類別製造過程的多變量指數加權移動平均監控計劃

國 立 交 通 大 學

統 計 學 研 究 所

碩 士 論 文

類別製造過程的多變量指數加權移動平均監控計劃

Multivariate Exponentially Weighted Moving Average

Monitoring Scheme for a Categorical Manufacturing Process

研 究 生：洪鏡婷

指導教授：陳志榮 博士

類別製造過程的多變量指數加權移動平

均監控計劃

Multivariate Exponentially Weighted Moving Average

Monitoring Scheme for a Categorical Manufacturing

Process

研 究 生：洪鏡婷 Student：Jing-Ting Hong

指導教授：陳志榮 博士 Advisor：Dr. Chih-Rung Chen

國 立 交 通 大 學

統 計 學 研 究 所

碩 士 論 文

A Thesis

Submitted to Institute of Statistics

College of Science

National Chiao Tung University

In Partial Fulfillment of the Requirements

for the Degree of Master

in

Statistics

June 2010

Hsinchu, Taiwan

類別製造過程的多變量指數加權移動平

均監控計劃

學生：洪鏡婷

指導教授：陳志榮

國立交通大學統計學研究所

摘

要

本文是發展一個方法去監控不同類別的資料 。首先介紹

Dirichlet-多項式模型，根據該模型提出了類別製造過程的多變量

指數加權移動平均管制圖並研究其相關的特性。最後用模擬的資

料來說明這個方法的實用性與適用性。

Multivariate Exponentially Weighted Moving Average

Monitoring Scheme for a Categorical Manufacturing

Process

Student：Jing-Ting Hong Advisor：Dr. Chih-Rung Chen

Institute of Statistics,

National Chiao Tung University

ABSTRACT

T

his paper is to develop a method to monitor the fractions of the tested

items

falling

into

different

categories

of

pass/fail

modes.

T

he

Dirichlet-compound multinomial model is first introduced and then a

multivariate exponentially weighted moving average control chart is proposed

for a categorical manufacturing process under the proposed model. Some

relevant properties of the proposed control chart are also investigated. Finally, a

simulation study is presented to show the usefulness and applicability of the

proposed methodology.

誌 謝

研究所兩年的光陰匆匆流逝，現在的我即將踏出校園邁入我人

生的另一個階段。

在系上學習的這兩年時光，有喜有樂、有苦有悲，感謝有師長

們的一路陪伴，讓我能從挫折中學會成長，感謝系上提供我們優秀

的學習環境，特別感謝我的指導教授－陳志榮教授，謝謝您不厭其

煩的指導我的論文，給予我許多的建議和協助。 也要謝謝我那些

可愛的同學們，謝謝你們一路的支持鼓勵，因為有你們，讓生活增

添了許多歡笑，讓我不覺得孤單。

還有，我要謝謝我的家人，給了我最大且最安全的後盾，讓我

無憂無慮的完成我的學業，我愛你們！

最後，我還要感謝古佳健先生，謝謝你給我滿滿的愛，在我背

後給我打氣加油、不斷的給我鼓勵，謝謝你！

感謝大家，這份榮耀是屬於你們的！

洪鏡婷 謹誌于

國立交通大學統計學研究所

中華民國九十九年六月

Contents

List of Tables ii 1 Introduction 1 1.1 Motivation . . . 1 1.2 Literature Review . . . 1 1.3 Outline . . . 5

2 Dirichlet-Compound Multinomial Model for a

Categorical Manufacturing Process 6

3 Multivariate Exponentially Weighted Moving

Average Monitoring Scheme 9

4 A Simulation Study 12

List of Tables

1 h for λ ∈ S0 and α0 = (85, 10, 5)T. . . 18 2 ARL1 for l = 0, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }, where α (1) 1 ≡ (80, 12.5, 7.5)T, α₁(2) ≡ (75, 15, 10)T_{, and α}(3) 1 ≡ (70, 20, 10)T. . . 18 3 ARL1 for l = 1, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 18 4 ARL1 for l = 2, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 19 5 ARL1 for l = 3, λ ∈ S 0 ; α1 ∈ {α(1)1 , α (2) 1 , α (3) 1 }. . . 19 6 ARL1 for l = 4, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 19 7 ARL1 for l = 5, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 20 8 ARL1 for l = 6, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 20 9 ARL1 for l = 7, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 20 10 ARL1 for l = 8, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 21 11 ARL1 for l = 9, λ ∈ S 0 ; α1 ∈ {α(1)1 , α (2) 1 , α (3) 1 }. . . 21 12 ARL1 for l = 10, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 21 13 ARL1 for l = 11, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 22 14 ARL1 for l = 12, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 22 15 ARL1 for l = 13, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 22 16 ARL1 for l = 14, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 23 17 ARL1 for l = 15, λ ∈ S 0 ; α1 ∈ {α(1)1 , α (2) 1 , α (3) 1 }. . . 23 18 ARL1 for l = 16, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 23 19 ARL1 for l = 17, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 24 20 ARL1 for l = 18, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 24 21 ARL1 for l = 19, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 24 22 ARL1 for l = 20, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 25 23 ARL1 for l = 21, λ ∈ S 0 ; α1 ∈ {α(1)1 , α (2) 1 , α (3) 1 }. . . 25 24 ARL1 for l = 36, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 25 25 ARL1 for l = 37, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 26 26 ARL1 for l = 38, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 26

27 ARL1 for l = 138, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 26 28 ARL1 for l = 139, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 27 29 ARL1 for l = 140, λ ∈ S 0 ; α1 ∈ {α(1)1 , α (2) 1 , α (3) 1 }. . . 27 30 ARL1 for l = 450, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 27 31 ARL1 for l = 500, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. . . 28

1

Introduction

1.1

Motivation

In the current international marketplace, quality improvement is a key point for main-taining competitive advantage. Statistical process control is an effective tool for achieving process stability and improving process capability through variation reduction. When a product item is tested, usually one has more information than just pass or fail. Often there are categories of failures. For instance, a product may have several categories of failure modes. Thus, in the paper, a multivariate exponentially weighted moving av-erage monitoring scheme is proposed for a categorical manufacturing process under the Dirichlet-compound multinomial model and then some relevant properties of the proposed monitoring scheme are also investigated.

