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

Authors: Wen-Chuan Lee; Jong-Wuu Wu; Ching-Wen Hong; Shie-Fan Hong
Journal: Journal of Applied Mathematics
 (2013)

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## 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 indexCL is utilized to measure the performance of product, where L is the lower specification limit. This study constructs a Bayesian estimator of CL 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 CL are utilized to construct the testing procedure for CL 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).


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## Body

## 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 CL 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 CL and proposed a hypothesis testing procedure for the complete sample from a one-parameter exponential distribution. In addition, the lifetime performance index CL 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 CL 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 CL 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 CL to evaluate product quality for exponential distribution under the progressively type II right censored sample. Lee et al. [7] also constructed an UMVUE of CL 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 CL 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 Xj will be called an upper record value if it exceeds in value all of the preceding observations, that is, if Xj>Xi, for every i<j. The sequence of record times Tn, n≥1 is defined as follows.LetT1=1 with probability 1, for n≥2, Tn=min{j:Xj>XTn-1}. A sequence of upper record values {XU(n)} is then defined by
(1)XU(n)=XTn,n=1,2,….
Let X~={XU(1),XU(2),…,XU(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 indexCL 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
(2)f(x)=xθ2e-x2/2θ2,(3)F(x)=1-e-x2/2θ2,
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 parameterθ 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 ofCL under a Rayleigh distribution with upper record values. The Bayesian estimators of CL are developed under symmetric and nonsymmetric loss functions. The estimators of CL are then utilized to develop a credible interval, respectively. The new testing procedures of credible interval for CL 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. Section2 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 CL with the non-Bayesian approach. Section 5 develops a hypothesis testing procedure for the lifetime performance index CL 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 CL to measure the larger-the-better quality characteristic. Then, CL is defined by
(4)CL=μ-Lσ,
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,CL can be defined as the lifetime performance index. If X comes from the Rayleigh distribution, then the lifetime performance index CL can be rewritten as
(5)CL=μ-Lσ=π/2θ-L(4-π)/2θ=π4-π-24-πLθ,CL=μ-Lσ=π/2θ-L(4-π)/2θ-∞<CL<π4-π,
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 productX 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
(6)Pr=P(X≥L)=e-(1/2)(π/2-(4-π)/2CL)2,Pr=P(X≥L)-∞<CL<π/(4-π).Obviously, a strictly increasing relationship exists between the conforming ratePr and the lifetime performance index CL. Table 1 lists various CL values and the corresponding conforming rate Pr by using STATISTICA software [13] (also see [14]).Table 1
The lifetime performance indexCL versus the conforming rate Pr.
C
L
P
r
C
L
P
r
C
L
P
r
-
∞
0.000000
0.40
0.611832
1.20
0.896627
-
4.50
0.000147
0.45
0.631685
1.25
0.909965
-
4.00
0.000551
0.50
0.651484
1.30
0.922511
-
3.50
0.001858
0.55
0.671182
1.35
0.934227
-
3.00
0.005628
0.60
0.690734
1.40
0.945076
-
2.50
0.015308
0.65
0.710094
1.45
0.955027
-
2.00
0.037404
0.70
0.729213
1.50
0.964047
-
1.50
0.082094
0.75
0.748044
1.55
0.972109
-
1.00
0.161849
0.80
0.766539
1.60
0.979187
-
0.50
0.286621
0.85
0.784648
1.65
0.985259
0.00
0.455938
0.90
0.802324
1.70
0.990306
0.15
0.513214
0.95
0.819518
1.75
0.994310
0.20
0.532718
1.00
0.836183
1.80
0.997261
0.25
0.552370
1.05
0.852271
1.85
0.999147
0.30
0.572132
1.10
0.867738
1.90
0.999963
0.35
0.591967
1.15
0.882538
1.91
0.999997For theCL values which are not listed in Table 1, the conforming rate Pr can be obtained through interpolation. In addition, since a one-to-one mathematical relationship exists between the conforming rate Pr and the lifetime performance index CL. Therefore, utilizing the one-to-one relationship between Pr and CL, lifetime performance index can be a flexible and effective tool, not only evaluating product performance, but also estimating the conforming rate Pr.


