Evaluating the Lifetime Performance Index Based on the Bayesian Estimation for the Rayleigh Lifetime Products with the Upper Record Values

Abstract

Quality management is very important for many manufacturing industries. Process capability analysis has been widely applied in the field of quality control to monitor the performance of industrial processes. Hence, the lifetime performance index 𝐶_𝐿 is utilized to measure the performance of product, where 𝐿 is the lower specification limit. This study constructs a Bayesian estimator of 𝐶_𝐿 under a Rayleigh distribution with the upper record values. The Bayesian estimations are based on squared-error loss function, linear exponential loss function, and general entropy loss function, respectively. Further, the Bayesian estimators of 𝐶_𝐿 are utilized to construct the testing procedure for 𝐶_𝐿 based on a credible interval in the condition of known 𝐿. The proposed testing procedure not only can handle nonnormal lifetime data, but also can handle the upper record values. Moreover, the managers can employ the testing procedure to determine whether the lifetime performance of the Rayleigh products adheres to the required level. The hypothesis testing procedure is a quality performance assessment system in enterprise resource planning (ERP).

1. Introduction

Process capability analysis is an effective means to measure the performance and potential capabilities of a process. Process capability indices (PCIs) are utilized to assess whether product quality meets the required level in manufacturing industries. For instance, the lifetime of electronic components exhibits a larger-the-better type of quality characteristic. Montgomery [1] proposed the process capability index 𝐶_𝐿 to evaluate the lifetime performance of electronic components, where 𝐿 is the lower specification limit. Tong et al. [2] constructed the uniformly minimum variance unbiased estimator (UMVUE) of 𝐶_𝐿 and proposed a hypothesis testing procedure for the complete sample from a one-parameter exponential distribution. In addition, the lifetime performance index 𝐶_𝐿 also is applied to evaluate the lifetime of product in the censored sample. For instance, Hong et al. [3, 4] constructed the lifetime performance index 𝐶_𝐿 to evaluate business performance under the right type II censored sample and proposed a confidence interval for Pareto’s distribution. Wu et al. [5] proposed a hypothesis testing procedure based on a maximum likelihood estimator (MLE) of 𝐶_𝐿 to evaluate the product quality for two-parameter exponential distribution under the right type II censored sample. Lee et al. [6] also proposed a hypothesis testing procedure based on a MLE of 𝐶_𝐿 to evaluate product quality for exponential distribution under the progressively type II right censored sample. Lee et al. [7] also constructed an UMVUE of 𝐶_𝐿 and developed a testing procedure for the performance index of products with the exponential distribution based on the type II right censored sample. All of the above 𝐶_𝐿 have been utilized to evaluate the quality performance for complete data and censored data. Nevertheless, record values often arise in industrial stress testing and other similar situations.

Record values and the associated statistics are of interest and important in many real-life applications. In industry and reliability studies, many products fail under stress. For example, a battery dies under the stress of time. But the precise breaking stress or failure point varies even among identical items. Hence, in such experiments, measurements may be made sequentially and only the record values (lower or upper) are observed. Record values arise naturally in many real-life applications involving data relating to weather, sports, economics, and life tests. According to the model of Chandler [8], there are some situations in lifetime testing experiments where the failure time of a product is recorded if it exceeds all preceding failure times. These recorded failure times are the upper record value sequence. In general, let 𝑋₁,𝑋₂,… be a sequence of independent and identically distributed (i.i.d.) random variables having the same distribution as the (population) random variable 𝑋 with the cumulative distribution function (c.d.f) 𝐹(𝑥) and the probability density function (p.d.f) 𝑓(𝑥). An observation 𝑋_𝑗 will be called an upper record value if it exceeds in value all of the preceding observations, that is, if 𝑋_𝑗>𝑋_𝑖, for every 𝑖<𝑗. The sequence of record times 𝑇_𝑛, 𝑛≥1 is defined as follows.

Let 𝑇₁=1 with probability 1, for 𝑛≥2, 𝑇_𝑛=min{𝑗∶𝑋_𝑗>𝑋_𝑇𝑛−1}. A sequence of upper record values {𝑋_𝑈(𝑛)} is then defined by 𝑋_𝑈(𝑛)=𝑋_𝑇𝑛,𝑛=1,2,…. (1) Let 𝑋_∼={𝑋_𝑈(1),𝑋_𝑈(2),…,𝑋_𝑈(𝑛)} be the first 𝑛 upper record values arising from a sequence of i.i.d. random variables 𝑋₁,𝑋₂,… with c.d.f. 𝐹(𝑥).

In this study, we consider the case of the upper record values instead of complete data or censored data. In order to evaluate the lifetime performance of nonnormal data with upper record values, the lifetime performance index 𝐶_𝐿 is utilized to measure product quality under the Rayleigh distribution with the upper record values. The Rayleigh distribution is a nonnormal distribution and a special case of the Weibull distribution, which provides a population model useful in several areas of statistics, including life testing and reliability whose age with time as its failure rate is a linear function of time. Bhattacharya and Tyagi [9] mentioned that in some clinical studies dealing with cancer patients the survival pattern follows the Rayleigh distribution. Cliff and Ord [10] showed that the Rayleigh distribution arises as the distribution of the distance between an individual and its nearest neighbor when the special pattern is generated by the Poisson process. Dyer and Whisenand [11] demonstrated the importance of this distribution in communication engineering (also see Soliman and Al-Aboud [12]). The p.d.f. and c.d.f. of the Rayleigh distribution are given, respectively, by 𝑓(𝑥)=𝑥𝜃²𝑒^{−𝑥2/2𝜃2},𝐹(𝑥)=1−𝑒^{−𝑥2/2𝜃2}, (2)(3) where 𝑥>0 and 𝜃>0, respectively. When 𝑋 comes from the Rayleigh distribution, the mean of 𝑋 is 𝐸(𝑋)=𝜋/2𝜃, and the standard deviation of 𝑋 is 𝜎=(4−𝜋)/2𝜃.

Bayesian and non-Bayesian approaches have been used to obtain the estimators of the parameter 𝜃 under the Rayleigh distribution with the upper record values. Soliman and Al-Aboud [12] compared the performance of the Bayesian estimators with non-Bayesian estimators such as the MLE and the best linear unbiased (BLUE) estimator. The Bayesian estimators are developed under symmetric and nonsymmetric loss functions. The symmetric loss function is squared-error (SE) loss function. The nonsymmetric loss function includes linear exponential (LINEX) and general entropy (GE) loss functions. In recent decades, the Bayesian viewpoint has received frequent attention for analyzing failure data and other time-to-event data and has been often proposed as a valid alternative to traditional statistical perspectives. The Bayesian approach to estimation of the parameters and reliability analysis allows prior subjective knowledge on lifetime parameters and technical information on the failure mechanism as well as experimental data. Bayesian methods usually require less sample data to achieve the same quality of inferences than methods based on sampling theory (also see [12]).