1.2

Literature Review

Statistical process control (SPC) refers to some statistical methods which are widely used to monitor and improve the quality and productivity of industrial processes and service operations. SPC primarily involves the implementation of control charts. The method of control charts is a graphical tool which is used to monitor processes in or-der to distinguish special significant causes of variation from general assignable causes of variation in processes. The Shewhart (Shewhart, 1931), cumulative sums (CUSUM) (Page, 1954), and exponentially weighted moving average (EWMA) (Roberts, 1959) con-trol charts are widely used in practice. Standard concon-trol chart usage involves phase I and phase II applications, with two different and distinct objectives. In phase I, a set of process data is gathered and analyzed all at once in a retrospective analysis, constructing trial control limits to determine if the process has been in control over the period of time where the data were collected, and to see if reliable control limits can be established to monitor future production. In phase II, we use the control chart to monitor the process by comparing the sample statistic for each successive sample as it is drawn from the process to the control limits.

The Shewhart control chart (Shewhart, 1931) monitors the process observations di-rectly. Suppose that {xtm: t = 1, 2, ... and m = 1, ..., n} are independent univariate

mea-surements, where n is a known positive integer and xt1, ..., xtn are identically distributed

for each t. Set ¯xt ≡ Pn_m=1xtm/n for t = 1, 2, .... Let µ0 denote the known real-valued

in-control process target and σ0 the known positive in-control process standard deviation.

When ¯xt ∼ N (µ0, σ02/n), the process is called in control at time t; otherwise, out of control

at time t. Then the Shewhart control chart is based on the values

wt= ¯xt− µ0 (1)

for t = 1, 2, .... Shewhart (1931) proposed the stopping time of the Shewhart control chart as the first time t such that

|wt| > L

σ0

√

n, (2)

where L = 3. In practice, L is often chosen to achieve a specified in-control average run length.

Page (1954) introduces the CUSUM chart as a sequential probability test. Suppose that {xtm: t = 1, 2, ... and m = 1, ..., n} are independent univariate measurements, where

n is a known positive integer and xt1, ..., xtn are identically distributed for each t. Set ¯xt≡

Pn

m=1xtm/n for t = 1, 2, .... Let µ0 denote the known real-valued in-control process target

and σ0 the known positive in-control process standard deviation. When ¯xt∼ N (µ0, σ02/n),

the process is called in control at time t; otherwise, out of control at time t. The CUSUM algorithm assigns equal weights to past observations, and its tabular form consists of two quantities, w_t+ = max0, w+_t−1+ (¯xt− µ0) − kσ0/ √ n (3) and w_t−= min0, w− t−1+ (¯xt− µ0) + kσ0/ √ n (4)

for t = 1, 2, ..., where w+₀ = w−₀ = 0 and k is the reference value which is often chosen about halfway between the target µ0 and the out-of-control mean value µ1 of interest.

Page (1954) proposed the stopping time of the CUSUM control chart as the first time t such that either w+_t or w−_t exceed the decision interval H, where H is chosen to achieve a specified in-control average run length, e.g., H = 5 when ARL0 = 465.

Roberts (1959) proposed a monitoring scheme which is based on the EWMA of the observations. The EWMA, originally called geometric moving average (GMA) in Roberts (1959), is briefly introduced as follows: Suppose that {xtm: t = 1, 2, ... and m =

1, ..., n} are independent univariate measurements, where n is a known positive integer and xt1, ..., xtn are identically distributed for each t. Set ¯xt ≡

Pn

m=1xtm/n for t = 1, 2, ....

Let µ0 denote the known in-control real-valued process target and σ0 the known positive

in-control process standard deviation. When ¯xt ∼ N (µ0, σ02/n), the process is called in

control at time t; otherwise, out of control at time t. Then the EWMA control chart is based on the values

wt ≡ (1 − λ)wt−1+ λ(¯xt− µ0) = λ(1 − λ)t−1(¯x1− µ0) + · · · + λ(1 − λ)(¯xt−1− µ0) + λ(¯xt− µ0) = λ t−1 X i=0 (1 − λ)i(¯xt−i− µ0) (5)

for t = 1, 2, ..., where w0 ≡ 0 and λ is a specified value in (0, 1]. If ¯xt∼ N (µ0, σ20/n), then

wt∼ N (0, σ2t), where σt= s λ [1 − (1 − λ)2t_] n(2 − λ) σ0 → s λ n(2 − λ) σ0 (6) as t → ∞. Roberts (1959) proposed the stopping time of the EWMA monitoring scheme as the first time t such that

|wt| > Lσt, (7)

where L = 3. In practice, L is often chosen to achieve a specified in control average run length. It is the same as the Shewhart control chart when λ = 1, and nearly the same as the CUSUM control chart when λ → 0. There have been numerous extensions and variations of the basic EWMA control chart.

Through modern technology that allows simultaneously monitoring all key quality characteristics during a manufacturing process, the monitored quality characteristics are

usually dependent each other. This is especially true for quality characteristics related to safety, fault detection and diagnosis, quality control, and process control. Joint moni-toring of quality characteristics ensures appropriate control of the overall process. Multi-variate SPC techniques have recently been applied to novel fields such as environmental monitoring and detection of computer intrusion. The purpose of multivariate on-line techniques is to investigate whether quality characteristics are simultaneously in control or not. Versions of the multivariate Shewhart, CUSUM, and EWMA control charts have been proposed under the multivariate normality assumption.

To incorporate the recent historical information, Lowry et al. (1992) proposed a mul-tivariate exponentially weighted moving average (MEWMA) control chart which is briefly introduced as follows: Suppose that {xtm: t = 1, 2, ... and m = 1, ..., n} are independent

p-variate measurements, where n is a known positive integer; xt1, ..., xtn are identically

distributed for each t; and p (≥ 2) is a known positive integer. Set ¯xt≡

Pn

m=1xtm/n for

t = 1, 2, .... Let µ0 denote the known p × 1 in-control process target vector in (−∞, ∞)p

and Σ0the known p×p positive definite in-control process covariance matrix. Let Np(µ, Σ)

denote the p-variate normal distribution with mean vector µ and covariance matrix Σ. When ¯xt ∼ Np(µ0, Σ0/n), the process is called in control at time t; otherwise, out of

control at time t. Lowry et al. (1992) proposed the MEWMA control chart as based on the p × 1 vectors wt ≡ (Ip− Λ)wt−1+ Λ(¯xt− µ0) = Λ(Ip− Λ)t−1(¯x1− µ0) + · · · + Λ(Ip− Λ)(¯xt−1− µ0) + Λ(¯xt− µ0) = Λ t−1 X i=0 (Ip− Λ)i(¯xt−i− µ0) (8)

for t = 1, 2, ..., where w0 ≡ 0p×1, the p × 1 vector (0, ..., 0)T; Ip denotes the identity matrix

of order p; and Λ is a specified diagonal matrix diag{λ1, ..., λp} with λ1, ..., λp ∈ (0, 1]. Set