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

This section will apply non-Bayesian approach to construct a maximum likelihood estimator (MLE) ofCL under a Rayleigh distribution with upper record values. The MLE of CL 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:CL≤c0 and the alternative hypothesis H1:CL>c0 are constructed. Based on the new hypothesis testing procedure, the lifetime performance of products is easy to assess.LetX be the lifetime of such a product, and X has a Rayleigh distribution with the p.d.f. as given by (2). Let X~=(XU(1),XU(2),…,XU(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 (XU(1),XU(2),…,XU(n)) is
(7)f(xU(n))∏i=1n-1f(xU(i))1-F(xU(i)),
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 (XU(1),XU(2),…,XU(n)) is given as
(8)L(θ∣xU(1),…,xU(n))={∏i=1nxU(i)θ2}exp(-xU(n)22θ2).
The natural logarithm of the likelihood function with (8) is
(9)lnL(θ∣xU(1),…,xU(n))=∑i=1nln(xU(i))-2nlnθ-xU(n)22θ2.
Upon differentiating (9) with respect to θ and equating the result to zero, the MLE of the parameter θ can be shown to be
(10)θ^MLE=XU(n)2n
(also see [12]). By using the invariance of MLE (see Zehna [17]), the MLE of CL can be written as given by
(11)C^L,MLE=π4-π-24-πLθ^MLE=π4-π-24-π·L·2nXU(n).Construct a statistical testing procedure to assess whether the lifetime performance index adheres to the required level. The one-sided confidence interval forCL is obtained using the pivotal quantity XU(n)2/θ2. By using the pivotal quantity XU(n)2/θ2, given the specified significance level α, the level 100(1-α)% one-sided confidence interval for CL can be derived as follows.Since the pivotal quantityXU(n)2/θ2~χ2n2, and χ2n,1-α2 function which represents the lower 1-α percentile of χ2n2, then
(12)P(XU(n)2θ2≤χ2n,1-α2)=1-α,⟹P(CL≥π4-π-(π4-π-C^L,MLE)(χ2n,1-α22n)1/2)=1-α,
where CL=π/(4-π)-2/(4-π)(L/θ). From (12), we obtain that a 100(1-α)% one-sided confidence interval for CL is
(13)CL≥π4-π-(π4-π-C^L,MLE)(χ2n,1-α22n)1/2,
where C^L,MLE is given by (11). Therefore, the 100(1-α)% lower confidence interval bound for CL can be written as
(14)LB_MLE=π4-π-(π4-π-C^L,MLE)(χ2n,1-α22n)1/2,
where C^L,MLE, α, and n denote the MLE of CL, 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 ofCL with C^L,MLE can be organized as follows.Step 1.
Determine the lower lifetime limitL for products and the performance index value c0, then the testing null hypothesis H0:CL≤c0 and the alternative hypothesis H1:CL>c0 are constructed.Step 2.
Specify a significance levelα.Step 3.
Given the upper record valuesx~=(xU(1),xU(2),…,xU(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 CL, 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 valuec0∉[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 valuec0∈[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 IndexCL with the Bayesian Approach

This section will apply the Bayesian approach to construct an estimator ofCL 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 CL 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:CL≤c0 and the alternative hypothesis H1:CL>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 IndexCL with the Bayesian Estimator under Squared-Error Loss Function

