Optimal step-stress test under type I progressive group-censoring with random removals

Journal of Statistical Planning and Inference 138 (2008) 817 – 826 www.elsevier.com/locate/jspi

Optimal step-stress test under type I progressive group-censoring with random removals Shuo-Jye Wua,∗ , Ying-Po Linb , Shyi-Tien Chenc a Department of Statistics, Tamkang University, Tamsui, Taipei 251, Taiwan b Graduate Institute of Management Sciences, Tamkang University, Tamsui, Taipei 251, Taiwan c Department of Safety, Health and Environmental Engineering, National Kaohsiung First University of Science and Technology, Kaohsiung 811,

Taiwan Received 8 September 2005; received in revised form 12 October 2006; accepted 11 February 2007 Available online 12 March 2007

Abstract Some traditional life tests result in no or very few failures by the end of test. In such cases, one approach is to do life testing at higher-than-usual stress conditions in order to obtain failures quickly. This paper discusses a k-level step-stress accelerated life test under type I progressive group-censoring with random removals. An exponential failure time distribution with mean life that is a log-linear function of stress and a cumulative exposure model are considered. We derive the maximum likelihood estimators of the model parameters and establish the asymptotic properties of the estimators. We investigate four selection criteria which enable us to obtain the optimum test plans. One is to minimize the asymptotic variance of the maximum likelihood estimator of the logarithm of the mean lifetime at use-condition, and the other three criteria are to maximize the determinant, trace and the smallest eigenvalue of Fisher’s information matrix. Some numerical studies are discussed to illustrate the proposed criteria. © 2007 Elsevier B.V. All rights reserved. Keywords: A-optimality; D-optimality; E-optimality; Grouped data; Maximum likelihood method; Progressive type I censoring; Variance-optimality

1. Introduction Censoring is very common in life tests. It usually applies when the distribution of exact lifetimes are known for only a portion of the products and the remainder of the lifetimes are known only to exceed certain values under a life test. The most common censoring schemes are type I censoring and type II censoring. These two censoring schemes have been studied rather extensively by a number of authors including Mann et al. (1974), Meeker and Escobar (1998), and Lawless (2003). If an experimenter desires to remove functioning units at points other than the final termination point of the life test, the above two censoring schemes will not be of use to the experimenter. The traditional censoring does not allow for units to be removed from test at the points other than the final termination point. This allowance may be desirable when a compromise between reduced time of experimentation and the observation of at least some extreme lifetimes is sought. This leads us into the area of progressive censoring. Statistical inferences on the parameters of failure time ∗ Corresponding author.

E-mail address: [email protected] (S.-J. Wu). 0378-3758/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2007.02.004