The main aim of this study will construct the Bayesian estimator of 𝐶_𝐿 under a Rayleigh distribution with upper record values. The Bayesian estimators of 𝐶_𝐿 are developed under symmetric and nonsymmetric loss functions. The estimators of 𝐶_𝐿 are then utilized to develop a credible interval, respectively. The new testing procedures of credible interval for 𝐶_𝐿 can be employed by managers to assess whether the products performance adheres to the required level in the condition of known 𝐿. The new proposed testing procedures can handle nonnormal lifetime data with upper record values. Moreover, we will evaluate the performance of the new proposed testing procedures with Bayesian and non-Bayesian approaches.

The rest of this study is organized as follows. Section 2 introduces some properties of the lifetime performance index for lifetime of product with the Rayleigh distribution. Section 3 discusses the relationship between the lifetime performance index and conforming rate. Section 4 develops a hypothesis testing procedure for the lifetime performance index 𝐶_𝐿 with the non-Bayesian approach. Section 5 develops a hypothesis testing procedure for the lifetime performance index 𝐶_𝐿 with the Bayesian approach. Section 6 discusses the Monte Carlo simulation algorithm of confidence (or credible) level. One numerical example and concluding remarks are made in Sections 7 and 8, respectively.

2. The Lifetime Performance Index

Montgomery [1] has developed a process capability index 𝐶_𝐿 to measure the larger-the-better quality characteristic. Then, 𝐶_𝐿 is defined by 𝐶_𝐿=𝜇−𝐿𝜎, (4) where 𝜇, 𝜎, and 𝐿 are the process mean, the process standard deviation, and the lower specification limit, respectively.

To assess the product performance of products, 𝐶_𝐿 can be defined as the lifetime performance index. If 𝑋 comes from the Rayleigh distribution, then the lifetime performance index 𝐶_𝐿 can be rewritten as 𝐶_𝐿=𝜇−𝐿𝜎=𝜋/2𝜃−𝐿()=4−𝜋/2𝜃𝜋−4−𝜋2𝐿4−𝜋𝜃,−∞<𝐶_𝐿<𝜋,4−𝜋 (5) where 𝜇=𝐸(𝑋)=𝜋/2𝜃 is the process mean, 𝜎=Var(𝑋)=(4−𝜋)/2𝜃 is the process standard deviation, and 𝐿 is the lower specification limit.

3. The Conforming Rate

If the lifetime of a product 𝑋 exceeds the lower specification limit 𝐿, then the product is defined as a conforming product. The ratio of conforming products is known as the conforming rate and can be defined as 𝑃_𝑟=𝑃(𝑋≥𝐿)=𝑒^{−(1/2)(𝜋/2−(4−𝜋)/2𝐶𝐿)2},−∞<𝐶_𝐿<𝜋/(4−𝜋). (6)

Obviously, a strictly increasing relationship exists between the conforming rate 𝑃_𝑟 and the lifetime performance index 𝐶_𝐿. Table 1 lists various 𝐶_𝐿 values and the corresponding conforming rate 𝑃_𝑟 by using STATISTICA software [13] (also see [14]).

Table 1 

The lifetime performance index 𝐶_𝐿 versus the conforming rate 𝑃_𝑟.

For the 𝐶_𝐿 values which are not listed in Table 1, the conforming rate 𝑃_𝑟 can be obtained through interpolation. In addition, since a one-to-one mathematical relationship exists between the conforming rate 𝑃_𝑟 and the lifetime performance index 𝐶_𝐿. Therefore, utilizing the one-to-one relationship between 𝑃_𝑟 and 𝐶_𝐿, lifetime performance index can be a flexible and effective tool, not only evaluating product performance, but also estimating the conforming rate 𝑃_𝑟.

4. Testing Procedure for the Lifetime Performance Index 𝐶_𝐿 with Non-Bayesian Approach

This section will apply non-Bayesian approach to construct a maximum likelihood estimator (MLE) of 𝐶_𝐿 under a Rayleigh distribution with upper record values. The MLE of 𝐶_𝐿 is then utilized to develop a new hypothesis testing procedure in the condition of known 𝐿. Assuming that the required index value of lifetime performance is larger than 𝑐₀, where 𝑐₀ denotes the target value, the null hypothesis 𝐻₀∶𝐶_𝐿≤𝑐₀ and the alternative hypothesis 𝐻₁∶𝐶_𝐿>𝑐₀ are constructed. Based on the new hypothesis testing procedure, the lifetime performance of products is easy to assess.

Let 𝑋 be the lifetime of such a product, and 𝑋 has a Rayleigh distribution with the p.d.f. as given by (2). Let 𝑋_∼=(𝑋_𝑈(1),𝑋_𝑈(2),…,𝑋_𝑈(𝑛)) be the first 𝑛 upper record values arising from a sequence of i.i.d. Rayleigh variables with p.d.f. as given by (2). Since the joint p.d.f. of (𝑋_𝑈(1),𝑋_𝑈(2),…,𝑋_𝑈(𝑛)) is 𝑓𝑥_𝑈(𝑛)^𝑛−1_𝑖=1𝑓𝑥_𝑈(𝑖)1−𝐹𝑥_𝑈(𝑖), (7) where 𝑓(𝑥) and 𝐹(𝑥) are the p.d.f. and c.d.f. of 𝑋, respectively (also see [12, 15, 16]). So, the likelihood function of (𝑋_𝑈(1),𝑋_𝑈(2),…,𝑋_𝑈(𝑛)) is given as 𝐿𝜃∣𝑥_𝑈(1),…,𝑥_𝑈(𝑛)=^𝑛_𝑖=1𝑥_𝑈(𝑖)𝜃²−𝑥exp^2𝑈(𝑛)2𝜃². (8) The natural logarithm of the likelihood function with (8) is ln𝐿𝜃∣𝑥_𝑈(1),…,𝑥_𝑈(𝑛)=^𝑛_𝑖=1ln𝑥_𝑈(𝑖)𝑥−2𝑛ln𝜃−^2𝑈(𝑛)2𝜃². (9) Upon differentiating (9) with respect to 𝜃 and equating the result to zero, the MLE of the parameter 𝜃 can be shown to be 𝜃_MLE=𝑋_𝑈(𝑛)√2𝑛 (10) (also see [12]). By using the invariance of MLE (see Zehna [17]), the MLE of 𝐶_𝐿 can be written as given by 𝐶_𝐿,MLE=𝜋−4−𝜋2𝐿4−𝜋𝜃_MLE=𝜋−4−𝜋2√4−𝜋⋅𝐿⋅2𝑛𝑋_𝑈(𝑛). (11)

Construct a statistical testing procedure to assess whether the lifetime performance index adheres to the required level. The one-sided confidence interval for 𝐶_𝐿 is obtained using the pivotal quantity 𝑋²_𝑈(𝑛)/𝜃². By using the pivotal quantity 𝑋²_𝑈(𝑛)/𝜃², given the specified significance level 𝛼, the level 100(1−𝛼)% one-sided confidence interval for 𝐶_𝐿 can be derived as follows.

