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

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 indexC 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).


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 reallife 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  1 ,  2 , . . .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   ,  ≥ 1 is defined as follows.
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  > 0 and  > 0, respectively.When  comes from the Rayleigh distribution, the mean of  is () = √/2, and the standard deviation of  is  = √(4 − )/2.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.

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  = () = √/2 is the process mean,  = √Var() = √(4 − )/2 is the process standard deviation, and  is the lower specification limit.

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   .

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  0 , where  0 denotes the target value, the null hypothesis  0 :   ≤  0 and the alternative hypothesis  1 :   >  0 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  ∼ = (  (1) ,  (2) , . . .,  () ) 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 ( (1) ,  (2) , . . .,  () ) is where () and () are the p.d.f. and c.d.f. of , respectively (also see [12,15,16]).So, the likelihood function of ( (1) ,  (2) , . . .,  () ) 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  2 () / 2 .By using the pivotal quantity  2 () / 2 , given the specified significance level , the level 100(1 − )% one-sided confidence interval for   can be derived as follows.
Since the pivotal quantity where   = √/(4 − ) − √2/(4 − )(/).From ( 12), we obtain that a 100(1 − )% one-sided confidence interval for   is where Ĉ,MLE is given by (11).Therefore, the 100(1−)% lower confidence interval bound for   can be written as where Ĉ,MLE , , 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 Ĉ,MLE can be organized as follows.
Step 1. Determine the lower lifetime limit  for products and the performance index value  0 , then the testing null hypothesis  0 :   ≤  0 and the alternative hypothesis  1 :   >  0 are constructed.
Step 2. Specify a significance level .
Step 4. The decision rule of statistical test is provided as follows: (1) if the performance index value  0 ∉ [LB MLE , ∞), then we will reject  0 .It is concluded that the lifetime performance index of products meets the required level; (2) if the performance index value  0 ∈ [LB MLE , ∞), then we do not reject  0 .It is concluded that the lifetime performance index of products does not meet the required level.

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   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  0 , where  0 denotes the target value, the null hypothesis  0 :   ≤  0 and the alternative hypothesis  1 :   >  0 are constructed.Based on the new hypothesis testing procedure, the lifetime performance of products is easy to assess.

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  > 0,  > 0, and  > 0.
The lifetime performance index   of Rayleigh products can be written as By using (5) and the Bayesian estimator θBS as (20), Ĉ,BS based on the Bayesian estimator θBS of  is given by where  =  2 () + ,  +  − 1/2 > 0, , 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 / 2 |  ∼ =( (1) ,..., () ) .
Since the pivotal quantity / where where   as the definition of (5).From (23), we obtain that a 100(1−)% one-sided credible interval for   is given by where Ĉ,BS is given by (22).Therefore, the 100(1 − )% lower credible interval bound for   can be written as where Ĉ,BS 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 Ĉ,BS can be organized as follows.
Step 1. Determine the lower lifetime limit  for products and the performance index value  0 , then the testing null hypothesis  0 :   ≤  0 and the alternative hypothesis  1 :   >  0 are constructed.
Step 2. Specify a significance level .
Step 4. The decision rule of statistical test is provided as follows: (1) if the performance index value  0 ∉ [LB BS , ∞), then we will reject  0 .It is concluded that the lifetime performance index of products meets the required level; (2) if the performance index value  0 ∈ [LB BS , ∞), then we do not reject  0 .It is concluded that the lifetime performance index of products does not meet the required level.
The value of  * that minimizes [(Δ 1 ) |  ∼ ] denoted by θBL is obtained by solving the equation: that is, θBL is the solution to the following equation: By using ( 18) and (30), we have where  =  2 () + .
Hence, the Bayesian estimator of  under the LINEX loss function is given by where  =  2 () + , , and  are the parameters of prior distribution with density as (15).
By using (5) and the Bayesian estimator θBL as (32), Ĉ,BL based on the Bayesian estimator θBL of  is given by where  =  2 () + , , 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 / 2 |  ∼ =( (1) ,..., () ) .
Step 1. Determine the lower lifetime limit  for products and the performance index value  0 , then the testing null hypothesis  0 :   ≤  0 and the alternative hypothesis  1 :   >  0 are constructed.
Step 2. Specify a significance level .
Step 4. The decision rule of statistical test is provided as follows: (1) if the performance index value  0 ∉ [LB BL , ∞), then we will reject  0 .It is concluded that the lifetime performance index of products meets the required level; (2) if the performance index value  0 ∈ [LB BL , ∞), then we do not reject  0 .It is concluded that the lifetime performance index of products does not meet the required level.

