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A novel Bayesian algorithm for evaluation lifetime performance of exponential model.

1. Introduction

With the progress of science and technology as well as more sophisticated and complex products, people want to be able to guarantee the life of the products to enhance the competitiveness of the product and stimulate consumer's purchasing desire. Then manufacturers need to work more than ever to improve the product quality and the reliability of the evaluation. Process capability index is an effective and convenient tool for quality assessment (He et al, 2008; Lino et al., 2016), which has become the most widely used statistical process control tool in enterprise for promoting quality assurance, reducing costs and improving customer's satisfaction. Many process capability indices, such as [C.sub.P], [C.sub.pk], [C.sub.pm] and [C.sub.pmk] have been put forward (Juran, 1974; Kane, 1986; Chan and Cheng, 1988; Pearn et al, 1992). The statistical inferences of these process capability indices have drawn great attention. For example, Shiau, Chiang and Hung (1999) discussed the Bayesian estimation of [C.sub.pm] and [C.sub.pmk] under the restriction, which the process meaning is equal to the midpoint of the two specification limits. Pearn and Wu (2005) discussed the Bayes test of [C.sub.pk] for a general situation without restriction on the process mean. Chen and Hsu (2016) proposed a likelihood ratio test to [C.sub.pk]. Baral and Anis (2015) developed a generalized confidence interval method to measure the process capability index [C.sub.pm] in presence of measurement errors. Macintyre (2015) studied the Bayesian estimation of the process capability indices for the inverse Rayleigh lifetime model.

In recent years process capability index has been widely used in process numerical strength and life of the product performance test, whose purpose lies in understanding process to meet specifications and quality requirements, and improve the manufacturing process as well.

Customers cherish the life of the product, and the longer the life of the product means the better quality, thus the quality of the product belongs to the larger-the-better type of the quality characteristics. With these regards, Montgomery (1985) proposed the use of a special unilateral specification process capability index, named as lifetime performance index [C.sub.L], to measure the product lifetime performance, where L is the lower bound of the specifications.

Most of the literature about the process ability index of statistical inference research assumes that the quality characteristics of the product obey a normal distribution. However, the life of the product often obeys the non-normal distribution, and even such distributions as exponential distribution, Pareto distribution and Weibull distribution. The process of competence evaluation is full of quality characteristics by following normal distribution under the assumption of the index. However, some quality characteristics don't obey normal distribution, but takes up the exponential distribution, Pareto distribution and Weibull distribution, in particular for the lifetime of the product, including electronic components, cameras, engine, and electrical appliances. Tong, Chen and Chen (2012) investigated the minimum variance unbiased estimator of the electronic component life under exponential distribution. Wu, Lee and Hou (2007) explored the maximum likelihood estimation of life performance under Rayleigh distribution, and developed a test procedure for evaluating the performance of the product. The studies mentioned above are all about the statistical inference problem of product process capability index under complete sample. However, when the reliability of the application of the products are being analyzed and improved, there is a need to do product sampling life experiment, because life test is usually destructive experiment, and such experiment is usually time-consuming and costly. Therefore, how fast and effectively can life test achieve, and how to save time and cost has become an important issue. In real life, due to the time constraints, manpower and cost considerations, the samples obtained are often referred to censored samples as they are incomplete. The progressively type II censored samples are expanding ones, which have attracted extensive attention from scholars in recent years (Ahmed, 2014). Yan and Liu (2012) proposed the product life obey index distribution of the fixed number of censored data under the lifetime performance index of P value test program, and take the example to illustrate the feasibility and effectiveness of the method. Wu, Chen and Chen (2013) under progressively type II censored life test and discussed the Rayleigh distribution product lifetime performance index of the maximum likelihood estimation, minimum variance without offset estimation and to further develop the corresponding product lifetime performance inspection procedures. Laumen and Cramer (2015) discussed the special index distribution of product family life performance of maximum likelihood estimation and testing procedure based on progressively II censored lifetime data. Lee, Hong and Wu (2015) discussed maximum likelihood estimation and hypothesis testing problem of lifetime performance index based on censored samples, which the lifetime of product from the normal distribution but sample data modeled by fuzzy numbers. All of the above are the statistical inference problem of product life performance under the classical statistical framework. However along with the progress of the manufacturing technology, the reliability of the products becomes increasing high, while the censoring of data is very small. At this time Bayesian method can better deal with the statistical inference problem with small sample case model. However, there are few studies on Bayesian statistical inference about life performance index, and it is necessary to conduct in-depth research. Recently, Lee et al. (2011) studied the Bayesian estimation and testing procedures of lifetime performance index under squared error loss for Rayleigh distribution product. Liu and Ren (2013) obtained Bayesian estimation of lifetime performance index for exponential product under progressively type II censored samples and analyzed the practical example to demonstrate the product quality performance of the proposed Bayesian test program.