Since the pivotal quantity 𝑋²_𝑈(𝑛)/𝜃²∼𝜒²_2𝑛, and 𝜒²_{2𝑛,1−𝛼} function which represents the lower 1−𝛼 percentile of 𝜒²_2𝑛, then 𝑃𝑋²_𝑈(𝑛)𝜃²≤𝜒²_{2𝑛,1−𝛼}𝐶=1−𝛼,⟹𝑃_𝐿≥𝜋−4−𝜋𝜋−𝐶4−𝜋^𝐿,MLE𝜒²_{2𝑛,1−𝛼}2𝑛^1/2=1−𝛼, (12) where 𝐶_𝐿=𝜋/(4−𝜋)−2/(4−𝜋)(𝐿/𝜃). From (12), we obtain that a 100(1−𝛼)% one-sided confidence interval for 𝐶_𝐿 is 𝐶_𝐿≥𝜋−4−𝜋𝜋−𝐶4−𝜋^𝐿,MLE𝜒²_{2𝑛,1−𝛼}2𝑛^1/2, (13) where 𝐶_𝐿,MLE is given by (11). Therefore, the 100(1−𝛼)% lower confidence interval bound for 𝐶_𝐿 can be written as LB_MLE=𝜋−4−𝜋𝜋−𝐶4−𝜋^𝐿,MLE𝜒²_{2𝑛,1−𝛼}2𝑛^1/2, (14) where 𝐶_𝐿,MLE, 𝛼, and 𝑛 denote the MLE of 𝐶_𝐿, the specified significance level, and the upper record sample of size, respectively.

The managers can use the one-sided confidence interval to determine whether the product performance adheres to the required level. The proposed testing procedure of 𝐶_𝐿 with 𝐶_𝐿,MLE can be organized as follows.

Step 1. Determine the lower lifetime limit 𝐿 for products and the performance index value 𝑐₀, then the testing null hypothesis 𝐻₀∶𝐶_𝐿≤𝑐₀ and the alternative hypothesis 𝐻₁∶𝐶_𝐿>𝑐₀ are constructed.

Step 2. Specify a significance level 𝛼.

Step 3. Given the upper record values 𝑥_∼=(𝑥_𝑈(1),𝑥_𝑈(2),…,𝑥_𝑈(𝑛)), the lower lifetime limit 𝐿, and the significance level 𝛼, then we can calculate the 100(1−𝛼)% one-sided confidence interval [LB_MLE,∞) for 𝐶_𝐿, where LB_MLE as the definition of (14).

Step 4. The decision rule of statistical test is provided as follows:(1)if the performance index value 𝑐₀∉[LB_MLE,∞), then we will reject 𝐻₀. It is concluded that the lifetime performance index of products meets the required level;(2)if the performance index value 𝑐₀∈[LB_MLE,∞), then we do not reject 𝐻₀. It is concluded that the lifetime performance index of products does not meet the required level.

5. Testing Procedure for the Lifetime Performance Index 𝐶_𝐿 with the Bayesian Approach

This section will apply the Bayesian approach to construct an estimator of 𝐶_𝐿 under a Rayleigh distribution with upper record values. The Bayesian estimators are developed under symmetric and nonsymmetric loss functions. The symmetric loss function is squared-error (SE) loss function. The nonsymmetric loss function includes linear exponential (LINEX) and general entropy (GE) loss functions. The Bayesian estimator of 𝐶_𝐿 is then utilized to develop a new hypothesis testing procedure in the condition of known 𝐿. Assuming that the required index value of lifetime performance is larger than 𝑐₀, where 𝑐₀ denotes the target value, the null hypothesis 𝐻₀∶𝐶_𝐿≤𝑐₀ and the alternative hypothesis 𝐻₁∶𝐶_𝐿>𝑐₀ are constructed. Based on the new hypothesis testing procedure, the lifetime performance of products is easy to assess.

5.1. Testing Procedure for the Lifetime Performance Index 𝐶_𝐿 with the Bayesian Estimator under Squared-Error Loss Function

Let 𝑋 be the lifetime of such a product, and 𝑋 has a Rayleigh distribution with the p.d.f. as given by (2). We consider the conjugate prior distribution of the form which is defined by the square-root inverted-gamma density as follows: 𝜋₁(𝜃)=𝑎^𝑏Γ(𝑏)2^𝑏−1𝜃^−2𝑏−1𝑒^{−𝑎/2𝜃2}, (15) where 𝜃>0, 𝑎>0, and 𝑏>0. 𝐸(𝜃)=𝑎2^1/2Γ(𝑏−1/2)Γ(𝑏),(𝜃)=𝑎Var2()−𝑎𝑏−12Γ(𝑏−1/2)Γ(𝑏)²,𝑏>1. (16) With record values, 𝑛 products (or items) take place on test. Consider (𝑋_𝑈(1),𝑋_𝑈(2),…,𝑋_𝑈(𝑛)) is the upper record values. We can obtain that the posterior p.d.f. of 𝜃∣(𝑋_𝑈(1),𝑋_𝑈(2),…,𝑋_𝑈(𝑛)) is given as 𝑝₁𝜃∣𝑥_𝑈(1),…,𝑥_𝑈(𝑛)=𝐿𝜃∣𝑥_𝑈(1),…,𝑥_𝑈(𝑛)𝜋₁(𝜃)∫^∞₀𝐿𝜃∣𝑥_𝑈(1),…,𝑥_𝑈(𝑛)𝜋₁(𝜃)=𝜃𝑑𝜃_{−2(𝑛+𝑏)−1}𝑥²_𝑈(𝑛)+𝑎^(𝑛+𝑏)𝑒^{−(𝑥𝑈(𝑛)+𝑎)/2𝜃2}Γ(𝑛+𝑏)2^{𝑛+𝑏−1}, (17) where 𝜃>0, and the likelihood function 𝐿(𝜃∣𝑥_𝑈(1),…,𝑥_𝑈(𝑛)) as (8).

Let 𝑡=𝑥²_𝑈(𝑛)+𝑎, then the posterior p.d.f. can be rewritten as 𝑝₁𝜃∣𝑥_𝑈(1),…,𝑥_𝑈(𝑛)=𝜃^{−2(𝑛+𝑏)−1}𝑡^(𝑛+𝑏)𝑒^{−𝑡/2𝜃2}Γ(𝑛+𝑏)2^{𝑛+𝑏−1}, (18) and we can show that 𝑡/𝜃²|_{𝑥∼=(𝑥𝑈(1),…,𝑥𝑈(𝑛))}∼𝜒²_2(𝑛+𝑏).

The Bayesian estimator 𝜃_BS of 𝜃 based on the squared error (SE) loss function 𝐿(𝜃^∗,𝜃)=(𝜃^∗−𝜃)² can be derived as follows: 𝐸𝜃∣𝑥_𝑈(1),…,𝑥_𝑈(𝑛)=^∞₀𝜃𝑝₁𝜃∣𝑥_𝑈(1),…,𝑥_𝑈(𝑛)=Γ()𝑑𝜃𝑛+𝑏−1/2Γ()𝑡𝑛+𝑏2^1/2, (19) where 𝑡=𝑥²_𝑈(𝑛)+𝑎, 𝑛+𝑏−1/2>0.

Hence, the Bayesian estimator of 𝜃 based on SE loss function is given by 𝜃_BS=Γ(𝑛+𝑏−1/2)Γ()𝑇𝑛+𝑏2^1/2, (20) where 𝑇=𝑋²_𝑈(𝑛)+𝑎, 𝑛+𝑏−1/2>0, 𝑎, and 𝑏 are the parameters of prior distribution with density as (15).