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distributions under progressive censoring have been studied by several authors such as Cohen (1963), Mann (1971), Gibbons and Vance (1983), Wong (1993), Balasooriya and Saw (1998), Balakrishnan and Aggarwala (2000), Wu and Chang (2003), Ng et al. (2004), and Lin et al. (2006). Note that in progressive censoring, the number of removals are all pre-fixed. However, in some practical situations, these numbers may occur at random. Yuen and Tse (1996) indicated that, for example, in some reliability experiments, an experimenter may decide that it is inappropriate or too dangerous to carry on the testing on some of the tested units even though these units have not failed. In these cases, the pattern of removal is random. Accelerated life testing (ALT) is often used for reliability analysis. Test units are run at higher-than-usual stress conditions in order to obtain failures quickly. A model relating life length to stress is fitted to the accelerated failure times and then extrapolated to estimate the failure time distribution under usual conditions. The stress loading in an ALT can be applied various ways. They include constant stress, step stress, and random stress. Nelson (1990, Chapter 1) discussed their advantages and disadvantages. In this study, we only consider the case of step stress. In step-stress scheme, a test unit is subjected to successively higher levels of stress. A test unit starts at a specified low stress for a specified length of time. If it does not fail, stress on it is raised and held a specified time. The stress is thus increased step by step until the test unit fails. Generally, all test units go through the same specified pattern of stress levels and test times. The simplest step-stress ALT uses only two stress levels and we call it simple step-stress ALT. The statistical inferences in this step-stress ALT have been investigated by several authors such as Miller and Nelson (1983), Nelson (1990), Tang et al. (1996), Khamis and Higgins (1998), Xiong (1998), Yeo and Tang (1999), and Gouno et al. (2004). In practice, it is often impossible to continuously observe the testing process, even with censoring. Instead, one can inspect the units only intermittently. That is, we can only record whether a test unit fails in an interval instead of measuring failure time exactly. Hence, data of this type are called grouped data. In the literature, grouped data have been studied by many researchers such as Cheng and Chen (1988), Lui et al. (1993), Chen and Mi (1996), Aggarwala (2001), Qian and Correa (2003), Xiong and Ming (2004), and Yang and Tse (2005). In this paper, we combine progressive censoring, ALT and grouped data to develop a step-stress ALT scheme under type I progressive group-censoring with random removals. An exponential failure time distribution at each level of stress is considered. Meeker and Escobar (1998, p. 79) pointed out that the exponential distribution is a popular distribution for some kinds of electronic components such as capacitors and high-quality integrated circuits. The exponential distribution is useful to describe failure times of components which are subject to wear out. Pal et al. (2006, p. 152) indicated that failure time of electric bulbs, batteries, appliances, and transistors, etc., can be modeled by exponential distribution. Therefore, this distribution is frequently discussed in reliability applications. The rest of this paper is organized as follows. In Section 2, we describe the model and some necessary assumptions. We use the maximum likelihood method to obtain the point estimators of the model parameters in Section 3. The problem of choosing the optimal length of inspection interval will also be addressed using four different optimization criteria in Section 4. The proposed methods are applied to a numerical example in Section 5. Some simulation results are presented in Section 6 and some discussion and conclusions are made in Section 7. 2. Model and assumptions Let us consider the following k-level step-stress ALT scheme with type I progressive group-censoring: n units are simultaneously placed on a life test at stress setting x1 , and run until time 1 , at which point the number of failed units n1 are counted and r1 surviving units are removed from the test; starting from time 1 , the n − n1 − r1 non-removed surviving units are put to a different stress setting x2 and run until time 1 + 2 , at which point the number of failures  n2 are counted and r2 surviving units are removed from the test, and so on. At time ki=1 i , the number of failed units   nk are counted and the remaining surviving rk = n − ki=1 ni − k−1 j =1 rj units are all removed, thereby terminating the test. This scheme may be depicted pictorially in Fig. 1. For any stress setting, the failure time distribution of the test unit has an exponential distribution. At stress setting xi , the mean lifetime i of a test unit is a log-linear function of xi . That is, ln i =  xi ,

i = 1, 2, . . . , k.

(1)

Here  = (0 , 1 , . . . , b ) is a (b + 1) × 1 vector of unknown parameters and stress setting xi = (1, xi1 , . . . , xib ) is a (b + 1) × 1 general covariate vector, i = 1, 2, . . . , k. The reason for considering such a stress setting is that some

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Fig. 1. k-level step-stress accelerated life test under type I progressive group-censoring.

accelerated tests involve more than one accelerating stress or an accelerating stress and other engineering variables such as batch, brand, country of origin, etc. The log-linear function is a common choice for the life–stress relationship because it includes both the power law and the Arrhenius law as special cases. Furthermore, failures occur according to a cumulative exposure model. That is, the remaining life of a unit depends only on the exposure it has seen, and the unit does not remember how the exposure was accumulated (see Miller and Nelson, 1983). From previous assumptions, the cumulative distribution function of the lifetime of a test unit under k-level step-stress test is ⎞ ⎛ i−1 i−1 i    ⎠ ⎝ for j ; i j < t  j , F (t) = F si−1 + t − j =1

j =1

j =1

where F (t; i ) = 1 − e−t/i ,

(2)

s0 = 0, and si−1 = (i /i−1 )(si−2 + i−1 ) is the solution of F (si−1 ; i ) = F (i−1 + si−2 ; i−1 ), i = 2, 3, . . . , k. Hence, the probability density function of a test unit is