LetX 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:
(15)π1(θ)=abΓ(b)2b-1θ-2b-1e-a/2θ2,
where θ>0, a>0, and b>0. 								(16)E(θ)=(a2)1/2Γ(b-1/2)Γ(b),Var(θ)=a2(b-1)-(a2)[Γ(b-1/2)Γ(b)]2,b>1.
With record values, n products (or items) take place on test. Consider (XU(1),XU(2),…,XU(n)) is the upper record values. We can obtain that the posterior p.d.f. of θ∣(XU(1),XU(2),…,XU(n)) is given as
(17)p1(θ∣xU(1),…,xU(n))=L(θ∣xU(1),…,xU(n))π1(θ)∫0∞L(θ∣xU(1),…,xU(n))π1(θ)dθ=θ-2(n+b)-1(xU(n)2+a)(n+b)e-(xU(n)+a)/2θ2Γ(n+b)2n+b-1,
where θ>0, and the likelihood function L(θ∣xU(1),…,xU(n)) as (8).Lett=xU(n)2+a, then the posterior p.d.f. can be rewritten as
(18)p1(θ∣xU(1),…,xU(n))=θ-2(n+b)-1t(n+b)e-t/2θ2Γ(n+b)2n+b-1,
and we can show that t/θ2|x~=(xU(1),…,xU(n))~χ2(n+b)2.The Bayesian estimatorθ^BS of θ based on the squared error (SE) loss function L(θ*,θ)=(θ*-θ)2 can be derived as follows:
(19)E(θ∣xU(1),…,xU(n))=∫0∞θp1(θ∣xU(1),…,xU(n))dθ=Γ(n+b-1/2)Γ(n+b)(t2)1/2,
where t=xU(n)2+a, n+b-1/2>0.Hence, the Bayesian estimator ofθ based on SE loss function is given by
(20)θ^BS=Γ(n+b-1/2)Γ(n+b)(T2)1/2,
where T=XU(n)2+a, n+b-1/2>0, a, and b are the parameters of prior distribution with density as (15).The lifetime performance indexCL of Rayleigh products can be written as
(21)CL=π4-π-24-πLθ,-∞<CL<π4-π.
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
(22)C^L,BS=π4-π-24-πLθ^BS=π4-π-24-π·L·[Γ(n+b-1/2)Γ(n+b)(T2)1/2]-1,
where T=XU(n)2+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 forCL 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 CL can be derived as follows.Since the pivotal quantityt/θ2|x~=(xU(1),…,xU(n))~χ2(n+b)2, and χ2(n+b),1-α2 function which represents the lower 1-α percentile of χ2(n+b)2, then
(23)aaP(tθ2≤χ2(n+b),1-α2∣x~)=1-α,wherex~=(xU(1),xU(2),…,xU(n)),⟹P((χ2(n+b),1-α22)1/2CL≥π4-π-(π4-π-C^L,BS)aaaaaa×Γ(n+b-1/2)Γ(n+b)(χ2(n+b),1-α22)1/2∣x~)=1-α,
where CL as the definition of (5).From (23), we obtain that a 100(1-α)% one-sided credible interval for CL is given by
(24)CL≥π4-π-(π4-π-C^L,BS)×Γ(n+b-1/2)Γ(n+b)(χ2(n+b),1-α22)1/2,
where C^L,BS is given by (22).Therefore, the100(1-α)% lower credible interval bound for CL can be written as
(25)LB_BS=π4-π-(π4-π-C^L,BS)×Γ(n+b-1/2)Γ(n+b)(χ2(n+b),1-α22)1/2,
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 ofCL with C^L,BS can be organized as follows.Step 1.
Determine the lower lifetime limitL for products and the performance index value c0, then the testing null hypothesis H0:CL≤c0 and the alternative hypothesis H1:CL>c0 are constructed.Step 2.
Specify a significance levelα.Step 3.
Given the parametersa and b of prior distribution, the upper record values x~=(xU(1),xU(2),…,xU(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 CL, 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 valuec0∉[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 valuec0∈[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 IndexCL with the Bayesian Estimator under Linear Exponential Loss Function

LetX 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])
(26)L(Δ)∝ecΔ-cΔ-1,
where Δ=(θ*-θ), c≠0.In this paper, we suppose that(27)Δ1=(θ*θ)2-1,
where θ* is an estimator of θ.By using (26)-(27), the posterior expectation of LINEX loss function L(Δ1) is
(28)E[L(Δ1)∣x~]=e-cE[ec(θ*/θ)2∣x~]-cE[(θ*θ)2-1∣x~]-1,
where x~=(xU(1),xU(2),…,xU(n)).The value ofθ* that minimizes E[L(Δ1)∣x~] denoted by θ^BL is obtained by solving the equation:
(29)dE[L(Δ1)∣x~]dθ*=e-cE[ec(θ*/θ)2·2cθ*θ2∣x~]-cE(2θ*θ2∣x~)=0;
that is, θ^BL is the solution to the following equation:
(30)E[θ^BLθ2ec(θ^BL/θ)2∣x~]=ecE(θ^BLθ2∣x~).
By using (18) and (30), we have
(31)2θ^BL(n+b)·tn+b(t-2cθ^BL2)n+b+1=ecθ^BL2(n+b)t,2θ^BL(n+b)aaaaaaaaaaaaaiawheret=xU(n)2+a.
Hence, the Bayesian estimator of θ under the LINEX loss function is given by
(32)θ^BL=[T2c(1-e-c/(n+b+1))]1/2,
where T=XU(n)2+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
(33)C^L,BL=π4-π-24-πLθ^BL=π4-π-24-πL[T2c(1-e-c/(n+b+1))]-1/2,
where T=XU(n)2+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 forCL 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 CL can be derived as follows.Since the pivotal quantityt/θ2|x~=(xU(1),…,xU(n))~χ2(n+b)2, and χ2(n+b),1-α2 function which represents the lower 1-α percentile of χ2(n+b)2, then
(34)aaP(tθ2≤χ2(n+b),1-α2∣x~)=1-α,wherex~=(xU(1),xU(2),…,xU(n)),⟹P([12c(1-e-c/(n+b+1))χ2(n+b),1-α2]1/2CL≥π4-π-(π4-π-C^L,BL)aaaaaa×[12c(1-e-c/(n+b+1))χ2(n+b),1-α2]1/2∣x~(π4-π-C^L,BL))=1-α,
where CL as the definition of (5).From (34), we obtain that a 100(1-α)% one-sided credible interval for CL is
(35)CL≥π4-π-(π4-π-C^L,BL)×[12c(1-e-c/(n+b+1))χ2(n+b),1-α2]1/2,
where C^L,BL is given by (33).Therefore, the100(1-α)% lower credible interval bound for CL can be written as
(36)LB_BL=π4-π-(π4-π-C^L,BL)×[12c(1-e-c/(n+b+1))χ2(n+b),1-α2]1/2,
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 ofCL with C^L,BL can be organized as follows.Step 1.
Determine the lower lifetime limitL for products and the performance index value c0, then the testing null hypothesis H0:CL≤c0 and the alternative hypothesis H1:CL>c0 are constructed.Step 2.
Specify a significance levelα.Step 3.
Given the parametersa and b of prior distribution, the parameter c of LINEX loss function, the upper record values x~=(xU(1),xU(2),…,xU(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 CL, 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 valuec0∉[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 valuec0∈[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 IndexCL with the Bayesian Estimator under General Entropy Loss Function

