GOODNESS OF FIT

In this section, the goodness of …t of the proposed model for a set of available historical in-control response vectors, fy¹; : : : ; y_Tg, generated in a manufacturing process is discussed. Recall that ( ^T₁; : : : ; ^T_T)^T, y (y^T₁; : : : ; y_T^T)^T, Y Yⁿ1 YⁿT, and F is the in-control prior c.d.f.

Consider the null hypothesis H0: 1; : : : ; _T ^i:i:d:F 2 fF : 2 g versus the alternative H1: 1; : : : ; _T ^i:i:d: F =2 fF : 2 g. Let F( ) denote the set of all prior c.d.f.’s on and let `(F ; y) denote the log-likelihood function of F given y.

Then

`(F ; y) log

" _T Y

t=1

f (y_t; F )

#

= XT

t=1

log[f (y_t;F )]

XT t=1

`(F ; y_t);

where

f (y_t; F ) = Z

f (y_tj ^t) dF ( _t):

Let WT(y) denote the corresponding likelihood ratio (LR) statistic given y.

Then

where ^F is the non-parametric MLE of F given y under H1 and ^ is the parametric MLE of under H0. Since it takes too much time to calculate the critical point for performing the LR test, an alternative goodness-of-…t test is proposed in the paper as follows:

Note that the empirical prior c.d.f. ~F with p.m.f. T ¹ PT

t=1 1_{f tg} converges to F in distribution as T ! 1 and that, for t 2 f1; : : : ; T g, the MLE y^t=n_t of t

given ytconverges to tas nt! 1. Since ¹; : : : ; _T are unobserved, the empirical prior c.d.f. ~F is unavailable. Thus, we utilize the estimated empirical prior c.d.f. F with p.m.f. T ¹ PT

t=1 1_fyt=ntg to estimate F . When all of n1; : : : ; n_T, and T tend to 1, F converges to F in distribution.

In the paper, consider the goodness-of-…t statistic

W_T(y) 2 h

`(F ; y)jF =F ` ^; y i

2 h

`(F ; y) ` ^; y i

: (47)

One way to calculate the critical point for performing the goodness-of-…t test is as follows: First simulate an i.i.d. sample fy⁽¹⁾; : : : ; y^(R)g, e.g., R = 50 000, from the estimated in-control marginal c.d.f. F_y; ⁰j ⁰= ^ ( F_y;^). Let (y(1); : : : ; y_(R)) be a permutation of (y⁽¹⁾; : : : ; y^(R)) such that W_T(y₍₁₎) : : : W_T(y_(R)). Let

be a known constant with 0 < < 1, e.g., 0:05. An approximate size 1 goodness-of-…t test is to reject H0 if and only if W_T(y) > W_T(y_{([R (1 )])}), where [R (1 )] is the largest integer less than or equal to R (1 ).

The corresponding values of W_T(y_{([R (1 )])})’s for Cases 1-4 in Section 4 are shown in Table 1, where k = 1, T = 300, n1 = : : : = n_T = 300, R = 50 000, and = 0:05. And the empirical c.d.f.’s of W_T(y)’s for Cases 1-4 in Section 4 are shown in Figures 1, where k = 1, T = 300, n1 = : : : = n_T = 300, R = 50 000.

Table 1: The values of W_T(y_{([R (1 )])})’s for Cases 1-4, where k = 1, T = 300, n₁ = : : : = n_T = n_t= 300, R = 50 000, and = 0:05.

Case 1 Case 2 Case 3 Case 4

W_T(y_{([R (1 )])}) 18.1 4.90 12.7 1.78

Figure 1: The empirical c.d.f.’s of W_T’s for Case 1-4, where k = 1, T = 300, n₁ = : : : = n_T = n_t= 300, and R = 50 000.

-5.0 0.0 5.0 10.0 15.0 20.0 25.0

W^*_T

0.0 0.2 0.4 0.6 0.8 1.0

Empirical c.d.f.

Case 1 Case 2 Case 3 Case 4

4.90 0.95

18.1

1.78 12.7

6

SIMPLIFICATION

In this section, the simpli…cation of the two-components mixture prior para-metric family to either the …rst or the second component prior parapara-metric family is discussed if the null hypothesis of the previous goodness-of-…t test is not rejected.

Let u 2 f1; 2g be …xed. Consider the null hypothesis H^u0: 1; : : : ; _T ^i:i:d:F 2 fF^{u; u}: u 2 ^ug versus the alternative H^u1: 1; : : : ; _T ^i:i:d: F 2 fF : 2 g.