⎧ 1 t ⎪ , 0 < t 1 , exp − ⎪ ⎪ ⎪ 1 1 ⎪

⎪ ⎪ 1 1 t − 1 ⎪ ⎪ ⎨ exp − , 1 < t 1 + 2 , −    2 1 2 f (t) = .. ⎪ ⎪ ⎪ . ⎪    ⎪ ⎪ ⎪ t − k−1 j k−1 1   ⎪ j =1 k−1 1 ⎪ , − − ··· − ⎩ exp − j =1 j < t < ∞. k k k−1 1 3. Maximum likelihood estimation Suppose a type I progressively group-censored sample is collected as described in Section 2, beginning with a random sample of n units failure time distribution. Let Ni be the number of units known to have failed  with anexponential i  ,  ] and let Ri be the number of surviving units being withdrawn from the test at time in the interval ( i−1 j =1 j j =1 j i j =1 j , for i = 1, 2, . . . , k. Note that Ni and Ri are random variables with observed values pending on the outcomes of life experiment. Then, we have the fact that Ni |ni−1 , . . . , n1 , ri−1 , . . . , r1 ∼ binomial(mi , Fi ())

(3)

Ri |ni , . . . , n1 , ri−1 , . . . , r1 ∼ uniform(0, mi − ni ),

(4)

and

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where  = (1 , 2 , . . . , k ) is the vector of the lengths of k inspection intervals, mi = n − number of non-removed surviving units at the beginning of the ith stage, and i

Fi () =

i−1 j =1 j ) − F ( j =1 j )  1 − F ( i−1 j =1 j )

F(

= 1 − e−i /i

i−1

j =1 nj

−

i−1

j =1 rj

is the

for i = 1, 2, . . . , k.

The likelihood function is then L=

k 

f (ni |n1 , . . . , ni−1 , r1 , . . . , ri−1 )f (ri |n1 , . . . , ni , r1 , . . . , ri−1 )

i=1

= ∝

 k   m i

i=1 k 

ni

[Fi ()]ni [1 − Fi ()]mi −ni

1 mi − n i + 1

(1 − e−i /i )ni (e−i /i )mi −ni .

i=1

Substituting for i the expression from (1) involving , the log-likelihood function can be written as ln L ∝

k 

[ni ln(1 − e−i e

− xi



) + (mi − ni )(−i e− xi )].

(5)

i=1

Thus, the equation to be solved for the maximum likelihood estimate (MLE) of  is k 



i e− xi [mi − ni (1 − e−i e

− xi

)−1 ]xi = 0,

(6)

i=1

where 0 is a (b + 1) × 1 zero vector. Since (6) cannot be solved analytically, some numerical methods such as Newton’s method must be employed. Choice of initial values is very important in most iteration procedures. Lawless (2003, p. 556) stated that optimization software usually has default initial values, but can accept input values. This can be useful if the software fails to locate a maximum on its own. In problems where the shape of likelihood function is not well understood, it is a good idea to see if different initial values lead to the same  . Under some mild regularity conditions, any of several maximum likelihood large-sample procedure might be used to make inferences about . One possibility is to employ the asymptotic normality to obtain the approximate distribution of . We now derive Fisher’s information matrix. From (5), we have     − x k i e− xi e−i e i ni j2 ln L  − xi −mi + 1− xi xi . = i e (7) − xi − xi −  e −  e j j i i 1−e 1−e i=1