The lifetime performance index 𝐶_𝐿 of Rayleigh products can be written as 𝐶_𝐿=𝜋−4−𝜋2𝐿4−𝜋𝜃,−∞<𝐶_𝐿<𝜋.4−𝜋 (21) By using (5) and the Bayesian estimator 𝜃_BS as (20), 𝐶_𝐿,BS based on the Bayesian estimator 𝜃_BS of 𝜃 is given by 𝐶_𝐿,BS=𝜋−4−𝜋2𝐿4−𝜋𝜃_BS=𝜋−4−𝜋2Γ()4−𝜋⋅𝐿⋅𝑛+𝑏−1/2Γ()𝑇𝑛+𝑏2^1/2−1, (22) where 𝑇=𝑋²_𝑈(𝑛)+𝑎, 𝑛+𝑏−1/2>0, 𝑎, and 𝑏 are the parameters of prior distribution with density as in (15).

We construct a statistical testing procedure to assess whether the lifetime performance index adheres to the required level. The one-sided credible confidence interval for 𝐶_𝐿 are obtained using the pivotal quantity 𝑡/𝜃²|_{𝑥∼=(𝑥𝑈(1),…,𝑥𝑈(𝑛))}. By using the pivotal quantity 𝑡/𝜃²|_{𝑥∼=(𝑥𝑈(1),…,𝑥𝑈(𝑛))}, given the specified significance level 𝛼, the level 100(1−𝛼)% one-sided credible interval for 𝐶_𝐿 can be derived as follows.

Since the pivotal quantity 𝑡/𝜃²|_{𝑥∼=(𝑥𝑈(1),…,𝑥𝑈(𝑛))}∼𝜒²_2(𝑛+𝑏), and 𝜒²_{2(𝑛+𝑏),1−𝛼} function which represents the lower 1−𝛼 percentile of 𝜒²_2(𝑛+𝑏), then 𝑃𝑡𝜃²≤𝜒²_{2()𝑛+𝑏,1−𝛼}∣𝑥_∼=1−𝛼,where𝑥_∼=𝑥_𝑈(1),𝑥_𝑈(2),…,𝑥_𝑈(𝑛),⟹𝑃𝐶_𝐿≥𝜋−4−𝜋𝜋−𝐶4−𝜋^𝐿,BS×Γ(𝑛+𝑏−1/2)Γ()𝜒𝑛+𝑏2^{22()𝑛+𝑏,1−𝛼1/2}∣𝑥_∼=1−𝛼, (23) where 𝐶_𝐿 as the definition of (5).

From (23), we obtain that a 100(1−𝛼)% one-sided credible interval for 𝐶_𝐿 is given by 𝐶_𝐿≥𝜋−4−𝜋𝜋−𝐶4−𝜋^𝐿,BS×Γ(𝑛+𝑏−1/2)Γ()𝜒𝑛+𝑏2^{22(𝑛+𝑏),1−𝛼1/2}, (24) where 𝐶_𝐿,BS is given by (22).

Therefore, the 100(1−𝛼)% lower credible interval bound for 𝐶_𝐿 can be written as LB_BS=𝜋−4−𝜋𝜋−𝐶4−𝜋^𝐿,BS×Γ(𝑛+𝑏−1/2)Γ()𝜒𝑛+𝑏2^{22()𝑛+𝑏,1−𝛼1/2}, (25) where 𝐶_𝐿,BS is given by (22), and 𝑏 is a parameter of prior distribution with density as (15).

The managers can use the one-sided credible interval to determine whether the product performance adheres to the required level. The proposed testing procedure of 𝐶_𝐿 with 𝐶_𝐿,BS can be organized as follows.

Step 1. Determine the lower lifetime limit 𝐿 for products and the performance index value 𝑐₀, then the testing null hypothesis 𝐻₀∶𝐶_𝐿≤𝑐₀ and the alternative hypothesis 𝐻₁∶𝐶_𝐿>𝑐₀ are constructed.

Step 2. Specify a significance level 𝛼.

Step 3. Given the parameters 𝑎 and 𝑏 of prior distribution, the upper record values 𝑥_∼=(𝑥_𝑈(1),𝑥_𝑈(2),…,𝑥_𝑈(𝑛)), the lower lifetime limit 𝐿, and the significance level 𝛼, then we can calculate the 100(1−𝛼)% one-sided credible interval [LB_BS,∞) for 𝐶_𝐿, where LB_BS as the definition of (25).

Step 4. The decision rule of statistical test is provided as follows:(1)if the performance index value 𝑐₀∉[LB_BS,∞), then we will reject 𝐻₀. It is concluded that the lifetime performance index of products meets the required level;(2)if the performance index value 𝑐₀∈[LB_BS,∞), then we do not reject 𝐻₀. It is concluded that the lifetime performance index of products does not meet the required level.

5.2. Testing Procedure for the Lifetime Performance Index 𝐶_𝐿 with the Bayesian Estimator under Linear Exponential Loss Function

Let 𝑋 be the lifetime of such a product, and 𝑋 has a Rayleigh distribution with the p.d.f. as given by (2). We consider the linear exponential (LINEX) loss function as (also see [18–23]) 𝐿(Δ)∝𝑒^𝑐Δ−𝑐Δ−1, (26) where Δ=(𝜃^∗−𝜃), 𝑐≠0.

In this paper, we suppose that Δ₁=𝜃^∗𝜃²−1, (27) where 𝜃^∗ is an estimator of 𝜃.

By using (26)-(27), the posterior expectation of LINEX loss function 𝐿(Δ₁) is 𝐸𝐿Δ₁∣𝑥_∼=𝑒^−𝑐𝐸𝑒^{𝑐(𝜃∗/𝜃)2}∣𝑥_∼𝜃−𝑐𝐸^∗𝜃²−1∣𝑥_∼−1, (28) where 𝑥_∼=(𝑥_𝑈(1),𝑥_𝑈(2),…,𝑥_𝑈(𝑛)).

The value of 𝜃^∗ that minimizes 𝐸[𝐿(Δ₁)∣𝑥_∼] denoted by 𝜃_BL is obtained by solving the equation: 𝑑𝐸𝐿Δ₁∣𝑥_∼𝑑𝜃^∗=𝑒^−𝑐𝐸𝑒^{𝑐(𝜃∗/𝜃)2}⋅2𝑐𝜃^∗𝜃²∣𝑥_∼−𝑐𝐸2𝜃^∗𝜃²∣𝑥_∼=0; (29) that is, 𝜃_BL is the solution to the following equation: 𝐸𝜃_BL𝜃²𝑒^𝑐(𝜃_BL/𝜃)2∣𝑥_∼=𝑒^𝑐𝐸𝜃_BL𝜃²∣𝑥_∼. (30) By using (18) and (30), we have 2𝜃_BL()⋅𝑡𝑛+𝑏^𝑛+𝑏𝑡−2𝑐𝜃²_BL^𝑛+𝑏+1=𝑒^𝑐𝜃_BL2(𝑛+𝑏)𝑡,where𝑡=𝑥²_𝑈(𝑛)+𝑎. (31) Hence, the Bayesian estimator of 𝜃 under the LINEX loss function is given by 𝜃_BL=𝑇2𝑐1−𝑒^{−𝑐/(𝑛+𝑏+1)1/2}, (32) where 𝑇=𝑋²_𝑈(𝑛)+𝑎, 𝑎, and 𝑏 are the parameters of prior distribution with density as (15).