In this paper, we study the estimation of CL in Bayesian approach, and the test procedure of CL will also be constructed based on Bayesian approach. In Section 2, some properties of the lifetime of product with the exponential distribution are introduced. Moreover, the relationship between the lifetime performance index and conforming rate is also discussed. Furthermore, the Bayesian estimator of CL on the basis of the conjugate Gamma prior distribution is also obtained under squared error loss and LINEX loss functions. A new Bayes hypothesis testing procedure is developed in Section 3, and a practical example and discussion is given in Section 4. Finally, a conclusion is given in Section 5.

2. Methods

2.1. Lifetime performance index

Let X be the lifetime of such a product whose lifetime distribution is exponential distribution with the following probability density function:

f (x |[theta]) = [theta]exp(-[theta]x), x > 0 (1)

where [theta] > 0 is the unknown scale parameter.

Obviously, a longer lifetime often implies a better product quality. In this case, the lifetime of components owns the character of the-larger-the-better quality. The lifetime is generally required to exceed some unit times, for example, we use L expressing the lower specification limit to both be economically profitable and satisfy customers. Montgometry (1985) proposed a capability index [C.sub.L] to measure the larger-the-better quality character. The index [C.sub.L] is defined as follows

[C.sub.L] = [mu] - L/[sigma] (2)

Here [mu] is the process mean, [sigma] is the process standard deviation, and L is the lower specification limit. The capability index [C.sub.L] is often called the lifetime performance index. Under the assumption of exponential distribution with pdf (1), we can easily derive the process mean [mu] = EX = 1/[theta] and the process standard deviation [sigma] = [square root of (Var(X))] = 1/[theta] Then the lifetime performance index [C.sub.L] can be rewritten as follows

[C.sub.L] = [mu] - L/[sigma] = 1/[theta] - L/1/[theta] = 1 - [theta]L (3)

The failure rate function r(x) is defined by

r(x) = f(x |[theta])/1 - F(x|[theta]) = [theta] exp(-[theta]x)/exp (-[theta]x) = [theta] (4)

From the equations (3) and (4), we can see that the larger the mean 1/[theta], the smaller the failure rate and larger the lifetime performance index [C.sub.L]. Therefore, the lifetime performance index [C.sub.L] reasonably and accurately represents the product performance of new products.

Moreover, if the lifetime of a product X exceeds the lower specification limit L , then conforming rate of the product can be defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Obviously, conforming rate [P.sub.r] and lifetime performance index [C.sub.L] have a strictly increasing relationship. The relationships of [P.sub.r] and [C.sub.L] are shown in Table 1 by a list of various values. For the [C.sub.L] values which are not listed in Table 1, the conforming rate [P.sub.r] can be obtained through interpolation. The conforming rate [P.sub.r] can be calculated by dividing the number of conforming products by total number of products sampled (Tong and Chen, 2012). To accurately estimation, Montgomery in 1985 suggested that the sample size must be large. However, a large sample size is usually not practical from the perspective of cost, since collecting the lifetime data of new products need much money. In addition, a complete sample is also not practical due to time limitation and/or other restrictions (such as material resources, mechanical or experimental difficulties, etc.) on data collection. Since a one-to-one mathematical relationship exists between the conforming rate [P.sub.r] and the lifetime performance index [C.sub.L]. Therefore, utilizing the one-to-one relationship between [P.sub.r] and [C.sub.L], lifetime performance index can be a flexible and effective tool, not only evaluating products performance, but also for estimating the conforming rate [P.sub.r].

2.2. Bayes estimation of lifetime performance index

As the population mean and standard deviation of the lifetime of production components are generally unknown, we need estimate them by some estimating methods. Let [X.sub.1], [X.sub.2], ..., [X.sub.n] represent the lifetime of sample from the exponential distribution with pdf (1), which the likelihood function corresponding to pdf (1) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Here x = ([x.sub.1], [x.sub.2], ..., [x.sub.n]) is the observation of X = ([X.sub.1], [X.sub.2], ..., [X.sub.n]) and t = [n.summation over (i=1)] [x.sub.i].