Let Wu;T(y)denote the LR statistic given y, where

W_u;T(y) 2

"

` ^; y sup

u2 u

XT t=1

`_u( _u; y_t)

#

2 ` ^; y sup

u2 ^u

`_u( _u; y) 2 h

` ^; y `_u ^

u; y i

(48)

with ^udenoting the MLE of ugiven y under the uth component prior parametric family.

One way to calculate the critical point for performing the LR test is as follows:

First simulate fy^(u;1); : : : ; y^(u;R)g, e.g., R = 50 000, from the estimated in-control marginal c.d.f. F_y;u; ⁰_uj ⁰_u= ^u ( F_{y;u; ^}

u). Let (y^(u)₍₁₎; : : : ; y^(u)_(R)) be a permutation of (y^(u;1); : : :, y^(u;R)) such that Wu;T(y^(u)₍₁₎) : : : Wu;T(y^(u)_(R)). Let be a known constant with 0 < < 1, e.g., 0:05. An approximate size 1 LR test is to reject Hu0 if and only if Wu;T(y) > W_u;T(y^(u)_{([R (1 )])}), where [R (1 )] is the largest integer less than or equal to R (1 ).

When both H10 and H20 are rejected, the proposed two-components mixture prior parametric family for the in-control prior distribution is selected. The cor-responding monitoring technique is developed in the following section.

When H10 is not rejected but H20 is rejected, the …rst component prior

para-metric family for the in-control prior distribution is selected. The corresponding monitoring technique is developed in Chen et al. (2004).

When H10is rejected but H20is not rejected, the second component prior para-metric family for the in-control prior distribution is selected. The corresponding monitoring technique is developed in Chen et al. (2005).

When neither H10 nor H20is rejected, the model selection technique developed in Chen and Liu (2005) can be utilized. The corresponding monitoring technique is developed in either Chen et al. (2004) or Chen et al. (2005).

The corresponding values of Wu;T(y^(u)_{([R (1 )])})’s for Cases 1-4 in Section 4 are shown in Table 2, where u 2 f1; 2g, k = 1, T = 300, n¹ = : : : = nT = 300, R = 50 000, and = 0:05. And the empirical c.d.f.’s of W1;T(y)’s and W2;T(y)’s for Cases 1-4 in Section 4 are shown in Figures 2 and 3, where k = 1, T = 300, n₁ = : : : = n_T = 300, R = 50 000.

Table 2: The values of Wu;T(y^(u)_{([R (1 )])})’s for Cases 1-4, where u 2 f1; 2g, k = 1, T = 300, n1 = : : : = n_T = n_t = 300, R = 50 000, and = 0:05.

Case 1 Case 2 Case 3 Case 4 W_1;T(y⁽¹⁾_{([R (1 )])}) 2.146 1.762 0.566 1.284 W_2;T(y⁽²⁾_{([R (1 )])}) 1.035 0.653 1.789 0.335

Figure 2: The empirical c.d.f.’s of W1;T’s for Case 1-4, where k = 1, T = 300,

7

A PROCESS MONITORING SCHEME

Let Pin denote the false-alarm rate, i.e., the probability that an out-of-control signal occurs when the manufacturing process is in control. Conventionally, Pin is taken to be 2 ( 3) ( 0:002 699 8), where is the c.d.f. of the standard normal distribution. In this section, utilizing the LR method, a Bayesian (or an empirical Bayes) monitoring scheme for the manufacturing process is proposed when F = F ⁰ 2 fF : 2 g for some known (or unknown) ⁰ 2 . The main reason for using the LR test is that it often has a higher power than other tests when the alternative hypothesis is true, which corresponds to a better detecting power in monitoring the process when the process is out of control.

In order to monitor the manufacturing process at time t (> T ), suppose that the response vector yt is observed. Then we are interested in testing whether or not the manufacturing process is in control at time t. Recall that F t is the prior c.d.f. of t and that F( ) is the set of all c.d.f.’s on .

7.1

A BAYESIAN MONITORING SCHEME

In this subsection, consider the case where F = F ⁰ 2 fF : 2 g for some known ⁰ 2 . To monitor the manufacturing process at time t, the null hypothesis H₀: F t = F ⁰ versus the alternative H1: F t 6= F ⁰, i.e., F t 2 F( )nfF ⁰g, is tested.