By using (5) and the Bayesian estimator 𝜃_BL as (32), 𝐶_𝐿,BL based on the Bayesian estimator 𝜃_BL of 𝜃 is given by 𝐶_𝐿,BL=𝜋−4−𝜋2𝐿4−𝜋𝜃_BL=𝜋−4−𝜋2𝐿𝑇4−𝜋2𝑐1−𝑒^{−𝑐/(𝑛+𝑏+1)−1/2}, (33) where 𝑇=𝑋²_𝑈(𝑛)+𝑎, 𝑎, and 𝑏 are the parameters of prior distribution with density as (15).

We construct a statistical testing procedure to assess whether the lifetime performance index adheres to the required level. The one-sided credible confidence interval for 𝐶_𝐿 is obtained using the pivotal quantity 𝑡/𝜃²|_{𝑥∼=(𝑥𝑈(1),…,𝑥𝑈(𝑛))}. By using the pivotal quantity 𝑡/𝜃²|_{𝑥∼=(𝑥𝑈(1),…,𝑥𝑈(𝑛))}, given the specified significance level 𝛼, the level 100(1−𝛼)% one-sided credible interval for 𝐶_𝐿 can be derived as follows.

Since the pivotal quantity 𝑡/𝜃²|_{𝑥∼=(𝑥𝑈(1),…,𝑥𝑈(𝑛))}∼𝜒²_2(𝑛+𝑏), and 𝜒²_{2(𝑛+𝑏),1−𝛼} function which represents the lower 1−𝛼 percentile of 𝜒²_2(𝑛+𝑏), then 𝑃𝑡𝜃²≤𝜒²_{2()𝑛+𝑏,1−𝛼}∣𝑥_∼=1−𝛼,where𝑥_∼=𝑥_𝑈(1),𝑥_𝑈(2),…,𝑥_𝑈(𝑛),⟹𝑃𝐶_𝐿≥𝜋−4−𝜋𝜋−𝐶4−𝜋^𝐿,BL×12𝑐1−𝑒^{−𝑐/(𝑛+𝑏+1)}𝜒²_{2()𝑛+𝑏,1−𝛼}^1/2∣𝑥_∼=1−𝛼, (34) where 𝐶_𝐿 as the definition of (5).

From (34), we obtain that a 100(1−𝛼)% one-sided credible interval for 𝐶_𝐿 is 𝐶_𝐿≥𝜋−4−𝜋𝜋−𝐶4−𝜋^𝐿,BL×12𝑐1−𝑒^{−𝑐/(𝑛+𝑏+1)}𝜒²_{2()𝑛+𝑏,1−𝛼}^1/2, (35) where 𝐶_𝐿,BL is given by (33).

Therefore, the 100(1−𝛼)% lower credible interval bound for 𝐶_𝐿 can be written as LB_BL=𝜋−4−𝜋𝜋−𝐶4−𝜋^𝐿,BL×12𝑐1−𝑒^{−𝑐/(𝑛+𝑏+1)}𝜒²_{2()𝑛+𝑏,1−𝛼}^1/2, (36) where 𝐶_𝐿,BL is given by (33), and 𝑏 is a parameter of prior distribution with density as (15).

The managers can use the one-sided credible interval to determine whether the product performance attains to the required level. The proposed testing procedure of 𝐶_𝐿 with 𝐶_𝐿,BL can be organized as follows.

Step 1. Determine the lower lifetime limit 𝐿 for products and the performance index value 𝑐₀, then the testing null hypothesis 𝐻₀∶𝐶_𝐿≤𝑐₀ and the alternative hypothesis 𝐻₁∶𝐶_𝐿>𝑐₀ are constructed.

Step 2. Specify a significance level 𝛼.

Step 3. Given the parameters 𝑎 and 𝑏 of prior distribution, the parameter 𝑐 of LINEX loss function, the upper record values 𝑥_∼=(𝑥_𝑈(1),𝑥_𝑈(2),…,𝑥_𝑈(𝑛)), the lower lifetime limit 𝐿, and the significance level 𝛼, then we can calculate the 100(1−𝛼)% one-sided credible interval [LB_BL,∞) for 𝐶_𝐿, where LB_BL as the definition of (36).

Step 4. The decision rule of statistical test is provided as follows:(1)if the performance index value 𝑐₀∉[LB_BL,∞), then we will reject 𝐻₀. It is concluded that the lifetime performance index of products meets the required level;(2)if the performance index value 𝑐₀∈[LB_BL,∞), then we do not reject 𝐻₀. It is concluded that the lifetime performance index of products does not meet the required level.

5.3. Testing Procedure for the Lifetime Performance Index 𝐶_𝐿 with the Bayesian Estimator under General Entropy Loss Function

Let 𝑋 be the lifetime of such a product, and 𝑋 has a Rayleigh distribution with the p.d.f. as given by (2). We consider the general entropy (GE) loss function (also see [12, 20]): 𝐿𝜃^∗∝𝜃,𝜃^∗𝜃^𝑞𝜃−𝑞log^∗𝜃−1, (37) whose minimum occurs at 𝜃^∗=𝜃.

The loss function is a generalization of entropy loss used by several authors (e.g., Dyer and Liu [24], Soliman [25], and Soliman and Elkahlout [26]) where the shape parameter 𝑞=1. When 𝑞>0, a positive error (𝜃^∗>𝜃) causes more serious consequences than a negative error. The Bayesian estimator 𝜃_BG of 𝜃 under GE loss function is given by 𝜃_BG=𝐸𝜃^−𝑞∣𝑥_𝑈(1),…,𝑥_𝑈(𝑛)^−1/𝑞. (38)

By using (18) and (38), then the Bayesian estimator 𝜃_BG of 𝜃 is derived as follows: 𝐸𝜃^−𝑞∣𝑥_𝑈(1),…,𝑥_𝑈(𝑛)=^∞₀𝜃^−𝑞𝑝₁𝜃∣𝑥_𝑈(1),…,𝑥_𝑈(𝑛)=𝑡𝑑𝜃2^−𝑞/2⋅Γ𝑛+𝑏+𝑞/2Γ(),𝑛+𝑏 (39) where 𝑡=𝑥²_𝑈(𝑛)+𝑎. Hence, the Bayesian estimator 𝜃_BG of 𝜃 under the GE loss function is given by 𝜃_BG=𝑇2^1/2Γ𝑛+𝑏+𝑞/2Γ(𝑛+𝑏)^−1/𝑞, (40) where 𝑇=𝑋²_𝑈(𝑛)+𝑎, 𝑎, and 𝑏 are the parameters of prior distribution with density as (15).

By using (5) and the Bayesian estimator 𝜃_BG as (40), 𝐶_𝐿,BG based on the Bayesian estimator 𝜃_BG of 𝜃 is given by 𝐶_𝐿,BG=𝜋−4−𝜋2𝐿4−𝜋𝜃_BG=𝜋−4−𝜋2⋅𝑇4−𝜋⋅𝐿2^−𝑞/2Γ𝑛+𝑏+𝑞/2Γ(𝑛+𝑏)^1/𝑞, (41) where 𝑇=𝑋²_𝑈(𝑛)+𝑎, 𝑎, and 𝑏 are the parameters of prior distribution with density as (15).