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  = 1.When  > 0, a positive error ( * > ) causes more serious consequences than a negative error.The Bayesian estimator θBG of  under GE loss function is given by By using ( 18) and (38), then the Bayesian estimator θBG of  is derived as follows: where  =  2 () + .Hence, the Bayesian estimator θBG of  under the GE loss function is given by θBG = (  2 ) where  =  2 () + , , and  are the parameters of prior distribution with density as (15).
By using (5) and the Bayesian estimator θBG as (40), Ĉ,BG based on the Bayesian estimator θBG of  is given by where  =  2 () + , , 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 / 2 |  ∼ =( (1) ,..., () ) .
Step 1. Determine the lower lifetime limit  for products and the performance index value  0 , then the testing null hypothesis  0 :   ≤  0 and the alternative hypothesis  1 :   >  0 are constructed.
Step 2. Specify a significance level .
Step 4. The decision rule of statistical test is provided as follows: (1) if the performance index value  0 ∉ [LB BG , ∞), then we will reject  0 .It is concluded that the lifetime performance index of products meets the required level; (2) if the performance index value  0 ∈ [LB BG , ∞), then we do not reject  0 .It is concluded that the lifetime performance index of products does not meet the required level.

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 − ) based on a 100(1 − )% one-sided confidence (or credible) interval of the lifetime performance index   .We considered  = 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 − ) for   under MLE is given in the following steps.
Step 2. For the values of prior parameters (, ), use (15) to generate  from the square-root inverted-gamma distribution.
(a) Repeat Step 2-Step 4 for 100 times, then we can get the 100 estimations of confidence level as follows: The results of simulation are summarized in Table 2 based on  = 1.0, the different value of sample size , prior parameter (, ) = (2, 5), (6, 1.5), and (2, 2) at  = 0.05, respectively.The scope of SMSE is between 0.00413 and 0.00593.
Step 2. For the values of prior parameters (, ), use (15) to generate  from the square-root inverted-gamma distribution.
(c) In Table 4, fix the size  and the prior parameter (, ), comparison the parameter of LINEX loss function  = {−0.5,0.5, 1.5} as follows: the values of SMSE with prior parameter  = 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 (, ), comparison of GE loss function parameter  = {3, 5, 8} as follows: the values of SMSE with prior parameter  = 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 (, ) = (6, 1.5) and (2, 5) is appropriate for the square-root inverted-gamma distribution.Raqab and Madi [28] and Lee [29] indicated that a oneparameter 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  0 :  comes from a Rayleigh distribution versus  1 :  does not come from a Rayleigh distribution is constructed under significance level  = 0.05.

Numerical Example
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: () = θMLE ≐ 54.834 and Var() = 0.2, giving the prior parameter values as ( = 6.014,  = 1.001).
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  0 = 0.90.The testing hypothesis  0 :   ≤ 0.90 versus  1 :   > 0.90 is constructed.
Step 4. We can calculate that the 95% lower confidence interval bound for   , where 10,0.95 10 ) So, the 95% one-sided confidence interval for Step 5.Because of the performance index value  0 = 0.90 ∉ [LB MLE , ∞), we reject the null hypothesis  0 :   ≤ 0.90.Thus, we can conclude that the lifetime performance index of data meets the required level.
(2) The proposed testing procedures of   with Ĉ,BS , Ĉ,BL , and Ĉ,BG are stated as follows.
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  0 = 0.90.The testing hypothesis  0 :   ≤ 0.90 versus  1 :   > 0.90 is constructed.
Step 3. Specify a significance level  = 0.05, the parameter of LINEX loss function  = 0.5, and the parameter of GE loss function  = 2.

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 (, ) = (6, 1.5) and (2, 5) are smaller than the SMSE of prior parameter (, ) = (2, 2).We suggest that the prior parameters (, ) = (6, 1.5) and (2, 5) are appropriate for the square-root inverted-gamma distribution based on the Bayesian estimators under the Rayleigh distribution.

Figure 1 :
Figure 1: Probability plot for failure times of 25 ball bearings data.

Table 1 :
The lifetime performance index   versus the conforming rate   .

Table 2 :
Average empirical confidence level (1 − ) for   under MLE when  = 0.05.denotes the sample size; the values in parentheses are sample mean square error of ( 1 − ).

Table 3 :
Average empirical credible level (1 − ) for   under SE loss function when  = 0.05. denotes the sample size; the values in parentheses are sample mean square error of ( 1 − ).

Table 4 :
Average empirical credible level (1 − ) for   under LINEX loss function when  = 0.05. denotes the sample size; the values in parentheses are sample mean square error of ( 1 − ).

Table 5 :
Average empirical credible level (1 − ) for   under GE loss function when  = 0.05. denotes the sample size; the values in parentheses are sample mean square error of ( 1 − ).