List all the elements of the sample space Y^nt of yt by fy⁽¹⁾t ; : : : ; y^(jY_t ^ntj)g, where jYⁿtj (= (n^t+ k)!=(n_t!k!)) is the number of elements in Yⁿt. Regard F _t as the unknown parameter of interest in F( ). Then the unknown parameter of inter-est is non-parametric. Let `(F _t; y_t) ( log[f (y_t; F _t)]) denote the log-likelihood

function of F t given yt. Note that

`(F _t; y_t) = log Z

f (y_tj ^t) dF _t( _t) log Z

sup

t2

f (y_tj ^t) dF _t( _t)

= log Z

f (y_tj ^t)j t=yt=nt dF _t( _t) = logh

f (y_tj ^t)j t=yt=nt

i

;

where the binomial/multinomial likelihood f (ytj ^t)for tgiven ytattains its max-imum at t = y_t=n_t: Thus, the MLE ^F _t of F t given yt has p.m.f. 1_fyt=ntg and

sup

F _t2F( )

`(F <sub>t</sub>; y<sub>t</sub>) = `(F _t; y_t)jF _t= ^F _t ` F^ _t; y_t = log f (y_tj ^t)j t=yt=nt :

Let W_t; ⁰(y_t) denote the corresponding LR statistic, where

W_t; ⁰(y_t) = 2 log f (y_tj ^t)j t=yt=nt ` ⁰; y_t (49)

with P (f0 < Wt; ⁰(y_t) <1g; Fyt; ⁰) = 1.

The size PinLR test and a control chart of monitoring the LR statistic W_t; ⁰(y_t) can be constructed as follows: Let (yt;(1); : : : ; y_t;(jYntj))be a permutation of (y_t⁽¹⁾; : : :, y^(jY_t ^ntj))such that W_t; ⁰(y_t;(1)) : : : W_t; ⁰(y_t;(jYntj)). Note that W_t; ⁰(yt)is a dis-crete random variable. If a deterministic upper control limit is used, a pre-speci…ed false-alarm rate Pin (2 (0; 1)), e.g., 2 ( 3), is nearly impossible to attain. How-ever, there is no problem to attain any pre-speci…ed false-alarm rate based on the concept of a randomized-upper-control-limit approach proposed in Shiau et al. (2005). To …nd the randomized upper control limit ( RU CL ⁰), we start ac-cumulating the right tail probability from W_t; ⁰(y_t;(jYntj))until we reach the …rst r

such that P (fWt; ⁰(y_t) W_t; ⁰(y_t;(r))g; Fyt; ⁰) P_in. Denote this r by m ⁰, i.e.,

m ⁰ = max r: P W_t; ⁰(y_t) W_t; ⁰ y_t;(r) ; F_yt; ⁰ P_in : (50)

If P (fWt; ⁰(y_t) W_t; ⁰(y_{t;(m 0)})g; Fyt; ⁰) = P_in, which is nearly impossible, then there is no need for randomization and W_t; ⁰(y_{t;(m 0)})is the upper control limit ( U CL ⁰). If P (fWt; ⁰(y_t) W_t; ⁰(y_{t;(m 0)})g; Fyt; ⁰) > P_in, then W_t; ⁰(y_{t;(m 0)}) = RU CL ⁰. Note that there may be more than one y_t;(r) such that W_t; ⁰(y_t;(r)) = RU CL ⁰. Let m ⁰_;L, m ⁰_;U 2 f1; : : : ; jYⁿtjg such that

W_t; ⁰ y_{t;(m 0;L 1)} < W_t; ⁰ y_{t;(m 0;L)} = RU CL ⁰ = W_t; ⁰ y_{t;(m 0;U)}

< W_t; ⁰ y_{t;(m 0;U+1)} ;

where W_t; ⁰(y_t;(0)) 0 and W_t; ⁰(y_t;(jYntj+1)) 1. Then the randomization is done by signaling an out-of-control alarm with probability

P_in; ⁰_{;RU CL} = P_in P (fWt; ⁰(y_t) > RU CL ⁰g; Fyt; ⁰) P (fWt; ⁰(y_t) = RU CL ⁰g; Fyt; ⁰)

=

P_in PjY_ntj

r=m _0;U+1 P (fWt; ⁰(y_t) = W_t; ⁰(y_t;(r))g; Fyt; ⁰) P^m 0;U

r=m _0;L P (fWt; ⁰(y_t) = W_t; ⁰(y_t;(r))g; Fyt; ⁰) :(51) This leads to

Pin = P W_t; ⁰(yt) > RU CL ⁰ ; F_yt; ⁰

+P_in; ⁰_{;RU CL} P W_t; ⁰(y_t) = RU CL ⁰ ; F_yt; ⁰

and 0 < P_in; ⁰_{;RU CL} 1. When P_in; ⁰_{;RU CL} = 1, there is no need for

randomiza-tion.