To obtain Fisher’s information, we need the expectation of (7). To get this, let us compute the expectations of Mi , Ni and Ri , for i = 1, 2, . . . , k. Since Mi+1 = Mi − Ni − Ri , i = 1, 2, . . . , k − 1, we can show that E(Mi ) =

nGi−1 () , 2i−1

E(Ni ) =

nGi−1 ()Fi () , 2i−1

E(Ri ) =

nGi () , 2i

and

S.-J. Wu et al. / Journal of Statistical Planning and Inference 138 (2008) 817 – 826

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 where G0 ()=1 and Gi ()= ij =1 [1−Fj ()], for i =1, 2, . . . , k. A detailed proof of these expectations is in Appendix. Hence, the Fisher’s information is I() = n

k 

Di ()xi xi ,

i=1

where Di () =

[fi ()]2 Gi−1 () , 2i−1 [1 − Fi ()]Fi ()

i = 1, 2, . . . , k,

and 

fi () = i e− xi e−i e

− xi

,

i = 1, 2, . . . , k.

For a moderate sample size n, the MLE   is approximately distributed as a multivariate normal with mean vector  and variance–covariance matrix I−1 (). Therefore, the approximate confidence intervals for 0 , 1 , . . . , k or the asymptotic joint confidence region for  can be easily obtained. 4. Optimal length of the inspection interval The main purpose of this paper is to study the choice of i , lengths of the inspection intervals, in a k-level step-stress ALT with type I progressive group-censoring. For simplicity of discussion, we consider the special case where i =  for all i. That is, the lengths of inspection intervals in k stages are all equal. We also assume that the mean lifetime i of a test unit is a log-linear function of accelerating stress only and there is no other engineering variables in this study. That is, xi = (1, xi ) and, hence ln i = 0 + 1 xi ,

i = 1, 2, . . . , k,

(8)

where 0 and 1 (< 0) are unknown parameters and x1 < x2 < · · · < xk . Therefore, the mean lifetime of a test unit at the lower stress is longer than that at the higher stress. We investigate four selection criteria which enable one to choose the optimal value of . Variance-optimality: The mean of the failure time distribution is an important characteristic and indispensable in reliability analysis. In step-stress setting, we need to estimate the mean lifetime at the use-condition with maximum precision. We can use the asymptotic variance of the logarithm of mean lifetime at use-condition as the criterion for selecting the optimal length of the inspection interval. In ALT, one is usually interested in estimating the reliability characteristics at design stress or use stress. Let x0 be the design stress. The mean lifetime 0 at use-condition can be written as 0 = e0 +1 x0 . By invariance property, the MLE of the logarithm of mean lifetime at use-condition is ln ˆ 0 = ˆ 0 + ˆ 1 x0 and, hence the asymptotic variance of the MLE of ln 0 is AVar(ln ˆ 0 ) = [1, x0 ]I−1 (0 , 1 )[1, x0 ] k Di ()(xi − x0 )2 = k ki=1 . n( i=1 j =1 Di ()Dj ()(xi − xj )2 )/2 The criterion function is then defined by k gV () = k ki=1 i=1

Di ()(xi − x0 )2

j =1 Di ()Dj ()(xi

− xj )2

.

The variance optimal  is obtained by minimizing gV ().

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D-optimality: Another optimal criterion is based on the determinant of the Fisher’s information matrix. It is known that the determinant |I(0 , 1 )| is proportional to the reciprocal of the volume of the asymptotic joint confidence region for (0 , 1 ) so that maximizing this determinant is equivalent to minimizing the volume of confidence region. Consequently, a larger value of the determinant of the Fisher’s information matrix would correspond to higher joint precision of the estimators of 0 and 1 . Motivated by this, the optimal length of the inspection interval is chosen so that gD () =

k  k 

Di ()Dj ()(xi − xj )2

i=1 j =1

is maximized. This is called the D-optimality criterion. A-optimality: The third optimal criterion is also known as trace criterion. It maximizes the sum of the diagonal entries of Fisher’s information matrix. This means that the A-optimality criterion does not implement all available information on the parameters. The criterion function is defined by gA () =

k 

Di ()(xi2 + 1).