Then we can easily show that the probability density function of T is

[f.sub.T] (t) = [[theta].sup.n]/[GAMMA](n) [t.sup.n-1] [e.sup.-[theta]t], t > 0 (7)

which is called Gamma distribution, noted by [GAMMA](n, [theta]). Assume that the conjugate prior distribution of [theta] is the Gamma prior distribution [GAMMA]([alpha], [beta]), with pdf

[pi]([theta]; [alpha], [beta]) = [[beta].sup.[alpha]]/[GAMMA] ([alpha]) [[theta].sup.[alpha]-1] [e.sup.-[beta][theta]], [alpha], [beta] > 0, [theta] > 0 (8)

In Bayesian statistical inference, the loss function plays an important role, and symmetric loss function, such as the squared error loss L([??], [theta]) = [([??] - [theta]).sup.2] is widely used, which assigns equal losses to overestimation and underestimation. However, in many practical situations, overestimation and underestimation will make different consequents. Thus under such conditions, symmetric loss functions are inappropriate, Zellner (1986) proposed an asymmetric loss function known as the LINEX loss function and expressed it as

L([DELTA]) = [e.sup.a[DELTA]] -a[DELTA]-1, a [not equal to] 0 (9)

where [DELTA] = [??] - [theta], and [??] is an estimator of [theta], and a is a constant, which represents the direction and degree of symmetry respectively. When a > 0 ,the overestimation is more serious than underestimation, and vice-versa. More details about Bayes estimation and loss functions can be found in paper of Zakerzadeh and Zahraie (2015), Ali S, Aslam Mand Kazmi S M A (2013), Galvan, J. B., Recarte, L. and Perez-Ilzarbe, M. J. (2014).

The posterior expectation of the LINEX loss function (5) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

The Bayes estimator of [theta] under the LINEX loss function can be solved by minimizing (10) and that is the Bayes estimator [[??].sub.BL] can be solved with the following form:

[[??].sub.BL] = 1/[alpha] ln[E([e.sup.-a[theta]] | X)] (11)

Assume that the expectation E([e.sup.-a[theta]] | X) exists and is finite.

Combining the prior distribution (8) with the likelihood function (6), the posterior density of [theta] can be derived as follows by using Bayes' theorem,

[pi]([theta] | x) [varies] [[theta].sup.n] [e.sup.-[theta]t] x [[beta].sup.[alpha]]/[GAMMA] ([alpha]) [[theta].sup.[alpha]-1] [e.sup.-[beta][theta]] [varies] [[theta].sup.n+[alpha]-1] [e.sup.-([beta]+t)[theta]] (12)

Obviously, the posterior distribution of [theta] is Gamma distribution, i.e. [theta]|X ~ [GAMMA](n + [alpha], [beta] + t).

Then, (i) under the squared error loss function, the Bayes estimator of [C.sub.L] is its posterior mean

[[??].sub.L] = E([C.sub.L] | X) = E(1 - [theta]L | X) = 1 - L x E([theta]|X) = 1 - L x n + [alpha]/[beta] + T (13)

(11) Under the LINEX loss function, the Bayes estimator of [C.sub.L] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

3. Results

This section will construct a testing procedure by Bayesian approach to assess whether the lifetime performance index adheres to the required level. Assume that the required index value of lifetime performance is larger than c, where c is the target value. First, we establish the null hypothesis [H.sub.0] : [C.sub.L] [less than or equal to] c and the alternative hypothesis [H.sub.0] : [C.sub.L] > c . Then the new proposed testing procedure using Bayesian approach about [C.sub.L] is as follows:

(i) Determine the lower lifetime limit L for the components and sample size are n.

(ii) Calculate the Bayesian estimation

[[??].sub.BL] = 1 - n + [alpha]/a ln T + [beta]/T + [beta] - aL,

Where T = [n.summation over (i=1)] [X.sub.i].

(iii) Calculate the posterior odds ratio

BF = P([H.sub.0] | X)/P([H.sub.1] | X) = P([X.sub.0] | X)/1 - P([H.sub.0] | X),

where P([H.sub.0] | X) = [[integral].sup.[infinity].sub.1-c/L] [pi] ([theta] | X) d[theta] and [pi] ([theta]|x) = [[beta].sup.n+[alpha]]/[GAMMA](n + [alpha]) [[theta].sup.n+[alpha]-1] [e.sup.-([beta]+t)[theta]]

(iv) The decision rules are provided as follows:

If [[??].sub.BL] > c and BF < 1, we reject to the null hypothesis [H.sub.0] : [C.sub.L] [less than or equal to] c, then it is concluded that the lifetime performance index or conforming rate of the products meets the required level;

If [[??].sub.BL] < c and BF > 1, we accept the null hypothesis [H.sub.0] : [C.sub.L] [less than or equal to] c, then it is concluded that the lifetime performance index or conforming rate of the products does not meet the required level.

4. Discussion

To illustrate the practicability and feasibility of the proposed testing method, a practical data set on the mileages at which n=19 military personnel carriers failed in service is analyzed, which is studied by many references (Grubbs,1971; Lawless, 2003). Table 2 will give the data set.