The monitoring scheme is as follows: If W_t; ⁰(y_t) > RU CL ⁰, then the null hypothesis H0: F t = F ⁰ is rejected and the manufacturing process is declared to be out of control at time t; if W_t; ⁰(y_t) < RU CL ⁰, then the null hypothesis H₀: F _t = F ⁰ is not rejected and the manufacturing process is declared to be in control at time t; if W_t; ⁰(y_t) = RU CL ⁰, then, with probability P_in; ⁰_{;RU CL}, the null hypothesis H0: F _t = F ⁰ is rejected and the manufacturing process is declared to be out of control at time t.

The corresponding values of RU CL ⁰’s and P_in; ⁰_{;RU CL}’s for Cases 1-4 in Sec-tion 4 are shown in Table 3, where k = 1, T = 300, n1 = : : : = nT = nt = 300, and Pin = 2 ( 3) ( 0:002 699 8).

Table 3: The values of RU CL ⁰’s and P_in; ⁰_{;RU CL}’s for Cases 1-4 in Section 4, where k = 1, T = 300, n1 = : : : = n_T = n_t = 300, and Pin = 2 ( 3) ( 0:002 699 8).

Case 1 Case 2 Case 3 Case 4

RU CL ⁰ 10.5 10.4 10.5 10.6

P_in; ⁰_{;RU CL} 0.547 0.537 0.765 0.120

7.2

AN EMPIRICAL BAYES MONITORING SCHEME

In this subsection, consider the case where F = F ⁰ 2 fF : 2 g for some unknown ⁰ 2 . To monitor the manufacturing process at time t (> T ), the null hypothesis H0: F t = F ⁰ versus the alternative H1: F t 6= F ⁰ is tested.

The LR statistic W_t; ⁰(y_t) proposed in Section 7.1 for known ⁰ can be esti-mated by W_t; ⁰(y_t)j ⁰= ^ ( W_t;^(y_t)), where ^ is the MLE of given y and

W_t;^(y_t) = 2 n

log f (y_tj ^t)j t=yt=nt ` ^; y_t o

(52)

with P (f0 < Wt;^(y_t) < 1g; Fyt; ⁰) = 1. Note that ^ = ⁰ + O_p(1=p T ) as T ! 1, which implies that Wt;^(y_t) = W_t; ⁰(y_t) + O_p(1=p

T )as T ! 1.

An empirical Bayes monitoring scheme can be constructed by replacing the unknown ⁰ by ^ in the Bayesian monitoring scheme described in Section 7.1, where RU CL ⁰ and P_in; ⁰_{;RU CL} are estimated by RU CL ⁰j ⁰= ^ ( RU CL_^) and P_in; ⁰_{;RU CL}j ⁰= ^ ( Pin; ^ ;RU CL), respectively.

To see how the additional estimation error resulting from the empirical Bayes approach a¤ects the performance of the monitoring scheme, the Kullback-Leibler divergence d(F ⁰; F^) between F ⁰ and F^ can be used as a measure of how close F^ is to F ⁰, where F^ F j = ^ and

d (F ⁰; F^)

Z

log ( _t; )j = ⁰

( _t; )j = ^

( _t; )j = ⁰d _t Z

log

"

( _t; ⁰) ( _t; ^)

#

( _t; ⁰) d _t: (53)

When there is no closed-form formula for d(F ⁰; F^), it can be numerically evaluated as follows: First simulate an i.i.d. sample f ⁽¹⁾1 ; : : : ; ^(R₁ ¹⁾g of size R¹, e.g., R1 = 50 000, from the in-control prior c.d.f. F ⁰ and then numerically

evalu-ate d(F ⁰; F^)by largest integer less than or equal to R .