i=1

The optimal inspection interval  is determined by maximizing gA (). E-optimality: This criterion maximizes the smallest non-zero eigenvalue of Fisher’s information matrix. This also means that not all available information are used. The criterion function is defined by  ⎤ ⎡  2   k k k k     gE () = Di ()(xi2 + 1) −  Di ()(xi2 + 1) − 2 ⎣ Di ()Dj ()(xi − xj )2 ⎦. i=1

i=1 j =1

i=1

The optimal length of inspection interval  is chosen so that gE () is maximized. 5. Numerical example To illustrate the use of the proposed method in this article, a data set is generated by using (3) and (4). Suppose that a step-stress test was run to estimate mean lifetime at a design stress x0 = 1. There are n = 30 test items simultaneously placed on a five-stage step-stress ALT. Each test item was stressed for 10 units of time each at stress levels of 1.1, 1.2, 1.3, 1.4 and 1.5. A progressively type I group-censored sample is generated from exponential distribution under above setting with 0 = 5 and 1 = −1. The step-stress pattern and data are presented in Table 1. Assume that life–stress relationship is a log-linear function defined in (8). Following Section 3, the MLEs of 0 and 1 are ˆ 0 = 5.18 and ˆ 1 = −0.90, respectively. The estimate of mean lifetime at design stress is 71.91 units of time. To find the optimal length of the inspection interval in a future five-stage step-stress ALT with type I progressive group-censoring, we assume that the step-stress pattern in Table 1 is used to construct such a life test. We also assume that the parameters (0 , 1 ) are equal to (5.18, −0.90), our MLEs obtained previously. The optimal lengths of the inspection intervals according to the variance-optimality, D-optimality, A-optimality, and E-optimality are ˜ V = 39.59, ˜ D = 35.94, ˜ A = 82.29, and ˜ E = 23.83 units of time, respectively. Table 1 Step-stress pattern and progressively type I group-censored data Stage

1

2

3

4

5

Stress level xi Number of failures ni Number of removals ri

1.1 5 7

1.2 2 8

1.3 1 1

1.4 1 2

1.5 1 2

S.-J. Wu et al. / Journal of Statistical Planning and Inference 138 (2008) 817 – 826