We construct a statistical testing procedure to assess whether the lifetime performance index adheres to the required level. The one-sided credible confidence interval for 𝐶_𝐿 is obtained using the pivotal quantity 𝑡/𝜃²|_{𝑥∼=(𝑥𝑈(1),…,𝑥𝑈(𝑛))}. By using the pivotal quantity 𝑡/𝜃²|_{𝑥∼=(𝑥𝑈(1),…,𝑥𝑈(𝑛))}, given the specified significance level 𝛼, the level 100(1−𝛼)% one-sided credible interval for 𝐶_𝐿 can be derived as follows.

Since the pivotal quantity 𝑡/𝜃²|_{𝑥∼=(𝑥𝑈(1),…,𝑥𝑈(𝑛))}∼𝜒²_2(𝑛+𝑏), and 𝜒²_{2(𝑛+𝑏),1−𝛼} function which represents the lower 1−𝛼 percentile of 𝜒²_2(𝑛+𝑏), then 𝑃𝑡𝜃²≤𝜒²_{2()𝑛+𝑏,1−𝛼}∣𝑥_∼=1−𝛼,where𝑥_∼=𝑥_𝑈(1),𝑥_𝑈(2),…,𝑥_𝑈(𝑛),⟹𝑃𝐶_𝐿≥𝜋−4−𝜋𝜋−𝐶4−𝜋^𝐿,BG×Γ𝑛+𝑏+𝑞/2Γ(𝑛+𝑏)^−1/𝑞𝜒²_{2()𝑛+𝑏,1−𝛼}2^1/2∣𝑥_∼=1−𝛼, (42) where 𝐶_𝐿 as the definition of (5).

From (42), we obtain that a 100(1−𝛼)% one-sided credible interval for 𝐶_𝐿 is 𝐶_𝐿≥𝜋−4−𝜋𝜋−𝐶4−𝜋^𝐿,BG×Γ𝑛+𝑏+𝑞/2Γ(𝑛+𝑏)^−1/𝑞𝜒²_{2()𝑛+𝑏,1−𝛼}2^1/2, (43) where 𝐶_𝐿,BG is given by (41).

Therefore, the 100(1−𝛼)% lower credible interval bound for 𝐶_𝐿 can be written as LB_BG=𝜋−4−𝜋𝜋−𝐶4−𝜋^𝐿,BG×Γ𝑛+𝑏+𝑞/2Γ(𝑛+𝑏)^−1/𝑞𝜒²_{2()𝑛+𝑏,1−𝛼}2^1/2, (44) where 𝐶_𝐿,BG is given by (41), 𝛼 is the specified significance level, and 𝑏 is a parameter of prior distribution with density as (15).

The managers can use the one-sided credible interval to determine whether the product performance attains to the required level. The proposed testing procedure of 𝐶_𝐿 with 𝐶_𝐿,BG can be organized as follows.

Step 1. Determine the lower lifetime limit 𝐿 for products and the performance index value 𝑐₀, then the testing null hypothesis 𝐻₀∶𝐶_𝐿≤𝑐₀ and the alternative hypothesis 𝐻₁∶𝐶_𝐿>𝑐₀ are constructed.

Step 2. Specify a significance level 𝛼.

Step 3. Given the parameters 𝑎 and 𝑏 of prior distribution, the parameter 𝑞 of GE loss function, the upper record values 𝑥_∼=(𝑥_𝑈(1),𝑥_𝑈(2),…,𝑥_𝑈(𝑛)), the lower lifetime limit 𝐿, and the significance level 𝛼, then we can calculate the 100(1−𝛼)% one-sided credible interval [LB_BG,∞) for 𝐶_𝐿, where LB_BG as the definition of (44).

Step 4. The decision rule of statistical test is provided as follows:(1)if the performance index value 𝑐₀∉[LB_BG,∞), then we will reject 𝐻₀. It is concluded that the lifetime performance index of products meets the required level;(2)if the performance index value 𝑐₀∈[LB_BG,∞), then we do not reject 𝐻₀. It is concluded that the lifetime performance index of products does not meet the required level.

6. The Monte Carlo Simulation Algorithm of Confidence (or Credible) Level

In this section, we will report the results of a simulation study for confidence (or credible) level (1−𝛼) based on a 100(1−𝛼)% one-sided confidence (or credible) interval of the lifetime performance index 𝐶_𝐿. We considered 𝛼=0.05 and then generated samples from the Rayleigh distribution with p.d.f. as in (2) with respect to the record values.(i) The Monte Carlo simulation algorithm of confidence level (1−𝛼) for 𝐶_𝐿 under MLE is given in the following steps.

Step 1. Given 𝑛, 𝑎, 𝑏, 𝐿, and 𝛼, where 𝑎>0, 𝑏>0.

Step 2. For the values of prior parameters (𝑎,𝑏), use (15) to generate 𝜃 from the square-root inverted-gamma distribution.

Step 3. (a)The generation of data (𝑍₁,𝑍₂,…,𝑍_𝑛) is by standard exponential distribution with parameter 1.(b)Set 𝑌²₁=2𝜃²𝑍₁ and 𝑌²_𝑖=2𝜃²𝑍_𝑖+𝑌²_𝑖−1, for 𝑖=2,…,𝑛. 𝑌₁=2𝜃²𝑍₁,𝑌_𝑖=2𝜃²𝑍_𝑖+𝑌²_𝑖−1,for𝑖=2,…,𝑛. (45)(𝑌₁,𝑌₂,…,𝑌_𝑛) are the upper record values from the Rayleigh distribution.(c)The value of LB_MLE is calculated by LB_MLE=𝜋−4−𝜋𝜋−𝐶4−𝜋^𝐿,MLE𝜒²_{2𝑛,1−𝛼}2𝑛^1/2, (46) where 𝐶_𝐿,MLE is given by (11) and 𝜒²_{2𝑛,1−𝛼} function which represents the lower 1−𝛼 percentile of 𝜒²_2𝑛.(d)If 𝐶_𝐿≥LB_MLE, then count = 1; else count = 0.

Step 4. (a)Step 3 is repeated 1000 times.(b)The estimation of confidence level (1−𝛼) is (1−𝛼)=(totalcount)/1000.

Step 5. (a)Repeat Step 2–Step 4 for 100 times, then we can get the 100 estimations of confidence level as follows: 1−𝛼₁,1−𝛼₂,…,1−𝛼₁₀₀. (47)(b)The average empirical confidence level 1−𝛼 of (1−𝛼)_𝑖, 𝑖=1,…,100; that is, 1−𝛼=(1/100)∑¹⁰⁰_𝑖=1(1−𝛼)_𝑖.(c)The sample mean square error (SMSE) of (1−𝛼)₁,(1−𝛼)₂,…,(1−𝛼)₁₀₀SMSE = (1/100)∑¹⁰⁰_𝑖=1[(1−𝛼)_𝑖−(1−𝛼)]².