The corresponding values of RU CL_{^ (y}

([R ]))’s and P_{in; ^ (y}

([R ]));RU CL 0.257 0.0513 0.552 0.00126 0.261

Case 2 = 0:1 = 0:3 = 0:5 = 0:7 = 0:9

RU CL_{^ (y}

([R ])) 10.6 10.6 10.4 10.6 10.4

P_{in; ^(y}

([R ]));RU CL 0.954 0.807 0.216 0.566 0.587

Case 3 = 0:1 = 0:3 = 0:5 = 0:7 = 0:9 RU CL_^(y

([R ])) 10.5 10.3 10.3 10.5 10.6

P_{in; ^ (y}

([R ]));RU CL 0.688 0.0511 0.433 0.840 0.602

Case 4 = 0:1 = 0:3 = 0:5 = 0:7 = 0:9

RU CL_{^ (y}

([R ])) 10.5 10.4 10.7 10.5 10.3

P_{in; ^ (y}

([R ]));RU CL 0.346 0.506 0.353 0.449 0.553

From Tables 3 and 4, it is easily seen that all of RU CL_^(y

([R ]))’s are close to RU CL ⁰, but P_{in; ^(y}

([R ]));RU CL’s are not necessarily close to P_in; ⁰_{;RU CL} for Cases 1-4.

8

AVERAGE RUN LENGTH BEHAVIOR

In this section, the performance of the proposed process monitoring scheme is studied in terms of the average run length. Let ARL0 denote the average run length for an out-of-control signal to occur when the manufacturing process is in control. Recall that Pin is the false-alarm rate, i.e., the probability that an out-of-control signal occurs when the manufacturing process is in out-of-control. Then ARL0 = 1=P_in. When Pin = 2 ( 3) ( 0:002 699 8), ARL0 = 1=[2 ( 3)] ( 370:40). Let ARL1 denote the average run length for an out-of-control signal to occur when the manufacturing process is out of control. Let Pout denote the correct-alarm rate, i.e., the probability that an out-of-control signal occurs when the manufacturing process is out of control. Similarly, ARL1 = 1=P_out.

8.1

A BAYESIAN APPROACH

In this subsection, consider the case where F = F ⁰ 2 fF : 2 g for some known ⁰ 2 . To monitor the manufacturing process at time t, the monitoring scheme proposed in Section 7.1 is used for the null hypothesis H0: F _t = F ⁰ versus the alternative H1: F t 6= F ⁰.

Set ARL0 ARL_0; ⁰, Pin P_in; ⁰, ARL1 ARL_1; ⁰_;F

t, and Pout P_out; ⁰_;F

t. When P_in; ⁰ is pre-speci…ed to be 2 ( 3) ( 0:002 699 8),

ARL_0; ⁰ = 1

P_in; ⁰ = 1

2 ( 3) 370:40: (55)

When F t 6= F ⁰,

P_out; ⁰_;F

t = P W_t; ⁰(y_t) > RU CL ⁰ ; F_yt

+P_in; ⁰_{;RU CL} P W_t; ⁰(yt) = RU CL ⁰ ; Fyt ; (56)

where all of W_t; ⁰(y_t), RU CL ⁰, and P_in; ⁰_{;RU CL} are de…ned in Section 7.1 and F_yt is de…ned in Section 3.

The corresponding values of P_out; ⁰_{;F i}’s, and ARL_1; ⁰_{;F i}’s for Cases 1-4 in Section 4 are shown in Tables 5 and 6, where k = 1, T = 300, n1 = : : : = n_T = nt = 300, and i 2 f1; 2; 3g.

Case 1: ¹ = (log(1=4); log(80); log(20); 2:210; log[1=(0:210)²])^T, ² = (log(1=

9); log(90); log(10); 1:552; log[1=(0:220)²])^T, and ³ = (log(4=21); log(80); log(20);

0:503; log[1=(0:216)²])^T.

Case 2: ¹ = (log(9=11); log(85); log(15); 0:510; log[1=(0:210)²])^T, ² = (log(11=9); log(72); log(18); 2:030; log[1=(0:210)²])^T, and ³ = (log(14=11); log(80);

log(20); 0:203; log[1=(0:202)²])^T.

Case 3: ¹ = (log(2=3); log(65); log(35); 0:110; log[1=(0:210)²])^T, ² = (log(1);

log(70); log(30); 2:005; log[1=(0:253)²])^T, and ³ = (log(3=2); log(60); log(40); 0:2 03; log[1=(0:202)²])^T.