823

Table 2 Optimal length of the inspection interval

1



k=2

k=3

˜ V

˜ D

˜ A

˜ E

˜ V

k=4

˜ D

˜ A

˜ E

˜ V

˜ D

˜ A

˜ E

100

0.5 0.6 0.7 0.8 0.9

75.47 79.70 82.93 85.45 87.46

71.68 77.74 82.65 86.69 90.06

88.69 98.62 106.15 112.06 116.81

65.55 69.66 72.92 75.57 77.76

52.15 59.26 65.10 69.85 73.73

43.81 52.23 59.43 65.59 70.88

57.25 69.37 79.80 88.60 95.99

40.29 47.02 52.59 57.22 61.12

45.70 49.10 54.60 60.08 64.95

32.23 39.29 46.74 53.73 60.05

51.38 59.45 68.57 77.64 85.93

28.47 34.69 40.82 46.35 51.24

200

0.5 0.6 0.7 0.8 0.9

150.94 159.39 165.85 170.90 174.92

143.35 155.49 165.30 173.38 180.12

177.37 197.24 212.31 224.11 233.61

131.10 139.31 145.84 151.13 155.51

104.30 118.52 130.20 139.70 147.46

87.63 104.45 118.86 131.19 141.76

114.50 138.74 159.59 177.19 191.98

80.58 94.04 105.18 114.44 122.23

91.41 98.20 109.21 120.15 129.91

64.46 78.57 93.48 107.46 120.09

102.75 118.89 137.13 155.27 171.86

56.95 69.39 81.63 92.71 102.49

300

0.5 0.6 0.7 0.8 0.9

226.41 239.09 248.78 256.35 262.38

215.03 233.23 247.96 260.07 270.18

266.06 295.86 318.46 336.17 350.42

196.65 208.97 218.76 226.70 233.27

156.45 177.78 195.30 209.55 221.19

131.44 156.68 178.29 196.78 212.64

171.75 208.11 239.39 265.79 287.97

120.87 141.07 157.76 171.67 183.35

137.11 147.30 163.81 180.23 194.86

96.69 117.86 140.22 161.19 180.14

154.13 178.34 205.70 232.91 257.79

85.42 104.08 122.45 139.06 153.73

400

0.5 0.6 0.7 0.8 0.9

301.88 318.79 331.71 341.80 349.83

286.70 310.97 330.61 346.76 360.24

354.75 394.48 424.62 448.23 467.22

262.20 278.63 291.68 302.27 311.03

208.61 237.04 260.39 279.40 294.92

175.25 208.91 237.72 262.38 283.53

229.00 277.48 319.19 354.38 383.96

161.16 188.09 210.35 228.89 244.46

182.82 196.41 218.42 240.31 259.81

128.93 157.15 186.96 214.92 240.18

205.51 237.79 274.27 310.54 343.72

113.89 138.78 163.26 185.42 204.97

500

0.5 0.6 0.7 0.8 0.9

377.35 398.48 414.63 427.25 437.29

358.38 388.71 413.26 433.45 450.30

443.43 493.10 530.77 560.28 584.03

327.76 348.29 364.60 377.84 388.78

260.76 296.30 325.49 349.25 368.65

219.06 261.13 297.15 327.97 354.41

286.25 346.85 398.99 442.98 479.95

201.45 235.11 262.94 286.11 305.58

228.52 245.51 273.02 300.38 324.76

161.16 196.44 233.70 268.65 300.23

256.89 297.24 342.84 388.18 429.65

142.37 173.47 204.08 231.77 256.21

6. Simulation results We conducted a numerical study to investigate the optimal length of the inspection interval. Consider the equi-spaced stress levels xi =x0 +id, i =1, 2, . . . , k, where d > 0 is the amount of stress increased at each stage. It is easy to see that with this choice, the relation between mean lifetimes of the ith and the (i + 1)th stages is i+1 = i , i = 1, 2, . . . , k − 1, where 0 <  < 1. Let ˜ V , ˜ D , ˜ A , and ˜ E be the optimal lengths of the inspection interval according to the varianceoptimality, D-optimality, A-optimality, and E-optimality criteria, respectively. Table 2 presents the results for k =2, 3, 4, when 1 equals 100, 200, 300, 400, 500 and  equals 0.5, 0.6, 0.7, 0.8, 0.9. The findings are summarized as follows: 1. For fixed 1 and , all ˜ V , ˜ D , ˜ A , and ˜ E decrease as k increases. This means that larger the number of stress levels is, it is desirable to have a short length of the inspection interval. Fig. 2 also shows this result for 1 = 100. 2. For k = 3, 4, we have ˜ E < ˜ D < ˜ V < ˜ A . However, for the simple step-stress case k = 2, ˜ D is larger than ˜ V when  is equal to 0.8 and 0.9. These results can also be found in Fig. 3. 3. The behavior of the optimal ’s as a function of mean lifetime at the first stage is interesting. For given k and , all of the ratios ˜ V /1 , ˜ D /1 , ˜ A /1 and ˜ E /1 are constant across values of 1 . Therefore, when the optimal length of the inspection interval ˜ V , ˜ D , ˜ A or ˜ E with respect to the mean lifetime 1 at the first stage is determined, it is easy to obtain that the optimal length of the inspection interval with respect to c1 is c˜V , c˜D , c˜A or c˜E where c is a known constant.This is true because 1 is a scale parameter of exponential distribution. 4. As shown in Fig. 3, for fixed 1 and k, all ˜ V , ˜ D , ˜ A , and ˜ E decrease as  decreases. That is, more severe the successive stages are (smaller the ), the optimal length of the inspection interval is shorter and, hence the experiment is stopped faster.