LetX 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]):
(37)L(θ*,θ)∝(θ*θ)q-qlog(θ*θ)-1,
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
(38)θ^BG=[E(θ-q∣xU(1),…,xU(n))]-1/q.By using (18) and (38), then the Bayesian estimator θ^BG of θ is derived as follows:
(39)E(θ-q∣xU(1),…,xU(n))=∫0∞θ-qp1(θ∣xU(1),…,xU(n))dθ=(t2)-q/2·Γ(n+b+q/2)Γ(n+b),
where t=xU(n)2+a. Hence, the Bayesian estimator θ^BG of θ under the GE loss function is given by
(40)θ^BG=(T2)1/2[Γ(n+b+q/2)Γ(n+b)]-1/q,
where T=XU(n)2+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
(41)C^L,BG=π4-π-24-πLθ^BG=π4-π-24-π·L·[(T2)-q/2Γ(n+b+q/2)Γ(n+b)]1/q,
where T=XU(n)2+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 forCL 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 CL can be derived as follows.Since the pivotal quantityt/θ2|x~=(xU(1),…,xU(n))~χ2(n+b)2, and χ2(n+b),1-α2 function which represents the lower 1-α percentile of χ2(n+b)2, then
(42)aaP(tθ2≤χ2(n+b),1-α2∣x~)=1-α,wherex~=(xU(1),xU(2),…,xU(n)),⟹P([Γ(n+b+q/2)Γ(n+b)]-1/q(χ2(n+b),1-α22)1/2CL≥π4-π-(π4-π-C^L,BG)aaaaaa×[Γ(n+b+q/2)Γ(n+b)]-1/q(χ2(n+b),1-α22)1/2∣x~)=1-α,
where CL as the definition of (5).From (42), we obtain that a 100(1-α)% one-sided credible interval for CL is
(43)CL≥π4-π-(π4-π-C^L,BG)×[Γ(n+b+q/2)Γ(n+b)]-1/q(χ2(n+b),1-α22)1/2,
where C^L,BG is given by (41).Therefore, the100(1-α)% lower credible interval bound for CL can be written as
(44)LB_BG=π4-π-(π4-π-C^L,BG)×[Γ(n+b+q/2)Γ(n+b)]-1/q(χ2(n+b),1-α22)1/2,
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 ofCL with C^L,BG can be organized as follows.Step 1.
Determine the lower lifetime limitL for products and the performance index value c0, then the testing null hypothesis H0:CL≤c0 and the alternative hypothesis H1:CL>c0 are constructed.Step 2.
Specify a significance levelα.Step 3.
Given the parametersa and b of prior distribution, the parameter q of GE loss function, the upper record values x~=(xU(1),xU(2),…,xU(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 CL, 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 valuec0∉[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 valuec0∈[LB_BG,∞), then we do not reject H0. It is concluded that the lifetime performance index of products does not meet the required level.