Gail and Gastwirth (1978) have proved the data set can be modeled by exponential model based on Gini statistics. Next, we will give the steps of the proposed testing procedure about [C.sub.L] as follows:

(i) From Table 2, we can get t = [n.summation over (i=1)] [x.sub.i] = 18963 and here we assume the lower lifetime limit L =171.0144. To deal with the military's concerns regarding operational performances, the conforming rate [P.sub.r] is required to exceed 0.80. Referring to Tablei, the value of [C.sub.L] is required to exceed 0.80. Thus, the target value of performance index is set at c =0.80, the testing hypothesis [H.sub.0] : [C.sub.L] [less than or equal to] 0.80 [left right arrow] [H.sub.0] : [C.sub.L] > 0.80 is constructed.

(ii) Under LINEX loss function, we get the Bayesian estimate [[??].sub.BL] = 0.8098;

(iii) Suppose that the prior parameter values [alpha] = 2.0 and [beta] = 3.0 , then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So, posterior odds ratio is BF = P([H.sub.0] | X)/1 - P([H.sub.0] | X) = 0.5934.

(V) Obviously, [[??].sub.BL] = 0.8098 > c = 0.80 and BF = 0.5934 < 1, then we can reject the null hypothesis [H.sub.0] : [C.sub.L] [less than or equal to] c . That is, we conclude that the lifetime performance index meets the required level.

5. Conclusions

Process capability indices are widely employed by manufactures to assess the performance and potentiality of their process. Lifetime performance index is one of the most important indices of all the proposed process capability indices. In this paper, Bayesian estimation and Bayesian test method of life performance index are presented. The new proposed testing method is easier than other classical approaches. This method can be similar to other life distribution, such as Rayleigh distribution, Lomax distribution. In practice, the method proposed in this paper is easy to use such programming software as Matlab to carry on the programming operation, and it can provide the enterprise with the simple and clear form. It can also provide reference for the enterprise engineers to evaluate whether the true performance of products meets the requirements.

Recebido/Submission: 12/07/2016

Aceitacao/Acceptance: 25/10/2016

Acknowledgments

This project is supported by JK Project of Special Project of Fujian Provincial Colleges and Universities Foundation in China (No. JK2015026), National Social Science Foundation of China (No. 14AGL003), National Science Foundation of China (No. 71661012), Soft Science Project of Fujian Natural Science Foundation of China (No. 2016R0074), Young Doctoral Project of Social Science Foundation of Fujian Province of China (No. FJ2016C188), and the General Project of Young and Middle-aged Teachers Education Scientific Research Projects of Fujian Province of China(No.JAS160294).

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Minglin Jiang (1), Xiaowei Lin (1), Haiping Ren (2) *

heatherj@163.com, lxw949@163.com

(1) School of Business, Minnan Normal University, Zhangzhou 363000, P.R. China,

(2) School of Software, Jiangxi University of Science and Technology, Nanchang 330013, P.R. China.
Table 1--The lifetime performance index [C.sub.L] versus the
conforming rate [P.sub.r]

[C.sub.L]     [P.sub.r]

-[infinity]   0.00000
-2.25         0.03877
-2.00         0.04979
-1-75         0.06393
-1.50         0.08209
-1.25         0.10540
-1.00         0.13534
-0.75         0.17377
-0.50         0.22313
-0.25         0.28650
0.000         0.36788
0.025         0.37719
0.050         0.38674
0.075         0.39657
0.100         0.40657

[C.sub.L]     [P.sub.r]

0.125         0.41686
0.225         0.46070
0.250         0.47237
0.275         0.48432
0.300         0.49659
0.325         0.50916
0.350         0.52205
0.375         0.53526
0.400         0.54881
0.425         0.56270
0.450         0.57695
0.475         0.59156
0.500         0.60653
0.525         0.62189
0.550         0.63763

[C.sub.L]     [P.sub.r]

0.575         0.65377
0.625         0.72253
0.700         0.74082
0.725         0.75957
0.750         0.77880
0.775         0.79852
0.800         0.81873
0.825         0.83946
0.850         0.86071
0.875         0.88250
0.900         0.90484
0.925         0.92774
0.950         0.95123
0.975         0.97531
1.000         1.00000

Table 2--The Data Set

162   200    271    320    393    508    539    629    706    777
884   1008   1101   1182   1463   1603   1984   2355   2880
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Author:Jiang, Minglin; Lin, Xiaowei; Ren, Haiping
Publication:RISTI (Revista Iberica de Sistemas e Tecnologias de Informacao)
Date:Nov 15, 2016
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