Σ0 ≡ (Σ0jj0)j,j0=1,...,p. If ¯xt∼ Np(µ0, Σ0/n), then wt∼ Np(0p×1, Σt), where Σt ≡ (Σ_tjj0) j,j0=1,...,p = 1 nΛ "_t−1 X i=0 (Ip− Λ)iΣ0(Ip− Λ)i # Λ → 1 nΛ " _∞ X i=0 (Ip− Λ)iΣ0(Ip − Λ)i # Λ (9)

as t → ∞ with Σ_tjj0 = λjλj01 − (1 − λj)t(1 − λj0)t Σ0jj0 n(λj + λj0 − λjλj0) → λjλj 0Σ 0jj0 n(λj+ λj0 − λjλj0) (10) as t → ∞. In particular, when λ1 = ... = λp = λ, Σt= λ [1 − (1 − λ)2t_] n(2 − λ) Σ0 → λ n(2 − λ) Σ0 (11) as t → ∞. Then the stopping time of the MEWMA monitoring scheme is the first time t such that

wT_tΣ−1_t wt (≡ Tt2) > h, (12)

where h is chosen to achieve a specified in-control average run length.

Consider a manufacturing process where each product units can be classified as one of k + 1 disjoint categories for some fixed k ∈ {1, 2, ...}. When the outcome is recorded as one of two categories, e.g., {pass, fail}, the data are called binary. When the outcome is recorded as one of k +1 disjoint categories for some k ∈ {2, 3, ...}, the data are called poly-tomous, e.g., {pass, the first defect type, ..., the kth defect type}. See, e.g., McCullagh and Nelder (1989). Several researchers have investigated categorical data in different situ-ations. Shiau et al. (2005) proposed the Dirichlet-compound multinomial empirical Bayes model to monitor the polytomous data. In the paper, we develop an MEWMA control chart for monitoring a manufacturing process under the Dirichlet-compound multinomial model.

1.3

Outline

The paper is organized as follows. In Section 2, the Dirichlet-compound multinomial model for a categorical manufacturing process is briefly introduced. In Section 3, a mul-tivariate exponentially weighted moving average control chart is proposed and then some relevant properties of the proposed control chart are also investigated. In Section 4, a simulation study is presented to illustrate the proposed methodology. Finally, comparison and conclusions are given in Section 5.

2

Dirichlet-Compound Multinomial Model for a

Categorical Manufacturing Process

Consider a manufacturing process which produces product units having k different types of defects for some known positive integer k. In a product unit, let pit denote the

probability of having the ith defect type at time t for i = 1, ..., k. Then 1−Pk

i=1pit(≡ p0t)

is the probability of having none of these k defect types at time t. For i = 1, ..., k, let xit

denote the number of tested product units having the ith defect type among ntrandomly

chosen tested product units at time t. Then nt−Pk_i=1xit (≡ x0t) is the number of tested

product units having none of these k defect types at time t. Set pt ≡ (p0t, p1t, ..., pkt)T

and xt ≡ (x0t, x1t, ..., xkt)T. Then pt ∈ P and xt ∈ Xt, where P ≡ {pt: p0t, p1t, ..., pkt ∈

(0, 1) with Pk

i=0pit = 1} and Xt ≡ {xt: x0t, x1t, ..., xkt∈ {0, 1, ..., nt} with

Pk

i=0xit = nt}.

Then the number of elements in Xtis (nt+ k)!/(nt! k!) (≡ |Xt|). Assume that xtgiven pt

is distributed as either binomial(nt; pt) for k = 1 or multinomial(nt; pt) for k ≥ 2. Then

the conditional probability mass function (p.m.f.) of xt given pt is

f (xt|pt) = nt! x0t! x1t! · · · xkt! px0t 0t p x1t 1t · · · p xkt kt · 1Xt(xt), (13)

where 1Xt(xt) = 1 for xt∈ Xt and 0 otherwise.

Suppose that pt is distributed as either beta(α) for k = 1 or Dirichlet(α) for k ≥ 2,

where α (≡ (α0, α1, ..., αk)T) is the unknown (k + 1) × 1 parameter vector in the parameter

space (0, ∞)k+1_{. Set α} s ≡

Pk

i=0αi. Then the probability density function (p.d.f.) of pt is

f (pt; α) = Γ(αs) Γ(α0) Γ(α1) · · · Γ(αk) pα0−1 0t p α1−1 1t · · · p αk−1 kt · 1P(pt), (14)

where 1P(pt) = 1 for pt ∈ P and 0 otherwise. Then pt given xt is distributed as either

beta(α0+ x0t, α1+ x1t) for k = 1 or Dirichlet(α0+ x0t, α1+ x1t, ..., αk+ xkt) for k ≥ 2 with

Eα(pt|xt) =  α0+ x0t αs+ nt ,α1+ x1t αs+ nt , ..., αk+ xkt αs+ nt T . (15)

See, e.g., p. 217 of Johnson et al. (1995) for the parametric family of beta distributions and p. 488 of Kotz et al. (2000) for the parametric family of Dirichlet distributions.

Then the p.m.f. of xt is f (xt; α) = f (xt, pt; α) f (pt|xt; α) = f (xt|pt)f (pt; α) f (pt|xt; α) = exp "_nt−1 X j=0 log j + 1 αs+ j  − k X i=0 xit−1 X j=0 log j + 1 αi+ j # · 1Xt(xt); (16)

see, e.g., pp. 80–81 of Johnson et al. (1997). Then, given xt, the likelihood function for α

is L(α; xt) = exp "_n t−1 X j=0 log j + 1 αs+ j  − k X i=0 xit−1 X j=0 log j + 1 αi+ j # , (17)

the log-likelihood function for α is

l(α; xt) = log [L(α; xt)] = nt−1 X j=0 log j + 1 αs+ j  − k X i=0 xit−1 X j=0 log j + 1 αi + j  , (18)

the score function for α is ∂l(α; xt) ∂α = x0t−1 X j=0 1 α0+ j , x1t−1 X j=0 1 α1+ j , ..., xkt−1 X j=0 1 αk+ j !T − nt−1 X j=0 1 αs+ j ! · 1(k+1)×1 ≡ S(α; xt) ≡ (S0(α; xt), S1(α; xt), ..., Sk(α; xt)) T , (19)

the observed Fisher information for α is

−∂ 2_{l(α; x} t) ∂α∂αT = diag (_x0t−1 X j=0 1 (α0+ j)2 , x1t−1 X j=0 1 (α1+ j)2 , ..., xkt−1 X j=0 1 (αk+ j)2 ) − "_n t−1 X j=0 1 (αs+ j)2 # · 1(k+1)×11T(k+1)×1 ≡ J(α; xt) ≡ (Jii0(α; xt))_i,i0 =0,1,...,k, (20)

and the expected Fisher information for α is

Covα(S(α; xt)) ≡ It(α) ≡ (Itii0(α))i,i0=0,1,...,k, (21)

where 1(k+1)×1 denotes the (k+1)×1 vector (1, ..., 1)T. Notice that Eα(S(α; xt)) = 0(k+1)×1