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

LetX 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:
(15)π1(θ)=abΓ(b)2b-1θ-2b-1e-a/2θ2,
where θ>0, a>0, and b>0. 								(16)E(θ)=(a2)1/2Γ(b-1/2)Γ(b),Var(θ)=a2(b-1)-(a2)[Γ(b-1/2)Γ(b)]2,b>1.
With record values, n products (or items) take place on test. Consider (XU(1),XU(2),…,XU(n)) is the upper record values. We can obtain that the posterior p.d.f. of θ∣(XU(1),XU(2),…,XU(n)) is given as
(17)p1(θ∣xU(1),…,xU(n))=L(θ∣xU(1),…,xU(n))π1(θ)∫0∞L(θ∣xU(1),…,xU(n))π1(θ)dθ=θ-2(n+b)-1(xU(n)2+a)(n+b)e-(xU(n)+a)/2θ2Γ(n+b)2n+b-1,
where θ>0, and the likelihood function L(θ∣xU(1),…,xU(n)) as (8).Lett=xU(n)2+a, then the posterior p.d.f. can be rewritten as
(18)p1(θ∣xU(1),…,xU(n))=θ-2(n+b)-1t(n+b)e-t/2θ2Γ(n+b)2n+b-1,
and we can show that t/θ2|x~=(xU(1),…,xU(n))~χ2(n+b)2.The Bayesian estimatorθ^BS of θ based on the squared error (SE) loss function L(θ*,θ)=(θ*-θ)2 can be derived as follows:
(19)E(θ∣xU(1),…,xU(n))=∫0∞θp1(θ∣xU(1),…,xU(n))dθ=Γ(n+b-1/2)Γ(n+b)(t2)1/2,
where t=xU(n)2+a, n+b-1/2>0.Hence, the Bayesian estimator ofθ based on SE loss function is given by
(20)θ^BS=Γ(n+b-1/2)Γ(n+b)(T2)1/2,
where T=XU(n)2+a, n+b-1/2>0, a, and b are the parameters of prior distribution with density as (15).The lifetime performance indexCL of Rayleigh products can be written as
(21)CL=π4-π-24-πLθ,-∞<CL<π4-π.
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
(22)C^L,BS=π4-π-24-πLθ^BS=π4-π-24-π·L·[Γ(n+b-1/2)Γ(n+b)(T2)1/2]-1,
where T=XU(n)2+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 forCL 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 CL can be derived as follows.Since the pivotal quantityt/θ2|x~=(xU(1),…,xU(n))~χ2(n+b)2, and χ2(n+b),1-α2 function which represents the lower 1-α percentile of χ2(n+b)2, then
(23)aaP(tθ2≤χ2(n+b),1-α2∣x~)=1-α,wherex~=(xU(1),xU(2),…,xU(n)),⟹P((χ2(n+b),1-α22)1/2CL≥π4-π-(π4-π-C^L,BS)aaaaaa×Γ(n+b-1/2)Γ(n+b)(χ2(n+b),1-α22)1/2∣x~)=1-α,
where CL as the definition of (5).From (23), we obtain that a 100(1-α)% one-sided credible interval for CL is given by
(24)CL≥π4-π-(π4-π-C^L,BS)×Γ(n+b-1/2)Γ(n+b)(χ2(n+b),1-α22)1/2,
where C^L,BS is given by (22).Therefore, the100(1-α)% lower credible interval bound for CL can be written as
(25)LB_BS=π4-π-(π4-π-C^L,BS)×Γ(n+b-1/2)Γ(n+b)(χ2(n+b),1-α22)1/2,
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 ofCL with C^L,BS can be organized as follows.Step 1.
Determine the lower lifetime limitL for products and the performance index value c0, then the testing null hypothesis H0:CL≤c0 and the alternative hypothesis H1:CL>c0 are constructed.Step 2.
Specify a significance levelα.Step 3.
Given the parametersa and b of prior distribution, the upper record values x~=(xU(1),xU(2),…,xU(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 CL, 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 valuec0∉[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 valuec0∈[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 IndexCL with the Bayesian Estimator under Linear Exponential Loss Function

