#### 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 is utilized to measure the performance of product, where is the lower specification limit. This study constructs a Bayesian estimator of 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 are utilized to construct the testing procedure for based on a credible interval in the condition of known . 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 to evaluate the lifetime performance of electronic components, where is the lower specification limit. Tong et al. [2] constructed the uniformly minimum variance unbiased estimator (UMVUE) of and proposed a hypothesis testing procedure for the complete sample from a one-parameter exponential distribution. In addition, the lifetime performance index 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 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 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 to evaluate product quality for exponential distribution under the progressively type II right censored sample. Lee et al. [7] also constructed an UMVUE of 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 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 be a sequence of independent and identically distributed (i.i.d.) random variables having the same distribution as the (population) random variable with the cumulative distribution function (c.d.f) and the probability density function (p.d.f) . An observation will be called an upper record value if it exceeds in value all of the preceding observations, that is, if , for every . The sequence of record times , is defined as follows.

Let with probability 1, for , . A sequence of upper record values is then defined by Let be the first upper record values arising from a sequence of i.i.d. random variables with c.d.f. .

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 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 where and , respectively. When comes from the Rayleigh distribution, the mean of is , and the standard deviation of is .

Bayesian and non-Bayesian approaches have been used to obtain the estimators of the parameter 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 under a Rayleigh distribution with upper record values. The Bayesian estimators of are developed under symmetric and nonsymmetric loss functions. The estimators of are then utilized to develop a credible interval, respectively. The new testing procedures of credible interval for can be employed by managers to assess whether the products performance adheres to the required level in the condition of known . 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 with the non-Bayesian approach. Section 5 develops a hypothesis testing procedure for the lifetime performance index 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 to measure the larger-the-better quality characteristic. Then, is defined by where , , and are the process mean, the process standard deviation, and the lower specification limit, respectively.

To assess the product performance of products, can be defined as the lifetime performance index. If comes from the Rayleigh distribution, then the lifetime performance index can be rewritten as where is the process mean, is the process standard deviation, and is the lower specification limit.

#### 3. The Conforming Rate

If the lifetime of a product exceeds the lower specification limit , 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

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

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

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

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

Let be the lifetime of such a product, and has a Rayleigh distribution with the p.d.f. as given by (2). Let be the first 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 is where and are the p.d.f. and c.d.f. of , respectively (also see [12, 15, 16]). So, the likelihood function of is given as The natural logarithm of the likelihood function with (8) is Upon differentiating (9) with respect to and equating the result to zero, the MLE of the parameter can be shown to be (also see [12]). By using the invariance of MLE (see Zehna [17]), the MLE of can be written as given by

Construct a statistical testing procedure to assess whether the lifetime performance index adheres to the required level. The one-sided confidence interval for is obtained using the pivotal quantity . By using the pivotal quantity , given the specified significance level , the level one-sided confidence interval for can be derived as follows.

Since the pivotal quantity , and function which represents the lower percentile of , then where . From (12), we obtain that a one-sided confidence interval for is where is given by (11). Therefore, the lower confidence interval bound for can be written as where , , and denote the MLE of , 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 with can be organized as follows.

*Step 1. *Determine the lower lifetime limit for products and the performance index value , then the testing null hypothesis and the alternative hypothesis are constructed.

*Step 2. *Specify a significance level .

*Step 3. *Given the upper record values , the lower lifetime limit , and the significance level , then we can calculate the one-sided confidence interval for , where as the definition of (14).

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

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

This section will apply the Bayesian approach to construct an estimator of 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 is then utilized to develop a new hypothesis testing procedure in the condition of known . Assuming that the required index value of lifetime performance is larger than , where denotes the target value, the null hypothesis and the alternative hypothesis 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 with the Bayesian Estimator under Squared-Error Loss Function

Let be the lifetime of such a product, and 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: where , , and . With record values, products (or items) take place on test. Consider is the upper record values. We can obtain that the posterior p.d.f. of is given as where , and the likelihood function as (8).

Let , then the posterior p.d.f. can be rewritten as and we can show that .

The Bayesian estimator of based on the squared error (SE) loss function can be derived as follows: where , .