(0, ..., 0)T. Then Itii(α) = X xt∈Xt xit−1 X j=0 1 αi+ j − nt−1 X j=0 1 αs+ j !2 f (xt; α) = X xt∈Xt "_xit−1 X j=0 1 (αi+ j)2 # f (xt; α) − nt−1 X j=0 1 (αs+ j)2 (22) for i = 0, 1, ..., k and I_tii0(α) = − nt−1 X j=0 1 (αs+ j)2 (23)

for i, i0 = 0, 1, ..., k with i 6= i0. Sometimes, |Xt| is very large at time t in a

man-ufacturing process, e.g., |Xt| = 82, 408, 626, 300 when nt = 200 and k = 6. In such

situations, it will take too much time to evaluate Itii(α)s by equation (18). One

possi-ble approach to evaluate Itii(α)s is the Monte Carlo method as follows: First generate

i.i.d. (p(1)T_t , x(1)T_t )T, ..., (p(r)T_t , x(r)T_t )T such that p(u)_t is sampled from Dirichlet(α) and x(u)_t given p(u)_t is sampled from multinomial(nt; p

(u)

t ) for u = 1, ..., r, where r is a large positive

integer, e.g., r = 100, 000. Then Itii(α) can be approximately evaluated by \Itii(α), where

\ Itii(α) ≡ 1 r r X u=1   x(u)_it −1 X j=0 1 αi+ j − nt−1 X j=0 1 αs+ j   2 (24) or 1 r r X u=1 x(u)_it −1 X j=0 1 (αi+ j)2 − nt−1 X j=0 1 (αs+ j)2 . (25)

3

Multivariate Exponentially Weighted Moving

Average Monitoring Scheme

In this section, a multivariate exponentially weighted moving average control chart is proposed for monitoring a categorical manufacturing process under the Dirichlet-compound multinomial model as follows: Suppose that {(pT

t, xTt)T: t ≥ 1} are independent (2k+2)×1

random vectors, where both pt and xt are described in Section 2. Let α0 (∈ (0, ∞)k+1)

denote the known (k + 1) × 1 in-control process parameter vector in phase I. The multi-variate exponentially weighted moving average (MEWMA) control chart is based on the (k + 1) × 1 vectors wt ≡ (Ik+1− R)wt−1+ R S(α0; xt) = R(Ik+1− R)t−1S(α0; x1) + · · · + R(Ik+1− R)S(α0; xt−1) + R S(α0; xt) = R t−1 X i=0 (Ik+1− R)iS(α0; xt−i) (26)

for t = 1, 2, ..., where w0 ≡ 0(k+1)×1; Ik+1 denotes the identity matrix of order k + 1; R is

a specified (k + 1) × (k + 1) positive definite covariance matrix such that Ik+1− R is

non-negative definite; and (Ik+1 − R)0 ≡ Ik+1. It follows from the eigenvalue decomposition

that R = P ΛPT where P is an orthogonal matrix, i.e., P PT = Ik+1, and Λ is a diagonal

matrix diag{λ0, λ1, ..., λk} with λ0, λ1, ..., λk∈ (0, 1]. Then

wt= P Λ t−1 X i=0 (Ik+1− Λ)iPTS(α0; xt−i) (27) for t = 1, 2, ....

When p1, ..., pt ∼ Dirichlet(α0) for some t = 1, 2, ..., the following properties hold:

(i) Eα0(wt) = 0(k+1)×1. (ii) Covα0(wt) = P Λ "_t−1 X i=0 (Ik+1− Λ)iPTIt−i(α0)P (Ik+1− Λ)i # ΛPT (≡ Σt). (28) (iii) Eα0(w T tΣ −1 t wt) = k + 1. Set T_t2 ≡ wT tΣ −1 t wt. (29)

(iv) (R−1ΣtR−1)−1/2R−1wt a.s. → "_t−1 X i=0 It−i(α0) #−1/2 t−1 X i=0 S(α0; xt−i) (30) and T_t2 a.s.→ "_t−1 X i=0 S(α0; xt−i) #T "_t−1 X i=0 It−i(α0) #−1 t−1 X i=0 S(α0; xt−i) (≡ Tt02) (31)

as max{λ0, λ1, ..., λk} → 0, where Tt02 is the score test statistic up to time t for testing

the null hypothesis H0: p1, ..., pt ∼ Dirichlet(α0) versus the alternative H1: p1, ..., pt ∼

Dirichlet(α) for some α 6= α0.

(v) If sup_t≥1 nt < ∞, then (R−1ΣtR−1)−1/2R−1wt d → Nk+1(0(k+1)×1, Ik+1) and Tt2 d → χ2 k+1 as max{λ0, λ1, ..., λk} → 0 and t → ∞.

(vi) If Λ = λ Ik+1 for some λ ∈ (0, 1], then

wt= λ t−1 X i=0 (1 − λ)iS(α0; xt−i), (32) T_t2 = "_t−1 X i=0 (1 − λ)iS(α0; xt−i) #T "_t−1 X i=0 (1 − λ)2iIt−i(α0) #−1 t−1 X i=0 (1 − λ)iS(α0; xt−i), (33) and Σt= λ2 t−1 X i=0 (1 − λ)2iIt−i(α0) (34)

for t = 1, 2, ..., where 00 _{≡ 1. In particular, if Λ = I}

k+1, then wt = S(α0; xt), Tt2 =

ST_(α

0; xt)It−1(α0)S(α0; xt) (≡ Tt12), and Σt= It(α0) for t = 1, 2, ..., where Tt12 is the score

test statistic at time t for testing the null hypothesis H0: pt ∼ Dirichlet(α0) versus the

alternative H1: pt ∼ Dirichlet(α) for some α 6= α0.