LetX 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])
(26)L(Δ)∝ecΔ-cΔ-1,
where Δ=(θ*-θ), c≠0.In this paper, we suppose that(27)Δ1=(θ*θ)2-1,
where θ* is an estimator of θ.By using (26)-(27), the posterior expectation of LINEX loss function L(Δ1) is
(28)E[L(Δ1)∣x~]=e-cE[ec(θ*/θ)2∣x~]-cE[(θ*θ)2-1∣x~]-1,
where x~=(xU(1),xU(2),…,xU(n)).The value ofθ* that minimizes E[L(Δ1)∣x~] denoted by θ^BL is obtained by solving the equation:
(29)dE[L(Δ1)∣x~]dθ*=e-cE[ec(θ*/θ)2·2cθ*θ2∣x~]-cE(2θ*θ2∣x~)=0;
that is, θ^BL is the solution to the following equation:
(30)E[θ^BLθ2ec(θ^BL/θ)2∣x~]=ecE(θ^BLθ2∣x~).
By using (18) and (30), we have
(31)2θ^BL(n+b)·tn+b(t-2cθ^BL2)n+b+1=ecθ^BL2(n+b)t,2θ^BL(n+b)aaaaaaaaaaaaaiawheret=xU(n)2+a.
Hence, the Bayesian estimator of θ under the LINEX loss function is given by
(32)θ^BL=[T2c(1-e-c/(n+b+1))]1/2,
where T=XU(n)2+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
(33)C^L,BL=π4-π-24-πLθ^BL=π4-π-24-πL[T2c(1-e-c/(n+b+1))]-1/2,
where T=XU(n)2+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 forCL 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 CL can be derived as follows.Since the pivotal quantityt/θ2|x~=(xU(1),…,xU(n))~χ2(n+b)2, and χ2(n+b),1-α2 function which represents the lower 1-α percentile of χ2(n+b)2, then
(34)aaP(tθ2≤χ2(n+b),1-α2∣x~)=1-α,wherex~=(xU(1),xU(2),…,xU(n)),⟹P([12c(1-e-c/(n+b+1))χ2(n+b),1-α2]1/2CL≥π4-π-(π4-π-C^L,BL)aaaaaa×[12c(1-e-c/(n+b+1))χ2(n+b),1-α2]1/2∣x~(π4-π-C^L,BL))=1-α,
where CL as the definition of (5).From (34), we obtain that a 100(1-α)% one-sided credible interval for CL is
(35)CL≥π4-π-(π4-π-C^L,BL)×[12c(1-e-c/(n+b+1))χ2(n+b),1-α2]1/2,
where C^L,BL is given by (33).Therefore, the100(1-α)% lower credible interval bound for CL can be written as
(36)LB_BL=π4-π-(π4-π-C^L,BL)×[12c(1-e-c/(n+b+1))χ2(n+b),1-α2]1/2,
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 ofCL with C^L,BL can be organized as follows.Step 1.
Determine the lower lifetime limitL for products and the performance index value c0, then the testing null hypothesis H0:CL≤c0 and the alternative hypothesis H1:CL>c0 are constructed.Step 2.
Specify a significance levelα.Step 3.
Given the parametersa and b of prior distribution, the parameter c of LINEX loss function, the upper record values x~=(xU(1),xU(2),…,xU(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 CL, 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 valuec0∉[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 valuec0∈[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 IndexCL with the Bayesian Estimator under General Entropy Loss Function

