1 C C A where F;1

n is the empirical quantile function. Grimshaw and

Alt (1997) proposed to applyQe(1:::k) to monitor the

pop-ulation quantile vector Q(1:::k). The asymptotic property

of Qe(1:::k) relies on the empirical quantiles F ;1

n (j)'s.

Suppose that we have a training sample yiji = 1:::nj =

1:::m that represents an in-control data set of m samples of size n from distribution F so that estimate of Q(1:::k)

and its covariance matrix  are available. Generally we let

Qej(1:::k) and ^j be estimates, respectively, based on

sam-pleyiji = 1:::nand deneQ0(1:::k) = 1 m Pm j=1Qej( 1:::k) and 0 = 1 m Pm j=1 ^j. Treated estimates Q 0 and 0 as true

values of Q(1:::k) and , the control statistic and upper

control limit proposed by Grimshaw and Alt (1997) are Control statistic Te =n(Qe(1:::k) ;Q 0(1:::k)) 0;1 0 (Qe( 1:::k) ;Q 0(1:::k)) UCLe = 2 

Unlike that the empirical quantile is constructed based on the cumulative distribution function, the so-called symmetric quantile of Chen and Chiang (1996) is formulated based on a folded distribution function. The folded cumulative function about a constant , known or unknwon, is

Gs(a) = P(jy;j  a)a 0:

Then, the symmetric quantile pair is dened as

fF (;) s ( )F(+) s ( )g= f;G ;1 s ( )+G;1 s ( )g where G;1

s ( ) = inffa : Gs(a)  g: If distribution function

F is continuous, the symmetric quantile pair satises =

P(F(;)

s ( )  y  F (+)

s ( )). If we further assume that F is

symmetric at , it can be seen that

fF (;) s ( )F(+) s ( )g =fF ;1(1 ; 2 )F;1(1 + 2 )g

the population classical quantiles and the symmetric quantiles are identical. This leads to the fact that sample type symmetric quantiles can play the role of the empirical quantiles to estimate the population quantiles F;1()0s.

For the random sample y1:::yn from distribution F. let ^

be an estimate of . We may dene the sample type sym-metric quantile pair as

fF (;) sn ( )F(+) sn ( )g= f^ ;G ;1 sn( )^+G;1 sn( )g

with sample folded distribution functionGsn(a) = _{n i} I( yi

^

j  a) and

G;1

sn( ) = inffa : Gsn(a)  g:

4. Results and Discussion

Considering a number ` decreasing percentages 1 > 2 >

::: > `, we dene its corresponding 2` symmetric quantile

vec-tor and population symmetric quantiles, respectively, as

Qsn( 1::: `) = 0 B B B B B B B B B B B B B @ F(;) sn ( 1) F(;) sn ( 2) ... F(;) sn ( `) F(+) sn ( `) F(+) sn ( `;1) ... F(+) sn ( 1) 1 C C C C C C C C C C C C C A and Qs( 1::: `) = 0 B B B B B B B B B B B B B @ F(;) s ( 1) F(;) s ( 2) ... F(;) s ( `) F(+) s ( `) F(+) s ( `;1) ... F(+) s ( 1) 1 C C C C C C C C C C C C C A :

From Chen and Chiang (1996), we may see thatn1=2(Q

sn( 1::: `) ;

Qs( 1::: `)) is asymptotically normal N2`(02`s) for some

matrix s that will be given explicitly latter. This further

im-plies that the following

n(Qsn( 1::: `) ;Qs( 1::: `)) 0;1 s (Qsn( 1::: `) ;Qs( 1::: `)) ! 2(2`)

holds asymptotically in distribution.

Again, from the training sample yiji = 1:::nj = 1:::m

n, we let Qs ( ::: `) = <sub>m j</sub> Qsnj( ::: `) and s = 1

m Pm

j=1 ^sj where Qsnj(

1::: `) and ^sj are estimates of,

respectively, Qs( 1::: `) and s. Let us denote these two

esti-mates by Qs0 and s0. Based on these estimates, we proposed

control statistic and upper control limit as Control statistic Ts =n(Qsn( 1::: `) ;Qs 0( 1::: `)) 0;1 s0 (Qsn( 1::: `) ;Qs 0( 1::: `)) UCL =2 (2`)

We observe that the estimation of population quantile vector by empirical quantiles is more ecient when the quantile per-centage is 0:6. However, it is impressed that it gains more preci-sion to use symmetric quantile to construct the quantile vector estimator when percentage is equal or more than 0:8. In fact, the case that when the underlying distribution is the Laplace one the estimator of quantile vector constructed by symmetric quantiles totally dominate the one by empirical quantiles.