(vii) If n1 = n2 = ..., then Σt = P Λ "_t−1 X i=0 (Ik+1− Λ)iPT I1(α0)P (Ik+1− Λ)i # ΛPT ≡ P Λ "_t−1 X i=0 (Ik+1− Λ)iI1∗(α0)(Ik+1− Λ)i # ΛPT = P λjλj01 − (1 − λj)t(1 − λj0)t I ∗ 1jj0(α0) λj + λj0 − λjλj0 ! j,j0=0,1,...,k PT → P λjλj 0I∗ 1jj0(α0) λj + λj0 − λjλj0 ! j,j0=0,1,...,k PT (35)

as t → ∞, where PTI1(α0)P ≡ I1∗(α0) ≡ (I_1jj∗ 0(α₀))_j,j0_=0,1,...,k.

(viii) If n1 = n2 = ... and Λ = λ Ik+1 for some λ ∈ (0, 1], then

Σt=

λ [1 − (1 − λ)2t_]

2 − λ I1(α0) → λ

2 − λI1(α0) (36) as t → ∞. In particular, if n1 = n2 = ... and Λ = Ik+1, then Σt = I1(α0) for t = 1, 2, ...,

w1, w2, ... are i.i.d., and T12, T22, ... are i.i.d.

Then the stopping time of the MEWMA monitoring scheme is the first time t such that

T_t2 > h, (37) where h is chosen to achieve a specified in-control average run length, e.g., 1/[2Φ(−3)] ≈ 370.4 with Φ(·) denoting the cumulative distribution function (c.d.f.) of the standard normal distribution.

4

A Simulation Study

In the former information, it accumulate the data up to time t when t = 1, 2, ..., l. Through the former information, this paper discusses the different l for each λ. The l is the past in-control data in phase I.

In order to study the performance of this quality control scheme, it compute the average run length. To evaluate the in-control average run length (≡ ARL0), it

con-siders the special case where k = 2; p1, p2, ... are sampled from Dirichlet(α0) with α0 =

(85, 10, 5)T; n1 = n2 = ... = 100; P = I3; and Λ = λ I3 for λ ∈ {0.01, 0.05, 0.10,

0.15, 0.20, 0.30, 0.40, 0.50, 1} (≡ S).

For λ ∈ S and ARL0 = 1/[2Φ(−3)], The h in equation (33) can be evaluated as

follows:

Step 1: Generate i.i.d. (pT

1, xT1)T, ..., (pT370, xT370)T such that ptis sampled from Dirichlet(α0)

and xt is sampled from multinomial(100; pt) for t = 1, 2, ..., 370.

Step 2: To sort the T_t2s such that T_t2(1), T_t2(2), ..., T_t2(370). Step 3: Choose the maximum T_t2(370) (≡ T_t∗2).

Repeat Steps 1–3 for 10,000 times independently. To sort the T_t∗2s such that T_t∗2(1), T_t∗2(2) , ...,T_t∗2(10,000). The initial value of h1 is T

∗2(5,000)

t . To compute the ARL (1)

0 with h1. If the

ARL(1)₀ is large than 1/[2Φ(−3)], then h2 is given such that h2 < h1, and compute the

ARL(2)₀ . If the ARL(1)₀ is smaller than 1/[2Φ(−3)], then h2 is given such that h2 > h1,

and compute the ARL(2)₀ . Until it finds the h such that ARL0 ≈ 1/[2Φ(−3)] for each λ;

see Table 1.

To evaluate the out-of-control average run length (≡ ARL1), it considers the

spe-cial case where k = 2; p1, ..., pl are sampled from Dirichlet(α0) and pl+1, pl+2, ... are

sampled from Dirichlet(α1) for some l ∈ {0, 1, 2, ..., 500} with α0 = (85, 10, 5)T, α1 ∈

{(80, 12.5, 7.5)T _{(≡ α}(1) 1 ), (75, 15, 10)T (≡ α (2) 1 ), (70, 20, 10)T (≡ α (3) 1 )}; n1 = n2 = ... = 100; P = I3; and Λ = λ I3 for λ ∈ S.

The ARL1 can be evaluated as follows:

Step 1: Generate i.i.d. (pT

1, xT1)T, ..., (pTl , xTl )T such that ptis sampled from Dirichlet(α0)

Step 2: If T_t2 > h for some t ∈ {1, 2, ..., l}, then return to Step 1.

Step 3: Generate i.i.d. (pT_l+1, xT_l+1)T, ..., (p_l+tT ∗, xT_l+t∗)T such that pt is sampled from

Dirichlet(α1) and then xt is sampled from multinomial(100; pt) for t = l + 1, l + 2, ..., l + t∗,

where l + t∗ is the first time t such that T2 t > h.

Repeat Steps 1–3 for 100,000 times independently. Then the ARL1 is approximated

evaluated by the average of 100,000 t∗s for each λ ∈ S.

For λ ∈ S and ARL0 = 1/[2Φ(−3)], the ARL1 is put in Tables 2–31. The proposed

MEWMA monitoring scheme is also compared with the following monitoring scheme: the stopping time of the monitoring scheme is the first time t such that T_t02 > h0, where Tt02 is

5

Comparison and Conclusions

In the paper, an MEWMA control chart is proposed for monitoring a manufacturing process in the Dirichlet-compound multinomial model. For each λ ∈ S ∪ {0} (≡ S0), it is seen that h increases as λ increases; see Table 1.