Case 4: ¹ = (log(4); log(70); log(20); 1:510; log[1=(0:210)²])^T, ² = (log(3);

log(88); log(22); 1:203; log[1=(0:220)²])^T, and ³ = (log(83=17); log(80); log(20);

1:203; log[1=(0:041)²])^T.

Table 5: The values of P_out; ⁰_{;F i}’s for Cases 1-4, where k = 1, T = 300, n1 = : : : = n_T = n_t = 300, and i 2 f1; 2; 3g.

Case 1 Case 2 Case 3 Case 4 P_out; ⁰_;F

1 0.0568 0.024 0.0487 0.0434 P_out; ⁰_{;F 2} 0.0153 0.0652 0.0929 0.0492 P_out; ⁰_;F

3 0.0169 0.0140 0.0135 0.0590

Table 6: The values of ARL_1; ⁰_{;F i}’s for Cases 1-4, where k = 1, T = 300, n1 = : : : = n_T = n_t= 300, and i 2 f1; 2; 3g.

Case 1 Case 2 Case 3 Case 4 ARL_1; ⁰_{;F 1} 17.6 41.2 20.5 23.1 ARL_1; ⁰_;F

2 65.3 15.3 10.8 20.3

ARL_1; ⁰_{;F 3} 59.1 71.6 74.2 16.9

8.2

AN EMPIRICAL BAYES APPROACH

In this subsection, consider the case where F = F ⁰ 2 fF : 2 g for some unknown ⁰ 2 . To monitor the manufacturing process at time t, the monitoring scheme proposed in Section 7.2 is used for the null hypothesis H0: F _t = F ⁰ versus the alternative H1: F t 6= F ⁰.

Set ARL0 ARL_0; 0;^, Pin P_in; 0;^, ARL1 ARL_1; 0;^ ;F _t, and Pout

P_out; 0;^ ;F _t. When ^ = ⁰, which is nearly impossible, we have P_in; ⁰_;^ = P_in; ⁰ and P_out; ⁰_{;^ ;F}

t

= P_out; ⁰_;F

t, where both P_in; ⁰ and P_out; ⁰_;F

t are de…ned in Sec-tion 8.1. When ^ 6= ⁰, we have

P_in; 0;^ = P n

W_t;^(y_t) > RU CL^

o

; F_yt; ⁰ +P_{in;^;RU CL} P n

W_t;^(y_t) = RU CL_^o

; F_yt; ⁰ (57)

and

P_out; 0;^ ;F _t = P n

W_t;^(y_t) > RU CL_^o

; F_yt +Pin; ^ ;RU CL P n

W_t;^(y_t) = RU CL^

o

; F_yt ; (58)

where all of W_t;^(y_t), RU CL^, and Pin; ^;RU CL are de…ned in Section 7.2 and both F_yt; ⁰ and Fyt are de…ned in Section 3.

To see how the additional estimation error resulting from the empirical Bayes approach a¤ects the performance of the average run length, the Kullback-Leibler divergence d(F ⁰; F^)between F ⁰ and F^ de…ned in Section 7.2 can be used as a measure of how close F^ is to F ⁰. See Section 7.2 for details.

The corresponding values of P ’s and ARL ’s for Cases

1-4 in Section 4 are shown in Table 7, where k = 1, T = 300, n1 = : : : = n_T = n_t = 300, R = R1 = 50 000, and 2 f0:1; 0:3; 0:5; 0:7; 0:9g.

Table 7: The values of P_in; ⁰_{;^ (y}

([R2 ]))’s and ARL_0; ⁰_;^(y

([R ]))’s for Cases 1-4, where k = 1, T = 300, n1 = : : : = n_T = n_t = 300, R = R1 = 50 000, and 2 f0:1; 0:3; 0:5; 0:7; 0:9g.