LetX 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]):
(37)L(θ*,θ)∝(θ*θ)q-qlog(θ*θ)-1,
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
(38)θ^BG=[E(θ-q∣xU(1),…,xU(n))]-1/q.By using (18) and (38), then the Bayesian estimator θ^BG of θ is derived as follows:
(39)E(θ-q∣xU(1),…,xU(n))=∫0∞θ-qp1(θ∣xU(1),…,xU(n))dθ=(t2)-q/2·Γ(n+b+q/2)Γ(n+b),
where t=xU(n)2+a. Hence, the Bayesian estimator θ^BG of θ under the GE loss function is given by
(40)θ^BG=(T2)1/2[Γ(n+b+q/2)Γ(n+b)]-1/q,
where T=XU(n)2+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
(41)C^L,BG=π4-π-24-πLθ^BG=π4-π-24-π·L·[(T2)-q/2Γ(n+b+q/2)Γ(n+b)]1/q,
where T=XU(n)2+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 forCL 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 CL can be derived as follows.Since the pivotal quantityt/θ2|x~=(xU(1),…,xU(n))~χ2(n+b)2, and χ2(n+b),1-α2 function which represents the lower 1-α percentile of χ2(n+b)2, then
(42)aaP(tθ2≤χ2(n+b),1-α2∣x~)=1-α,wherex~=(xU(1),xU(2),…,xU(n)),⟹P([Γ(n+b+q/2)Γ(n+b)]-1/q(χ2(n+b),1-α22)1/2CL≥π4-π-(π4-π-C^L,BG)aaaaaa×[Γ(n+b+q/2)Γ(n+b)]-1/q(χ2(n+b),1-α22)1/2∣x~)=1-α,
where CL as the definition of (5).From (42), we obtain that a 100(1-α)% one-sided credible interval for CL is
(43)CL≥π4-π-(π4-π-C^L,BG)×[Γ(n+b+q/2)Γ(n+b)]-1/q(χ2(n+b),1-α22)1/2,
where C^L,BG is given by (41).Therefore, the100(1-α)% lower credible interval bound for CL can be written as
(44)LB_BG=π4-π-(π4-π-C^L,BG)×[Γ(n+b+q/2)Γ(n+b)]-1/q(χ2(n+b),1-α22)1/2,
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 ofCL with C^L,BG can be organized as follows.Step 1.
Determine the lower lifetime limitL for products and the performance index value c0, then the testing null hypothesis H0:CL≤c0 and the alternative hypothesis H1:CL>c0 are constructed.Step 2.
Specify a significance levelα.Step 3.
Given the parametersa and b of prior distribution, the parameter q of GE loss function, the upper record values x~=(xU(1),xU(2),…,xU(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 CL, 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 valuec0∉[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 valuec0∈[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 CL. 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 CL under MLE is given in the following steps.Step 1.
Givenn, 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,…,Zn) is by standard exponential distribution with parameter 1.
(b)
SetY12=2θ2Z1 and Yi2=2θ2Zi+Yi-12, for i=2,…,n. 										(45)Y1=2θ2Z1,Yi=2θ2Zi+Yi-12,fori=2,…,n.(Y1,Y2,…,Yn) are the upper record values from the Rayleigh distribution.
(c)
The value ofLB_MLE is calculated by
(46)LB_MLE=π4-π-(π4-π-C^L,MLE)(χ2n,1-α22n)1/2,
where C^L,MLE is given by (11) and χ2n,1-α2 function which represents the lower 1-α percentile of χ2n2.
(d)
IfCL≥LB_MLE, then count = 1; else count = 0.Step 4.
(a)
Step3 is repeated 1000 times.
(b)
The estimation of confidence level(1-α) is (1-α^)=(totalcount)/1000.Step 5.
(a)
Repeat Step2–Step 4 for 100 times, then we can get the 100 estimations of confidence level as follows:
(47)(1-α^)1,(1-α^)2,…,(1-α^)100.
(b)
The average empirical confidence level1-α¯ of (1-α^)i, i=1,…,100; that is, 1-α¯=(1/100)∑i=1100(1-α^)i.
(c)
The sample mean square error (SMSE) of(1-α^)1,(1-α^)2,…,(1-α^)100SMSE = (1/100)∑i=1100[(1-α^)i-(1-α)]2.The results of simulation are summarized in Table2 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 CL under MLE when α=0.05.
n
a
=
2
,
b
=
5
a
=
6
,
b
=
1.5
a
=
2
,
b
=
2
5
0.93653 (0.00462)
0.93482 (0.00467)
0.93577 (0.00507)
10
0.93643 (0.00449)
0.93581 (0.00413)
0.93450 (0.00593)
15
0.93523 (0.00489)
0.93557 (0.00503)
0.93463 (0.00516)
n denotes the sample size; the values in parentheses are sample mean square error of (1-α^).Step 1.
Givenn, 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,…,Zn) is by standard exponential distribution with parameter 1.
(b)
SetY12=2θ2Z1 and Yi2=2θ2Zi+Yi-12, for i=2,…,n. 										(48)Y1=2θ2Z1,Yi=2θ2Zi+Yi-12,fori=2,…,n.(Y1,Y2,…,Yn) are the upper record values from the Rayleigh distribution.
(c)
The values ofLB_BS, LB_BL, and LB_BG are calculated by (25), (36), and (44), respectively.
(d)
IfCL≥LB_, then count = 1; else count = 0, where LB_=LB_BS or LB_BL or LB_BG.Step 4.
(a)
Step3 is repeated 1000 times.
(b)
The estimation of credible level(1-α) is (1-α^)=(totalcount)/1000.Step 5.
(a)
Repeat Step2–Step 4 for 100 times, then we can get the 100 estimations of credible level as follows:
(49)(1-α^)1,(1-α^)2,…,(1-α^)100.
(b)
The average empirical credible level1-α¯ of (1-α^)i, i=1,…,100; that is, 1-α¯=(1/100)∑i=1100(1-α^)i.
(c)
The sample mean square error (SMSE) of(1-α^)1,(1-α^)2,…,(1-α^)100SMSE= (1/100)∑i=1100[(1-α^)i-(1-α)]2.Based onL=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 sizen, 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 Table4, 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 parameterc=0.5, 1.5, and -1.5 are the same as the values of SMSE in Table 3.(d)
In Table5, fix the prior parameter (a,b), comparison of GE loss function parameter q={3,5,8} as follows:the values of SMSE with prior parameterq=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 CL under SE loss function when α=0.05.
n
a
=
2
,
b
=
5
a
=
6
,
b
=
1.5
a
=
2
,
b
=
2
5
0.94357 (0.00386)
0.93760 (0.00386)
0.93925 (0.00395)
10
0.94244 (0.00359)
0.93704 (0.00359)
0.93664 (0.00462)
15
0.94096 (0.00346)
0.93619 (0.00426)
0.93635 (0.00467)
n denotes the sample size; the values in parentheses are sample mean square error of (1-α^).Table 4
Average empirical credible level(1-α) for CL under LINEX loss function when α=0.05.
n
c
a
=
2
,
b
=
5
a
=
6
,
b
=
1.5
a
=
2
,
b
=
2
−0.5
0.94357 (0.00386)
0.93760 (0.00386)
0.93925 (0.00395)
5
0.5
0.94357 (0.00386)
0.93760 (0.00386)
0.93925 (0.00395)
1.5
0.94357 (0.00386)
0.93760 (0.00386)
0.93925 (0.00395)
−0.5
0.94244 (0.00359)
0.93704 (0.00359)
0.93664 (0.00462)
10
0.5
0.94244 (0.00359)
0.93704 (0.00359)
0.93664 (0.00462)
1.5
0.94244 (0.00359)
0.93704 (0.00359)
0.93664 (0.00462)
−0.5
0.94096 (0.00346)
0.93619 (0.00426)
0.93635 (0.00467)
15
0.5
0.94096 (0.00346)
0.93619 (0.00426)
0.93635 (0.00467)
1.5
0.94096 (0.00346)
0.93619 (0.00426)
0.93635 (0.00467)
n denotes the sample size; the values in parentheses are sample mean square error of (1-α^).Table 5
Average empirical credible level(1-α) for CL under GE loss function when α=0.05.
n
q
a
=
2
,
b
=
5
a
=
6
,
b
=
1.5
a
=
2
,
b
=
2
3
0.94357 (0.00386)
0.93760 (0.00386)
0.93925 (0.00395)
5
5
0.94357 (0.00386)
0.93760 (0.00386)
0.93925 (0.00395)
8
0.94357 (0.00386)
0.93760 (0.00386)
0.93925 (0.00395)
3
0.94244 (0.00359)
0.93704 (0.00359)
0.93664 (0.00462)
10
5
0.94244 (0.00359)
0.93704 (0.00359)
0.93664 (0.00462)
8
0.94244 (0.00359)
0.93704 (0.00359)
0.93664 (0.00462)
3
0.94096 (0.00346)
0.93619 (0.00426)
0.93635 (0.00467)
15
5
0.94096 (0.00346)
0.93619 (0.00426)
0.93635 (0.00467)
8
0.94096 (0.00346)
0.93619 (0.00426)
0.93635 (0.00467)
n denotes the sample size; the values in parentheses are sample mean square error of (1-α^).