For 0 ≤ l ≤ 4 and λ ∈ S0, the ARL1 increases as λ increases; see Tables 2–6. Then λ

is chosen as 0. For l ≥ 5 and α1 ∈ {α (1) 1 , α (2) 1 , α (3)

1 }, consider the following three cases:

Case 1: α1 = α(1)1 . When 5 ≤ l ≤ 20, the MEWMA monitoring scheme with λ = 0

has the smallest ARL1 for λ ∈ S

0

; see Tables 7–22. When 21 ≤ l ≤ 36, the MEWMA monitoring scheme with λ = 0.05 has the smallest ARL1 for λ ∈ S

0

; see Tables 23–24. When 38 ≤ l ≤ 500, the MEWMA monitoring scheme with λ = 0.1 has the smallest ARL1 for λ ∈ S

0

; see Tables 26–31. Case 2: α1 = α

(2)

1 . When 5 ≤ l ≤ 11, the MEWMA monitoring scheme with λ = 0

has the smallest ARL1 for λ ∈ S

0

; see Tables 7–13. When 12 ≤ l ≤ 138, the MEWMA monitoring scheme with λ = 0.15 has the smallest ARL1 for λ ∈ S

0

; see Tables 14–27. When 140 ≤ l ≤ 500, the MEWMA monitoring scheme with λ = 0.2 has the smallest ARL1 for λ ∈ S

0

; see Tables 29–31. Case 3: α1 = α

(3)

1 . When 5 ≤ l ≤ 8, the MEWMA monitoring scheme with λ = 0 has

the smallest ARL1 for λ ∈ S

0

; see Tables 7–10. But when the 9 ≤ l ≤ 500, the MEWMA monitoring scheme with λ = 0.3 has the smallest ARL1 for λ ∈ S

0

; see Tables 11–31. For these three cases, it is seen that when α1 is faraway from α0, e.g., α1 = α

(3) 1 , the

ARL1 is smaller. When α1 is close to α0, e.g., α1 = α (1)

1 , the ARL1 is bigger.

Compare λ = 0.05 and 0.01: Case 1: α1 = α

(1)

1 . When 0 ≤ l ≤ 15, the ARL1 for λ = 0.01 is smaller than taht for

λ = 0.05, so λ = 0.01 is better than λ = 0.05; see Tables 2–17. When 16 ≤ l ≤ 500, the ARL1 for λ = 0.05 is smaller than taht for λ = 0.01, so λ = 0.05 is better than λ = 0.01;

see Tables 18–31. Case 2: α1 = α

(2)

1 . When 0 ≤ l ≤ 12, the ARL1 for λ = 0.01 is smaller than taht for

ARL1 for λ = 0.05 is smaller than taht for λ = 0.01, so λ = 0.05 is better than λ = 0.01;

see Tables 14–31.

Case 3: α1 = α(3)1 . When 0 ≤ l ≤ 11, the ARL1 for λ = 0.01 is smaller than taht for

λ = 0.05, so λ = 0.01 is better than λ = 0.05; see Tables 2–13. When 12 ≤ l ≤ 500, the ARL1 for λ = 0.05 is smaller than taht for λ = 0.01, so λ = 0.05 is better than λ = 0.01;

see Tables 14–31.

So, when l is large, the weight λ = 0.05 is better than λ = 0.01.

In general, the λ of EWMA in the interval 0.05 ≤ λ ≤ 0.25 works well in practice, with λ = 0.05, λ = 0.10, λ = 0.20 being popular choices. In this paper, it suggests a multivariate exponentially weighted moving average control chart for some relevant properties of the proposed control chart are also investigated in the Dirichlet-compound multinomial model for a categorical manufacturing process. It is found by simulation that the different α1 and l have different weight such that the ARL1 is smallest when λ ∈ S

0

. According to accumulating different in-control data up to time l in phase I, λ will be different. This can be taken as a reference.

References

[1] Hotelling, H. Multivariate quality control. Techniques of Statistical Analysis, ed. by Eisenhart, C., Hastay, M. W., and Wallis, W. A. (McGraw-Hill, New York, 1947).

[2] Johnson, N. L., Kotz, S., and Balakrishnan, N. Discrete Multivariate Distributions. John Wiley & Sons, New York (1997).

[3] Kotz, S., Balakrishnan, N., and Johnson, N. L. Continuous Multivariate Distributions, Vol. 1, 2nd ed. John Wiley& Sons, New York (2000).

[4] Lowry, C. A., Woodall, W. H., Champ, C. W., and Rigdon, S. E. A multivariate exponential weighted moving average control chart, Technometrics, 34, 46–53 (1992).

[5] McCullagh, P. and Nelder, J. A. Generalized Linear Models, 2nd ed. Chapman and Hall, London (1989).

[6] Montgomery, D. C. Introduction to Statistical Quality Control , 5th ed. John Wiley & Sons, New York (2005).

[7] Page, E. S. Continuous inspection schemes, Biometrika, 41, 100–115 (1954).

[8] Pham, H. (Ed.) Springer Handbook of Engineering Statistics. Springer-Verlag, Lon-don (2006).

[9] Roberts, S. W. Control chart tests based on geometric moving averages, Technometrics, 1, 239–250 (1959).

[10] Shewhart, W. A. Economic Control of Quality of Manufactured Product . Van Nos-trand, New York (1931).

[11] Shiau, J.-J. H., Chen, C.-R., and Feltz, C. J. An empirical Bayes process monitoring technique for polytomous data, Quality and Reliability Engineering International , 21, 13–28 (2005).

[12] Sturm, G. W., Feltz, C. J., and Yousry, M. A. An empirical Bayes strat-egy for analysing manufacturing data in real time, Quality and Reliability Engineering International , 7, 159–167 (1991).

Table 1: h for λ ∈ S0 and α0 = (85, 10, 5)T.

λ 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 h 6.53 8.33 11.96 14.79 17.02 19.08 22.45 25.42 27.73 34.34

* denotes the smallest ARL1.

Table 2: ARL1 for l = 0, λ ∈ S

0 ; α1 ∈ {α(1)1 , α (2) 1 , α (3) 1 }, where α (1) 1 ≡ (80, 12.5, 7.5)T, α(2)₁ ≡ (75, 15, 10)T_{, and α}(3) 1 ≡ (70, 20, 10)T. λ l = 0 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T 5.00∗ 6.19 8.32 10.10 12.00 14.08 18.64 23.47 27.90 45.20 (75, 15, 10)T _1.94∗ _{2.19 2.62 2.96 3.22 3.49 3.96 4.52 5.13 8.80} (70, 20, 10)T 1.30∗ 1.39 1.54 1.66 1.77 1.86 1.99 2.16 2.28 3.32

Table 3: ARL1 for l = 1, λ ∈ S

0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 1 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T 6.00∗ 7.20 9.29 10.94 12.71 14.76 19.13 23.75 28.03 45.2 (75, 15, 10)T _2.37∗ _{2.65 3.14 3.44 3.65 3.86 4.26 4.75 5.29 8.80} (70, 20, 10)T _1.53∗ _{1.64 1.84 1.96 2.05 2.11 2.21 2.31 2.41 3.32}