Case 1 = 0:1 = 0:3 = 0:5 = 0:7 = 0:9

P_in; 0;^ (y([R ])) 0.00169 0.00334 0.00123 0.00374 0.00132

ARL_0; 0;^ (y_{([R ])}) 590.7 299.8 812.9 267.5 759.5

Case 2 = 0:1 = 0:3 = 0:5 = 0:7 = 0:9

P_in; 0;^ (y([R ])) 0.00281 0.00320 0.00204 0.00332 0.00372

ARL_0; 0;^ (y_{([R ])}) 356 313 490 301 269

Case 3 = 0:1 = 0:3 = 0:5 = 0:7 = 0:9

P_in; 0;^ (y_{([R ])}) 0.00324 0.00202 0.00373 0.00247 0.00146

ARL_0; 0;^ (y([R ])) 309 495 268 406 685

Case 4 = 0:1 = 0:3 = 0:5 = 0:7 = 0:9

P_in; 0;^ (y_{([R ])}) 0.00282 0.00445 0.00162 0.00279 0.00569

ARL_0; 0;^ (y_{([R ])}) 355.1 225.0 615.6 358.9 175.7

The corresponding values of P_out; ⁰_;^(y

([R ]));F i’s and ARL_1; ⁰_{;^ (y}

([R ]));F i’s for Cases 1-4 in Section 4 are shown in Tables 8-10, where k = 1, T = 300, n1 = : : : = n_T = n_t = 300, R = R1 = 50 000, 2 f0:1; 0:3; 0:5; 0:7; 0:9g, and i 2 f1; 2; 3g.

Table 8: The values of P_out; ⁰_{;^ (y );F} ’s and ARL_1; ⁰_{;^ (y );F} ’s for Cases

1-4, where k = 1, T = 300, n1 = : : : = n_T = n_t = 300, R = R1 = 50 000, and 2 f0:1; 0:3; 0:5; 0:7; 0:9g.

Case 1 = 0:1 = 0:3 = 0:5 = 0:7 = 0:9

P_out; 0;^ (y_{([R ])});F 1 0.0409 0.0763 0.0192 0.0999 0.0318

ARL_1; 0;^(y_{([R ])});F 1 24.4 13.1 52.0 10.0 31.4

Case 2 = 0:1 = 0:3 = 0:5 = 0:7 = 0:9

P_out; 0;^ (y_{([R ])});F 1 0.0296 0.0288 0.0130 0.0199 0.0401

ARL_1; 0;^(y([R ]));F 1 33.7 34.8 77.0 50.2 25.0

Case 3 = 0:1 = 0:3 = 0:5 = 0:7 = 0:9

P_out; 0;^ (y([R ]));F 1 0.0647 0.0327 0.0365 0.0606 0.0451

ARL_1; 0;^(y_{([R ])});F 1 15.5 30.5 27.4 16.5 22.2

Case 4 = 0:1 = 0:3 = 0:5 = 0:7 = 0:9

P_out; 0;^ (y_{([R ])});F 1 0.0463 0.0520 0.0190 0.0520 0.0268

ARL_1; 0;^ (y([R ]));F 1 21.6 19.2 52.6 19.2 37.3

Table 9: The values of P_out; ⁰_{;^ (y}

([R ]));F 2’s and ARL_1; ⁰_{;^ (y}

([R ]));F 2’s for Cases 1-4, where k = 1, T = 300, n1 = : : : = n_T = n_t = 300, R = R1 = 50 000, and

2 f0:1; 0:3; 0:5; 0:7; 0:9g.

Case 1 = 0:1 = 0:3 = 0:5 = 0:7 = 0:9

P_out; 0;^(y_{([R ])});F 2 0.0126 0.0184 0.00802 0.0218 0.0108

ARL_1; 0;^(y_{([R ])});F 2 79.5 54.4 125 46.0 92.8

Case 2 = 0:1 = 0:3 = 0:5 = 0:7 = 0:9 P_out; 0;^ (y([R ]));F 2 0.0808 0.0778 0.0323 0.0515 0.110

ARL_1; 0;^(y_{([R ])});F 2 12.4 12.9 31.0 19.4 9.11

Case 3 = 0:1 = 0:3 = 0:5 = 0:7 = 0:9

P_out; 0;^ (y_{([R ])});F 2 0.0776 0.0938 0.136 0.0497 0.0371

ARL_1; 0;^(y([R ]));F 2 12.9 10.7 7.36 20.1 26.9

Case 4 = 0:1 = 0:3 = 0:5 = 0:7 = 0:9

P_out; 0;^ (y([R ]));F 2 0.0525 0.0586 0.0223 0.0586 0.0310

ARL_1; 0;^(y_{([R ])});F 2 19.1 17.1 44.8 17.1 32.2

Table 10: The values of P_out; ⁰_{;^ (y}

([R ]));F 3’s and ARL_1; ⁰_{;^ (y}

([R ]));F 3’s for Cases 1-4, where k = 1, T = 300, n1 = : : : = n_T = n_t = 300, R = R1 = 50 000, and

2 f0:1; 0:3; 0:5; 0:7; 0:9g.