The results of simulation are summarized in Table 2 based on 𝐿=1.0, the different value of sample size 𝑛, prior parameter (𝑎,𝑏)=(2,5), (6,1.5), and (2,2) at 𝛼=0.05, respectively. The scope of SMSE is between 0.00413 and 0.00593.

Table 2 

Average empirical confidence level (1−𝛼) for 𝐶_𝐿 under MLE when 𝛼=0.05.

Step 1. Given 𝑛, 𝑎, 𝑏, 𝐿, and 𝛼, where 𝑎>0, 𝑏>0.

Step 2. For the values of prior parameters (𝑎,𝑏), use (15) to generate 𝜃 from the square-root inverted-gamma distribution.

Step 3. (a)The generation of data (𝑍₁,𝑍₂,…,𝑍_𝑛) is by standard exponential distribution with parameter 1.(b)Set 𝑌²₁=2𝜃²𝑍₁ and 𝑌²_𝑖=2𝜃²𝑍_𝑖+𝑌²_𝑖−1, for 𝑖=2,…,𝑛. 𝑌₁=2𝜃²𝑍₁,𝑌_𝑖=2𝜃²𝑍_𝑖+𝑌²_𝑖−1,for𝑖=2,…,𝑛. (48)(𝑌₁,𝑌₂,…,𝑌_𝑛) are the upper record values from the Rayleigh distribution.(c)The values of LB_BS, LB_BL, and LB_BG are calculated by (25), (36), and (44), respectively.(d)If 𝐶_𝐿≥LB, then count = 1; else count = 0, where LB=LB_BS or LB_BL or LB_BG.

Step 4. (a)Step 3 is repeated 1000 times.(b)The estimation of credible level (1−𝛼) is (1−𝛼)=(totalcount)/1000.

Step 5. (a)Repeat Step 2–Step 4 for 100 times, then we can get the 100 estimations of credible level as follows: 1−𝛼₁,1−𝛼₂,…,1−𝛼₁₀₀. (49)(b)The average empirical credible level 1−𝛼 of (1−𝛼)_𝑖, 𝑖=1,…,100; that is, 1−𝛼=(1/100)∑¹⁰⁰_𝑖=1(1−𝛼)_𝑖.(c)The sample mean square error (SMSE) of (1−𝛼)₁,(1−𝛼)₂,…,(1−𝛼)₁₀₀SMSE= (1/100)∑¹⁰⁰_𝑖=1[(1−𝛼)_𝑖−(1−𝛼)]².

Based on 𝐿=1.0, the different value of sample size 𝑛, prior parameter (𝑎,𝑏)={(2,5),(6,1.5),(2,2)}, and 𝛼=0.05, the results of simulation are summarized in Tables 3–5 under SE loss function, LINEX loss function with parameter 𝑐={−0.5,0.5,1.5}, and GE loss function with parameter 𝑞={3,5,8}, respectively. The following points can be drawn.(a) All of the SMSEs are small enough and the scope of SMSE is between 0.00346 and 0.00467.(b)Fix the size 𝑛, comparison of prior parameter (𝑎,𝑏)=(2,5), (6,1.5), and (2,2) as follows:(i) the values of SMSE with prior parameter (𝑎,𝑏)=(6,1.5) and (2,5) are smaller than (𝑎,𝑏)=(2,2), respectively;(ii) a comparison of prior parameter (𝑎,𝑏)=(2,5), and (2,2); that is, fix prior parameter 𝑎, if prior parameter 𝑏 increases, then it can be seen that the SMSE will decrease.(c)In Table 4, fix the size 𝑛 and the prior parameter (𝑎,𝑏), comparison the parameter of LINEX loss function 𝑐={−0.5,0.5,1.5} as follows: the values of SMSE with prior parameter 𝑐=0.5, 1.5, and −1.5 are the same as the values of SMSE in Table 3.(d)In Table 5, fix the prior parameter (𝑎,𝑏), comparison of GE loss function parameter 𝑞={3,5,8} as follows: the values of SMSE with prior parameter 𝑞=3,5, and 8 are the same as the values of SMSE in Table 3.Hence, these results from simulation studies illustrate that the performance of our proposed method is acceptable. Moreover, we suggest that the prior parameter (𝑎,𝑏)=(6,1.5) and (2,5) is appropriate for the square-root inverted-gamma distribution.

Table 3 

Average empirical credible level (1−𝛼) for 𝐶_𝐿 under SE loss function when 𝛼=0.05.

Table 4 

Average empirical credible level (1−𝛼) for 𝐶_𝐿 under LINEX loss function when 𝛼=0.05.

Table 5 

Average empirical credible level (1−𝛼) for 𝐶_𝐿 under GE loss function when 𝛼=0.05.

7. Numerical Example

In this section, we propose the new hypothesis testing procedures to a real-life data. In the following example, we discuss a real-life data for 25 ball bearings to illustrate the use of the new hypothesis testing procedures in the lifetime performance of ball bearings. The proposed testing procedures not only can handle nonnormal lifetime data, but also can handle the upper record values.

Example. The data is the failure times of 25 ball bearings in endurance test. The 25 observations are the number of million revolutions before failure for each of the ball bearings. The data come from Caroni [27] as follows:

67.80, 67.80, 67.80, 68.64, 33.00, 68.64, 98.64, 128.04, 42.12, 28.92, 45.60, 51.84, 55.56, 173.40, 48.48, 17.88, 93.12, 54.12, 41.52, 51.96, 127.92, 84.12, 105.12, 105.84, and 68.88.

Raqab and Madi [28] and Lee [29] indicated that a one-parameter Rayleigh distribution is acceptable for these data. In addition, we also have a way to test the hypothesis that the failure data come from the Rayleigh distribution. The testing hypothesis 𝐻₀∶𝑋 comes from a Rayleigh distribution versus 𝐻₁∶𝑋 does not come from a Rayleigh distribution is constructed under significance level 𝛼=0.05.

From a probability plot of Figure 1 by using the Minitab Statistical Software, we can conclude the operational lifetimes data of 25 ball bearings from the Rayleigh distribution which is the Weibull distribution with the shape 2 (also see Lee et al. [14]). For informative prior, we use the prior information: 𝐸(𝜃)=𝜃_MLE≐54.834 and Var(𝜃)=0.2, giving the prior parameter values as (𝑎=6.014,𝑏=1.001).

Figure 1 

Probability plot for failure times of 25 ball bearings data.

Here, if only the upper record values have been observed, these are {𝑥_𝑈(𝑖),𝑖=1,…,5(=𝑛)}={67.80,68.64,98.64,128.04,173.40}.(1) The proposed testing procedure of 𝐶_𝐿 with 𝐶_𝐿,MLE is stated as follows.

Step 1. The record values from the above data are (𝑥_𝑈(𝑖),𝑖=1,2,3,4,5)=(67.80,68.64,98.64,128.04,173.40).

Step 2. The lower specification limit 𝐿 is assumed to be 23.37. The deal with the product managers’ concerns regarding lifetime performance and the conforming rate 𝑃_𝑟 of products is required to exceed 80 percent. Referring to Table 1, 𝐶_𝐿 is required to exceed 0.90. Thus, the performance index value is set at 𝑐₀=0.90. The testing hypothesis 𝐻₀∶𝐶_𝐿≤0.90 versus 𝐻₁∶𝐶_𝐿>0.90 is constructed.