The average run length (ARL), representing the average num-ber of samples taken before an action signal is given, is the most popular technique in evaluating a control chart or comparaison of alternative control charts. In the case of process being in-control, both ARL0s by symmetric quantiles and by empirical

quantiles are the expected number 200 for setting  = 0:05. Surprisingly ARL0s of symmetric quantiles are all smaller than

the corresponding ARL0s of empirical quantiles. This indicates

that the symmetric quantiles based control chart can detect the distributional shift with smaller number of samples.

In application, it is more interesting in comparing these two control charts when the coverage probability of the charts is xed at 0:9973. In this consideration, ARL0s are expected to be

370 when the process is in-control. We then com pute ARL0s

for two quantile control charts with signicance level 0:0027. We see that in this setting of coverage interval the symmet-ric quantiles based control chart is still more ecient than the empirical quantiles based control chart in dectection of distri-butional shift.

6. references

Chao, M. T. and Cheng, S. W. (1996). Semicircle control chart for variables data. Quality Engineering.

8

Chen, L.-A. and Chiang, Y. C. (1996). Symmetric type quan-tile and trimmed means for location and linear regression model. Journal of Nonparametric Statistics. 7, 171-185. Cheng, S. W. and Thaga, K. (2005). Multivariate Max-CUSUM

chart. Quality Technology and Quantitative Management International.

2

Grimshaw, S. D. and Alt, F. B. (1997). Control charts for quantile function values. Journal of Quality Technology,

29

Janacek, G. J. and Meikle, S. E. (1997). Control charts based on medians. The Statistician. 46, 19-31.

Kanji, G. K. and Arif, O. H. (2000). Median ranki control chart by quantile approach. Journal of Applied Statistics.

27

Kim, S. J. (1992). The metrically trimmed means as a robust estimator of location, Annals of Statistics. 20, 1534-1547. Liu, R. Y. and Tang, J. (1996). Control charts for dependent

and independent measurements based on bootstrap meth-ods. Journal of the American Statistical Association,

91

Repco, J. (21986). Process capability plot. The Proceedings of the 330th EQQC Conference. 373-381.

Ruppert, D. and Carroll, R.J. (1980). Trimmed least squares estimation in the linear model. Journal of American Sta-tistical Association

75

Shiling, E. G. and Nelson, P. R. (1976). The eect of non-normality on the control limits of X control chart. Journal of Quality Technology,

8

Spiring, F. A. and Cheng, S. W. (1998). An alternative vari-ables control chart: The univariate and multivariate case.

Statistica Sinica.

8

Van Nuland, Y. (1992). ISO 9002 and the circle technique.

Quality Engineering.

5

6. Self Evaluation

This study presents an alternative nonparametric control chart. This symmetric quantile based control chart inherits the adavantage of ecient estimation of population quantiles in re-sulting ecient estimators of population control limits. The average run length study also show that this new control chart

is appealing in application. In the future research, this new control chart is desired to be expanded to the multivariate dis-tribution.

申請人：陳鄰安  單位：交通大學統計所  國科會計畫名稱：近似容忍區間  95‐2118‐M‐009‐007‐  執行期限：2006/08/01─2007/10/31  出國期間：2007 年 8 月 7 日至 2007 年 9 月 4 日  研究地點：School of Public Health, U. of Texas.  研究成果：本次研究我們完成了在存分析相關分配的近似容忍區間之理論分析並    建立相當有價值的資料分析。同時也完成兩項相關議題：  (a) 參考區間(Reference interval)的無母數估計。  傳統上無母數方法估計參考區間是用經驗分位數(Empirical  quantile)來估計的。我們驗證由對稱分位數(Symmetric quantile,  Chen and Chiang (1996, J. of Nonparametric Statistics))具有較小變異 之優點。  (b) 參考區間之診斷  一般醫院採用參考區間來診斷病人，參考區間是醫師用來診斷病 人有病沒病的重要根據。一般醫院採用學術界、衛生單位或大醫 院所建立的參考區間之前必須用統計方法診斷其是否對本地區居 民眾有代表性。傳統方法皆非常粗糙。本計劃將建立由概似函數 方法的一種統計診斷方法。