Case 1 = 0:1 = 0:3 = 0:5 = 0:7 = 0:9

P_out; 0;^ (y_{([R ])});F 3 0.0108 0.0181 0.0113 0.0157 0.00902

ARL_1; 0;^(y_{([R ])});F 3 92.6 55.4 88.9 63.5 110.9

Case 2 = 0:1 = 0:3 = 0:5 = 0:7 = 0:9

P_out; 0;^ (y_{([R ])});F 3 0.0121 0.0152 0.0139 0.0195 0.0120

ARL_1; 0;^(y_{([R ])});F 3 82.8 65.9 72.1 51.3 83.2

Case 3 = 0:1 = 0:3 = 0:5 = 0:7 = 0:9 P_out; 0;^ (y([R ]));F 3 0.0199 0.00786 0.00911 0.0182 0.0121

ARL_1; 0;^ (y_{([R ])});F 3 50.3 127 109 55.1 82.6

Case 4 = 0:1 = 0:3 = 0:5 = 0:7 = 0:9

P_out; 0;^ (y_{([R ])});F 3 0.0627 0.0695 0.0282 0.0695 0.0383

ARL_1; 0;^ (y([R ]));F 3 16.0 14.4 35.5 14.4 26.1

From Tables 8-10, there is no pattern that the values of P_in; ⁰_;^ and P_out; ⁰_{;^ ;F}

i

for Case 1-Case 4 increase as the Kullback-Leibler divergence d(F ⁰; F^)increases, where k = 1, T = 300, n1 = : : : = n_T = n_t = 300, R = R1 = 50 000, and i2 f1; 2; 3g.

9

CONCLUSIONS

In the paper, …rst, a two-components mixture prior parametric family for the in-control prior distribution is proposed in a manufacturing process. Then an em-pirical Bayes approach is proposed when there are available in-control categorical data generated from the manufacturing process. As an illustration, an example of the proposed empirical Bayes model is introduced. For the purpose of model building, the goodness of …t and the simpli…cation of the proposed model are dis-cussed. Utilizing the likelihood ratio method, both Bayesian and empirical Bayes monitoring techniques are proposed as the main purpose of the paper. Finally, the performance of the proposed process monitoring scheme is studied in terms of the average run length to show the robustness of the methodology.

APPENDIX

All of nodes and weights of the Hermite polynomial of 32 degrees are shown in the following table. This table is obtained from the following website:

http://www.efunda.com/math/num_integration/…ndgausshermite.cfm

No. i abscissas xi weights wi

1 7:12581390983 7:31067642754 10 ²³ 2 6:40949814928 9:23173653482 10 ¹⁹ 3 5:81222594946 1:19734401957 10 ¹⁵ 4 5:27555098664 4:21501019491 10 ¹³ 5 4:77716450334 5:93329148347 10 ¹¹ 6 4:30554795347 4:09883215841 10 ⁹ 7 3:85375548542 1:57416779440 10 ⁷ 8 3:41716749282 3:65058512533 10 ⁶ 9 2:99249082501 5:41658405999 10 ⁵ 10 2:57724953773 5:36268365495 10 ⁴ 11 2:16949918361 3:65489032677 10 ³ 12 1:76765410946 1:75534288315 10 ² 13 1:37037641095 6:04581309559 10 ² 14 0:97650046359 1:51269734077 10 ¹ 15 0:58497876544 2:77458142303 10 ¹ 16 0:19484074157 3:75238352593 10 ¹

No. i abscissas xi weights wi

17 0:19484074157 3:75238352593 10 ¹ 18 0:58497876544 2:77458142303 10 ¹ 19 0:97650046359 1:51269734077 10 ¹ 20 1:37037641095 6:04581309559 10 ² 21 1:76765410946 1:75534288315 10 ² 22 2.16949918361 3:65489032677 10 ³ 23 2:57724953773 5:36268365495 10 ⁴ 24 2:99249082501 5:41658405999 10 ⁵ 25 3:41716749282 3:65058512533 10 ⁶ 26 3:85375548542 1:57416779440 10 ⁷ 27 4:30554795347 4:09883215841 10 ⁹ 28 4:77716450334 5:93329148347 10 ¹¹ 29 5:27555098664 4:21501019491 10 ¹³ 30 5:81222594946 1:19734401957 10 ¹⁵ 31 6:40949814928 9:23173653482 10 ¹⁹ 32 7:12581390983 7:31067642754 10 ²³

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