Table 4: ARL1 for l = 2, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 2 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _6.64∗ _{7.94 9.83 11.31 13.06 14.94 19.23 24.10 27.87 45.20} (75, 15, 10)T 2.64∗ 2.96 3.44 3.69 3.86 4.04 4.38 4.82 5.03 8.80 (70, 20, 10)T _1.69∗ _{1.82 2.03 2.14 2.19 2.24 2.29 2.37 2.44 3.32}

Table 5: ARL1 for l = 3, λ ∈ S

0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 3 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _7.17∗ _{8.36 10.19 11.69 13.24 15.05 19.38 24.26 28.03 45.20} (75, 15, 10)T 2.89∗ 3.22 3.66 3.87 3.99 4.11 4.41 4.84 5.31 8.80 (70, 20, 10)T _1.82∗ _{1.97 2.17 2.25 2.28 2.30 2.33 2.37 2.44 3.32}

Table 6: ARL1 for l = 4, λ ∈ S

0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 4 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _7.56∗ _{8.90 10.52 11.76 13.24 15.27 19.52 24.07 27.93 45.20} (75, 15, 10)T _3.09∗ _{3.42 3.84 4.01 4.08 4.18 4.43 4.85 5.32 8.80} (70, 20, 10)T 1.94∗ 2.10 2.29 2.34 2.34 2.35 2.35 2.38 2.44 3.32

Table 7: ARL1 for l = 5, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 5 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _8.01∗ _{9.24 10.85 11.94 13.58 15.18 19.57 24.12 28.19 45.20} (75, 15, 10)T 3.27∗ 3.63 3.99 4.10 4.13 4.23 4.46 4.87 5.37 8.80 (70, 20, 10)T _2.05∗ _{2.21 2.38 2.41 2.40 2.37 2.35 2.38 2.45 3.32}

Table 8: ARL1 for l = 6, λ ∈ S

0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 6 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _8.40∗ _{9.55 11.14 12.05 13.49 15.21 19.49 24.14 28.09 45.20} (75, 15, 10)T 3.45∗ 3.78 4.12 4.16 4.18 4.22 4.44 4.85 5.34 8.80 (70, 20, 10)T _2.16∗ _{2.31 2.46 2.46 2.42 2.39 2.37 2.38 2.44 3.32}

Table 9: ARL1 for l = 7, λ ∈ S

0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 7 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _8.80∗ _{9.95 11.23 12.07 13.46 15.33 19.37 24.16 28.44 45.20} (75, 15, 10)T _3.58∗ _{3.93 4.23 4.22 4.20 4.24 4.44 4.85 5.36 8.80} (70, 20, 10)T 2.23∗ 2.40 2.53 2.50 2.43 2.40 2.37 2.38 2.44 3.32

Table 10: ARL1 for l = 8, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 8 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _9.10∗ _{10.10 11.31 12.14 13.43 15.30 19.17 24.17 27.59 45.20} (75, 15, 10)T 3.75∗ 4.09 4.33 4.27 4.22 4.27 4.44 4.85 5.37 8.80 (70, 20, 10)T _2.33∗ _{2.49 2.60 2.53 2.46 2.40 2.36 2.38 2.44 3.32}

Table 11: ARL1 for l = 9, λ ∈ S

0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 9 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _9.40∗ _{10.53 11.51 12.39 13.58 15.28 19.64 24.01 27.79 45.20} (75, 15, 10)T 3.91∗ 4.21 4.41 4.29 4.22 4.23 4.45 4.84 5.36 8.80 (70, 20, 10)T _{2.40 2.57 2.65 2.56 2.48 2.40 2.36}∗ _{2.38 2.45 3.32}

Table 12: ARL1 for l = 10, λ ∈ S

0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 10 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _9.68∗ _{10.79 11.62 12.26 13.55 15.35 19.35 23.99 28.01 45.20} (75, 15, 10)T _4.01∗ _{4.35 4.48 4.34 4.23 4.25 4.46 4.86 5.34 8.80} (70, 20, 10)T 2.50 2.64 2.69 2.58 2.47 2.40 2.35∗ 2.37 2.45 3.32

Table 13: ARL1 for l = 11, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 11 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _10.03∗ _{11.03 11.79 12.31 13.57 15.33 19.55 24.17 28.23 45.20} (75, 15, 10)T 4.16∗ 4.45 4.55 4.36 4.27 4.25 4.46 4.85 5.33 8.80 (70, 20, 10)T _{2.55 2.71 2.75 2.59 2.48 2.41 2.36}∗ _{2.38 2.44 3.32}

Table 14: ARL1 for l = 12, λ ∈ S

0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 12 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _10.24∗ _{11.31 11.82 12.35 13.55 15.31 19.60 24.19 28.19 45.20} (75, 15, 10)T 4.27 4.57 4.59 4.38 4.24∗ 4.26 4.44 4.85 5.35 8.80 (70, 20, 10)T _{2.64 2.78 2.77 2.61 2.48 2.41 2.36}∗ _{2.39 2.44 3.32}

Table 15: ARL1 for l = 13, λ ∈ S

0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 13 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _10.53∗ _{11.54 11.93 12.41 13.56 15.29 19.32 24.23 28.40 45.20} (75, 15, 10)T _{4.40 4.68 4.65 4.39 4.26}∗ _{4.27 4.44 4.85 5.36 8.80} (70, 20, 10)T 2.70 2.85 2.80 2.61 2.47 2.42 2.35∗ 2.38 2.44 3.32

Table 16: ARL1 for l = 14, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 14 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _10.70∗ _{11.63 11.97 12.40 13.56 15.31 19.23 24.15 28.32 45.20} (75, 15, 10)T 4.48 4.77 4.71 4.41 4.27∗ 4.28 4.43 4.86 5.34 8.80 (70, 20, 10)T _{2.76 2.92 2.81 2.62 2.47 2.41 2.37}∗ _{2.38 2.44 3.32}

Table 17: ARL1 for l = 15, λ ∈ S

0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 15 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _10.96∗ _{11.81 12.03 12.39 13.57 15.33 19.37 24.17 28.28 45.20} (75, 15, 10)T 4.59 4.86 4.74 4.42 4.27∗ 4.28 4.45 4.86 5.35 8.80 (70, 20, 10)T _{2.82 2.97 2.84 2.62 2.48 2.41 2.35}∗ _{2.38 2.43 3.32}