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S.-J. Wu et al. / Journal of Statistical Planning and Inference 138 (2008) 817 – 826 ρ = 0.6

120

120

100

100 optimal length

optimal length

ρ = 0.5

80 60 40 20

80 60 40 20

0

0 2

3 k

4

2

4

ρ = 0.8

120

120

100

100 optimal length

optimal length

ρ = 0.7

3 k

80 60 40 20

80 60 40 20

0

0 2

3 k

4

2

3 k

4

ρ = 0.9 120

optimal length

100 80 60 40 20 0 2

3 k

4

Fig. 2. Optimal length of inspection interval for fixed . ˜ V , ◦; ˜ D , ; ˜ A , +; ˜ E , ×.

7. Discussion and conclusions In this paper, we derive the MLEs for the parameters on the k-level step-stress model in ALT with type I progressive group-censoring.An exponential failure time distribution with mean lifetime which is a log-linear function of stress and a cumulative exposure model are assumed. We study the problem of selecting the optimal length of the inspection interval based on variance-optimality, D-optimality, A-optimality and E-optimality criteria. We also investigate a simulation study to illustrate the proposed four criteria, and some interesting findings are discussed. In practice, the values of 0 , 1 , . . . , k are usually unknown. We have to use the information in the past history or data from a pilot test to get their estimates. Then, the optimal step-stress test plans can be obtained by using the proposed methods. The idea of planning the optimal step-stress test under progressive type I group-censoring scheme is a new one. We explored some of the results for samples from exponential distribution. Further study into more complicated failure time distributions such as Weibull and Gamma will be investigated in the future.

S.-J. Wu et al. / Journal of Statistical Planning and Inference 138 (2008) 817 – 826 k=3

120

120

100

100 optimal length

optimal length

k=2

825

80 60 40 20

80 60 40 20

0

0 0.5

0.6

0.7 

0.8

0.9

0.8

0.9

0.5

0.6

0.7 

0.8

0.9

k=4 120

optimal length

100 80 60 40 20 0 0.5

0.6

0.7 

Fig. 3. Optimal length of inspection interval for fixed k. ˜ V , ◦; ˜ D , ; ˜ A , +; ˜ E , ×.

Acknowledgments The authors wish to thank theAssociate Editor and two referees for valuable suggestions which led to the improvement of this paper. The first author is also grateful to Drs. N. Balakrishnan and C.-T. Lin for inspiring him to do this research. The work is partially supported by the National Science Council of ROC Grant NSC 92-2118-M-032-008. Appendix A. Expectations of Mi , Ni and Ri When i = 1, we have M1 = n, N1 ∼ binomial(m1 , F1 ()) and R1 |n1 ∼ uniform(0, m1 − n1 ). Therefore, E(M1 ) = n, E(N1 ) = m1 F1 () = nF 1 (), and

 E(R1 ) = E

M1 − N1 2

 =

n[1 − F1 ()] . 2

When i =2, we know that M2 =M1 −N1 −R1 , N2 |n1 , r1 ∼ binomial(m2 , F2 ()) and R2 |n2 , n1 , r1 ∼ uniform(0, m2 − n2 ). Hence, E(M2 ) = E(M1 − N1 − R1 ) = E(N2 ) = E(M2 )F2 () =

n[1 − F1 ()] , 2

n[1 − F1 ()]F2 () , 2

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and

 E(R2 ) = E

M2 − N2 2

 =

n[1 − F1 ()][1 − F2 ()] . 22

It is easy to obtain that, by induction, E(Mi ) =

nGi−1 () , 2i−1

E(Ni ) =

nGi−1 ()Fi () , 2i−1

E(Ri ) =

nGi () , 2i

and

where G0 () = 1 and Gi () =

i

j =1 [1 − Fj ()],

for i = 1, 2, . . . , k.

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