Hence, the Bayesian estimator of based on SE loss function is given by where , , , and are the parameters of prior distribution with density as (15).

The lifetime performance index of Rayleigh products can be written as By using (5) and the Bayesian estimator as (20), based on the Bayesian estimator of is given by where , , , and 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 are obtained using the pivotal quantity . By using the pivotal quantity , given the specified significance level , the level one-sided credible interval for can be derived as follows.

Since the pivotal quantity , and function which represents the lower percentile of , then where as the definition of (5).

From (23), we obtain that a one-sided credible interval for is given by where is given by (22).

Therefore, the lower credible interval bound for can be written as where is given by (22), and 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 with can be organized as follows.

*Step 1. *Determine the lower lifetime limit for products and the performance index value , then the testing null hypothesis and the alternative hypothesis are constructed.

*Step 2. *Specify a significance level .

*Step 3. *Given the parameters and of prior distribution, the upper record values , the lower lifetime limit , and the significance level , then we can calculate the one-sided credible interval for , where as the definition of (25).

*Step 4. *The decision rule of statistical test is provided as follows:(1)if the performance index value , then we will reject . It is concluded that the lifetime performance index of products meets the required level;(2)if the performance index value , then we do not reject . 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 with the Bayesian Estimator under Linear Exponential Loss Function

Let be the lifetime of such a product, and 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]) where , .

In this paper, we suppose that where is an estimator of .

By using (26)-(27), the posterior expectation of LINEX loss function is where .

The value of that minimizes denoted by is obtained by solving the equation: that is, is the solution to the following equation: By using (18) and (30), we have Hence, the Bayesian estimator of under the LINEX loss function is given by where , , and are the parameters of prior distribution with density as (15).

By using (5) and the Bayesian estimator as (32), based on the Bayesian estimator of is given by where , , and 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 is obtained using the pivotal quantity . By using the pivotal quantity , given the specified significance level , the level one-sided credible interval for can be derived as follows.

Since the pivotal quantity , and function which represents the lower percentile of , then where as the definition of (5).

From (34), we obtain that a one-sided credible interval for is where is given by (33).

Therefore, the lower credible interval bound for can be written as where is given by (33), and 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 with can be organized as follows.

*Step 1. *Determine the lower lifetime limit for products and the performance index value , then the testing null hypothesis and the alternative hypothesis are constructed.

*Step 2. *Specify a significance level .

*Step 3. *Given the parameters and of prior distribution, the parameter of LINEX loss function, the upper record values , the lower lifetime limit , and the significance level , then we can calculate the one-sided credible interval for , where as the definition of (36).

*Step 4. *The decision rule of statistical test is provided as follows:(1)if the performance index value , then we will reject . It is concluded that the lifetime performance index of products meets the required level;(2)if the performance index value , then we do not reject . 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 with the Bayesian Estimator under General Entropy Loss Function

Let be the lifetime of such a product, and 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]): whose minimum occurs at .

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 . When , a positive error causes more serious consequences than a negative error. The Bayesian estimator of under GE loss function is given by

By using (18) and (38), then the Bayesian estimator of is derived as follows: where . Hence, the Bayesian estimator of under the GE loss function is given by where , , and are the parameters of prior distribution with density as (15).

By using (5) and the Bayesian estimator as (40), based on the Bayesian estimator of is given by where , , and 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 is obtained using the pivotal quantity . By using the pivotal quantity , given the specified significance level , the level one-sided credible interval for can be derived as follows.

Since the pivotal quantity , and function which represents the lower percentile of , then where as the definition of (5).

From (42), we obtain that a one-sided credible interval for is where is given by (41).

Therefore, the lower credible interval bound for can be written as where is given by (41), is the specified significance level, and 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 with can be organized as follows.

*Step 1. *Determine the lower lifetime limit for products and the performance index value , then the testing null hypothesis and the alternative hypothesis are constructed.

*Step 2. *Specify a significance level .

*Step 3. *Given the parameters and of prior distribution, the parameter of GE loss function, the upper record values , the lower lifetime limit , and the significance level , then we can calculate the one-sided credible interval for , where as the definition of (44).