Table 18: ARL1 for l = 16, λ ∈ S

0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 16 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _11.22∗ _{12.02 12.00 12.38 13.58 15.29 19.24 24.26 28.17 45.20} (75, 15, 10)T _{4.68 4.95 4.77 4.42 4.26}∗ _{4.27 4.44 4.87 5.35 8.80} (70, 20, 10)T 2.88 3.05 2.87 2.63 2.47 2.42 2.33∗ 2.37 2.44 3.32

Table 19: ARL1 for l = 17, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 17 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _11.50∗ _{12.37 12.16 12.38 13.56 15.31 19.30 24.03 28.25 45.20} (75, 15, 10)T 4.81 5.07 4.79 4.43 4.26∗ 4.27 4.44 4.85 5.34 8.80 (70, 20, 10)T _{2.95 3.08 2.90 2.64 2.48 2.42 2.34}∗ _{2.38 2.44 3.32}

Table 20: ARL1 for l = 18, λ ∈ S

0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 18 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _11.77∗ _{12.61 12.18 12.38 13.55 15.34 19.17 23.92 28.33 45.20} (75, 15, 10)T 4.91 5.13 4.82 4.43 4.19∗ 4.26 4.45 4.85 5.35 8.80 (70, 20, 10)T _{3.01 3.11 2.91 2.64 2.49 2.41 2.33}∗ _{2.38 2.45 3.32}

Table 21: ARL1 for l = 19, λ ∈ S

0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 19 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _12.05∗ _{12.64 12.22 12.52 13.56 15.32 19.23 24.29 28.39 45.20} (75, 15, 10)T _{5.01 5.25 4.82 4.44 4.25}∗ _{4.27 4.44 4.86 5.37 8.80} (70, 20, 10)T 3.07 3.15 2.93 2.63 2.48 2.42 2.36∗ 2.39 2.45 3.32

Table 22: ARL1 for l = 20, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 20 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _12.12∗ _{12.84 12.23 12.49 13.58 15.33 19.33 24.03 28.24 45.20} (75, 15, 10)T 5.10 5.26 4.90 4.45 4.25∗ 4.28 4.44 4.85 5.36 8.80 (70, 20, 10)T _{3.09 3.21 2.94 2.63 2.49 2.42 2.37}∗ _{2.39 2.44 3.32}

Table 23: ARL1 for l = 21, λ ∈ S

0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 21 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _{12.42 13.01 12.27}∗ _{12.43 13.57 15.29 19.20 23.94 28.13 45.20} (75, 15, 10)T 5.17 5.38 4.91 4.44 4.26∗ 4.27 4.45 4.87 5.35 8.80 (70, 20, 10)T _{3.20 3.28 2.95 2.64 2.50 2.43 2.36}∗ _{2.37 2.44 3.32}

Table 24: ARL1 for l = 36, λ ∈ S

0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 36 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _{15.63 14.09 12.34}∗ _{12.45 13.62 15.35 19.40 24.17 28.18 45.20} (75, 15, 10)T _{5.98 6.08 4.95 4.44 4.26}∗ _{4.30 4.44 4.86 5.34 8.80} (70, 20, 10)T 3.77 3.69 3.01 2.65 2.50 2.43 2.34∗ 2.38 2.44 3.32

Table 25: ARL1 for l = 37, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 37 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _{15.76 14.21 12.46}∗ _12.46∗ _{13.65 15.37 19.43 24.19 28.20 45.20} (75, 15, 10)T 6.11 6.14 4.99 4.44 4.27∗ 4.29 4.44 4.86 5.36 8.80 (70, 20, 10)T _{3.82 3.73 3.03 2.66 2.50 2.43 2.35}∗ _{2.38 2.44 3.32}

Table 26: ARL1 for l = 38, λ ∈ S

0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 38 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _{16.15 14.37 12.51 12.47}∗ _{13.63 15.39 19.47 24.04 28.17 45.20} (75, 15, 10)T 6.24 6.19 5.00 4.44 4.27∗ 4.31 4.44 4.87 5.33 8.80 (70, 20, 10)T _{3.90 3.76 3.04 2.65 2.50 2.43 2.36}∗ _{2.37 2.44 3.32}

Table 27: ARL1 for l = 138, λ ∈ S

0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 138 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _{26.84 18.16 12.63 12.39}∗ _{13.60 15.30 19.42 23.94 28.19 45.20} (75, 15, 10)T _{11.32 7.78 5.07 4.47 4.28}∗ _{4.31 4.44 4.85 5.35 8.80} (70, 20, 10)T 6.71 4.67 3.07 2.65 2.49 2.41 2.35∗ 2.38 2.45 3.32

Table 28: ARL1 for l = 139, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 139 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _{26.91 18.17 12.63 12.41}∗ _{13.59 15.31 19.43 24.06 28.17 45.20} (75, 15, 10)T 11.36 7.79 5.07 4.46 4.26∗ 4.26∗ 4.45 4.86 5.36 8.80 (70, 20, 10)T _{6.74 4.68 3.06 2.65 2.50 2.42 2.36}∗ _{2.38 2.44 3.32}

Table 29: ARL1 for l = 140, λ ∈ S

0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 140 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _{27.04 18.18 12.64 12.42}∗ _{13.58 15.31 19.47 24.07 28.17 45.20} (75, 15, 10)T 11.38 7.79 5.06 4.45 4.32 4.21∗ 4.45 4.86 5.36 8.80 (70, 20, 10)T _{6.75 4.68 3.05 2.66 2.48 2.42 2.37}∗ _{2.39 2.44 3.32}

Table 30: ARL1 for l = 450, λ ∈ S

0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 450 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _{45.17 19.08 12.75 12.61}∗ _{13.55 15.38 19.45 24.34 28.52 45.20} (75, 15, 10)T _{19.36 8.34 5.11 4.54 4.47 4.32}∗ _{4.46 5.09 5.39 8.80} (70, 20, 10)T 11.71 4.87 3.10 2.68 2.51 2.42 2.33∗ 2.41 2.40 3.32

Table 31: ARL1 for l = 500, λ ∈ S 0 ; α1 ∈ {α (1) 1 , α (2) 1 , α (3) 1 }. λ l = 500 0.00 0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50 1.00 (80, 12.5, 7.5)T _{48.04 19.30 12.80 12.67}∗ _{13.57 15.42 19.44 24.38 28.49 45.20} (75, 15, 10)T 20.21 8.42 5.19 4.53 4.49 4.36∗ 4.47 5.13 5.42 8.80 (70, 20, 10)T _{12.09 4.89 3.11 2.66 2.52 2.40 2.31}∗ _{2.43 2.36 3.32}