Step 3. Specify a significance level 𝛼=0.05.

Step 4. We can calculate that the 95% lower confidence interval bound for 𝐶_𝐿, where LB_MLE=𝜋−4−𝜋𝜋−𝐶4−𝜋^𝐿,MLE𝜒²_{2𝑛,1−𝛼}2𝑛^1/2=𝜋−4−𝜋𝜋𝜒4−𝜋−1.26251254^210,0.9510^1/2=1.03285. (50)
So, the 95% one-sided confidence interval for 𝐶_𝐿 is [LB_MLE,∞)=[1.03285,∞).

Step 5. Because of the performance index value 𝑐₀=0.90∉[LB_MLE,∞), we reject the null hypothesis 𝐻₀∶𝐶_𝐿≤0.90.

Thus, we can conclude that the lifetime performance index of data meets the required level.(2) The proposed testing procedures of 𝐶_𝐿 with 𝐶_𝐿,BS, 𝐶_𝐿,BL, and 𝐶_𝐿,BG are stated as follows.

Step 1. The record values from the above data are (𝑥_𝑈(𝑖),𝑖=1,2,3,4,5)=(67.80,68.64,98.64,128.04,173.40).

Step 2. The lower specification limit 𝐿 is assumed to be 23.37. The deal with the product managers’ concerns regarding lifetime performance and the conforming rate 𝑃_𝑟 of products is required to exceed 80 percent. Referring to Table 1, 𝐶_𝐿 is required to exceed 0.90. Thus, the performance index value is set at 𝑐₀=0.90. The testing hypothesis 𝐻₀∶𝐶_𝐿≤0.90 versus 𝐻₁∶𝐶_𝐿>0.90 is constructed.

Step 3. Specify a significance level 𝛼=0.05, the parameter of LINEX loss function 𝑐=0.5, and the parameter of GE loss function 𝑞=2.

Step 4. We can calculate the 95% lower credible interval bound for 𝐶_𝐿, where LB_BS=𝜋−4−𝜋𝜋−𝐶4−𝜋^𝐿,BS×Γ(𝑛+𝑏−1/2)Γ()𝜒𝑛+𝑏2^{22(𝑛+𝑏),1−𝛼1/2}=𝜋−4−𝜋𝜋×Γ()4−𝜋−1.2460746855+1.001−1/2Γ()𝜒5+1.0012^{22(5+1.001),0.951/2}=0.96984. (51)
So, the 95% one-sided credible interval for 𝐶_𝐿 is [LB_BS,∞)=[0.96984,∞). LB_BL=𝜋−4−𝜋𝜋−𝐶4−𝜋^𝐿,BL×12𝑐1−𝑒^{−𝑐/(𝑛+𝑏+1)}𝜒²_{2(𝑛+𝑏),1−𝛼}^1/2=𝜋−4−𝜋𝜋×14−𝜋−1.1295610592(0.5)1−𝑒^{−0.5/(5+1.001+1)}𝜒²_{2(5+1.001),0.95}^1/2=0.96984. (52)
So, the 95% one-sided credible interval for 𝐶_𝐿 is [LB_BL,∞)=[0.96984,∞). LB_BG=𝜋−4−𝜋𝜋−𝐶4−𝜋^𝐿,BG×Γ𝑛+𝑏+𝑞/2Τ(𝑛+𝑏)^−1/𝑞𝜒²_{2(𝑛+𝑏),1−𝛼}2^1/2=𝜋−4−𝜋𝜋×Γ()4−𝜋−1.2004329985+1.001+2/2Γ(5+1.001)^−1/2𝜒²_{2(5+1.001),0.95}2^1/2=0.96984. (53)
So, the 95% one-sided credible interval for 𝐶_𝐿 is [LB_BG,∞)=[0.96984,∞).

Step 5. Because of the performance index value 𝑐₀=0.90∉[LB_BS,∞)=[LB_BL,∞)=[LB_BG,∞), we reject the null hypothesis 𝐻₀∶𝐶_𝐿≤0.90.

Thus, we can conclude that the lifetime performance index of data meets the required level. In addition, by using the prior parameter values (𝑎,𝑏)=(6,1.5) based on the simulation studies, we also obtain that 𝑐₀=0.90∉[LB_BS,∞)=[LB_BL,∞)=[LB_BG,∞)=0.94033. So, we also reject the null hypothesis 𝐻₀∶𝐶_𝐿≤0.90. Hence, the lifetime performance index of data meets the required level.

8. Conclusions

Montgomery [1] proposed a process capability index 𝐶_𝐿 for larger-the-better quality characteristic. The assumption of most process capability is normal distribution, but it is often invalid. In this paper, we consider that Rayleigh distribution is the special case of Weibull distribution.

Moreover, in lifetime data testing experiments, the experimenter may not always be in a position to observe the life times of all the products put to test. In industry and reliability studies, many products fail under stress, for example, an electronic component ceases to function in an environment of too high temperature, and a battery dies under the stress of time. So, in such experiments, measurement may be made sequentially and only the record values are observed.

In order to let the process capability, indices can be effectively used. This study constructs MLE and Bayesian estimator of 𝐶_𝐿 under assuming the conjugate prior distribution and SE loss function, LINEX loss function, and GE loss function based on the upper record values from the Rayleigh distribution. The Bayesian estimator of 𝐶_𝐿 is then utilized to develop a credible interval in the condition of known 𝐿.

This study also provides a table of the lifetime performance index with its corresponding conforming rate based on the Rayleigh distribution. That is, the conforming rate and the corresponding 𝐶_𝐿 can be obtained.

Further, the Bayesian estimator and posterior distribution of 𝐶_𝐿 are also utilized to construct the testing procedure of 𝐶_𝐿 which is based on a credible interval. If you want to test whether the products meet the required level, you can utilize the proposed testing procedure which is easily applied and an effectively method to test in the condition of known 𝐿. Numerical example is illustrated to show that the proposed testing procedure is effectively evaluating whether the true performance index meets requirements.

In addition, these results from simulation studies illustrate that the performance of our proposed method is acceptable. According to SMSE, the Bayesian approach is smaller than the non-Bayesian approach, we suggest that the Bayesian approach is better than the non-Bayesian approach. Then, the SMSEs of prior parameters (𝑎,𝑏)=(6,1.5) and (2,5) are smaller than the SMSE of prior parameter (𝑎,𝑏)=(2,2). We suggest that the prior parameters (𝑎,𝑏)=(6,1.5) and (2,5) are appropriate for the square-root inverted-gamma distribution based on the Bayesian estimators under the Rayleigh distribution.

Acknowledgments

The authors thank the referees for their suggestions and helpful comments on revising the paper. This research was partially supported by the National Science Council, China (Plans nos. NSC 101-2221-E-158-003, NSC 99-2118-M-415-001, and NSC 100-2118-M-415-002). Moreover, J. W. Wu and C. W. Hong have a direct financial relation with the National Science Council, Taiwan, but other authors do not a direct financial relation with the National Science Council, Taiwan.