*Step 4. *The decision rule of statistical test is provided as follows:(1)if the performance index value , then we will reject . It is concluded that the lifetime performance index of products meets the required level;(2)if the performance index value , then we do not reject . 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 based on a one-sided confidence (or credible) interval of the lifetime performance index . We considered 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 for under MLE is given in the following steps.

*Step 1. *Given , , , , and , where , .

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

*Step 3. *(a)The generation of data is by standard exponential distribution with parameter .(b)Set and , for . are the upper record values from the Rayleigh distribution.(c)The value of is calculated by
where is given by (11) and function which represents the lower percentile of .(d)If , then count = 1; else count = 0.

*Step 4. *(a)Step 3 is repeated 1000 times.(b)The estimation of confidence level is .

*Step 5. *(a)Repeat Step 2–Step 4 for 100 times, then we can get the 100 estimations of confidence level as follows:
(b)The average empirical confidence level of , ; that is, .(c)The sample mean square error (SMSE) of = .

The results of simulation are summarized in Table 2 based on , the different value of sample size , prior parameter , , and at , respectively. The scope of SMSE is between 0.00413 and 0.00593.

*Step 1. *Given , , , , and , where , .

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

*Step 3. *(a)The generation of data is by standard exponential distribution with parameter .(b)Set and , for . are the upper record values from the Rayleigh distribution.(c)The values of , , and are calculated by (25), (36), and (44), respectively.(d)If , then count = 1; else count = 0, where or or .

*Step 4. *(a)Step 3 is repeated 1000 times.(b)The estimation of credible level is .

*Step 5. *(a)Repeat Step 2–Step 4 for 100 times, then we can get the 100 estimations of credible level as follows:
(b)The average empirical credible level of , ; that is, .(c)The sample mean square error (SMSE) of = .

Based on , the different value of sample size , prior parameter , and , the results of simulation are summarized in Tables 3–5 under SE loss function, LINEX loss function with parameter , and GE loss function with parameter , 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 , comparison of prior parameter , , and as follows:(i) the values of SMSE with prior parameter and are smaller than , respectively;(ii) a comparison of prior parameter , and ; that is, fix prior parameter , if prior parameter increases, then it can be seen that the SMSE will decrease.(c)In Table 4, fix the size and the prior parameter , comparison the parameter of LINEX loss function as follows: the values of SMSE with prior parameter , , and are the same as the values of SMSE in Table 3.(d)In Table 5, fix the prior parameter , comparison of GE loss function parameter as follows: the values of SMSE with prior parameter , and 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 and is appropriate for the square-root inverted-gamma distribution.

#### 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 comes from a Rayleigh distribution versus does not come from a Rayleigh distribution is constructed under significance level .

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: and , giving the prior parameter values as .

Here, if only the upper record values have been observed, these are .(1) The proposed testing procedure of with is stated as follows.

*Step 1. *The record values from the above data are .

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

*Step 3. *Specify a significance level .

*Step 4. *We can calculate that the lower confidence interval bound for , where

So, the one-sided confidence interval for is .

*Step 5. *Because of the performance index value , we reject the null hypothesis .

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

*Step 1. *The record values from the above data are .

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

*Step 3. *Specify a significance level , the parameter of LINEX loss function , and the parameter of GE loss function .

*Step 4. *We can calculate the lower credible interval bound for , where

So, the one-sided credible interval for is .

So, the one-sided credible interval for is .

So, the one-sided credible interval for is .

*Step 5. *Because of the performance index value , we reject the null hypothesis .

Thus, we can conclude that the lifetime performance index of data meets the required level. In addition, by using the prior parameter values based on the simulation studies, we also obtain that . So, we also reject the null hypothesis . Hence, the lifetime performance index of data meets the required level.

#### 8. Conclusions

Montgomery [1] proposed a process capability index 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 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 is then utilized to develop a credible interval in the condition of known .

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 can be obtained.

Further, the Bayesian estimator and posterior distribution of are also utilized to construct the testing procedure of 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 . 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 and are smaller than the SMSE of prior parameter . We suggest that the prior parameters and 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.