## 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 Figure1 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{xU(i),i=1,…,5(=n)}={67.80,68.64,98.64,128.04,173.40}.(1)
The proposed testing procedure ofCL with C^L,MLE is stated as follows.Step 1.
The record values from the above data are(xU(i),i=1,2,3,4,5)=(67.80,68.64,98.64,128.04,173.40).Step 2.
The lower specification limitL is assumed to be 23.37. The deal with the product managers’ concerns regarding lifetime performance and the conforming rate Pr of products is required to exceed 80 percent. Referring to Table 1, CL is required to exceed 0.90. Thus, the performance index value is set at c0=0.90. The testing hypothesis H0:CL≤0.90 versus H1:CL>0.90 is constructed.Step 3.
Specify a significance levelα=0.05.Step 4.
We can calculate that the95% lower confidence interval bound for CL, where
(50)LB_MLE=π4-π-(π4-π-C^L,MLE)(χ2n,1-α22n)1/2=π4-π-(π4-π-1.26251254)(χ10,0.95210)1/2=1.03285.
So, the95% one-sided confidence interval for CL is [LB_MLE,∞)=[1.03285,∞).Step 5.
Because of the performance index valuec0=0.90∉[LB_MLE,∞), we reject the null hypothesis H0:CL≤0.90.Thus, we can conclude that the lifetime performance index of data meets the required level.(2)
The proposed testing procedures ofCL 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(xU(i),i=1,2,3,4,5)=(67.80,68.64,98.64,128.04,173.40).Step 2.
The lower specification limitL is assumed to be 23.37. The deal with the product managers’ concerns regarding lifetime performance and the conforming rate Pr of products is required to exceed 80 percent. Referring to Table 1, CL is required to exceed 0.90. Thus, the performance index value is set at c0=0.90. The testing hypothesis H0:CL≤0.90 versus H1:CL>0.90 is constructed.Step 3.
Specify a significance levelα=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 the95% lower credible interval bound for CL, where
(51)LB_BS=π4-π-(π4-π-C^L,BS)×Γ(n+b-1/2)Γ(n+b)(χ2(n+b),1-α22)1/2=π4-π-(π4-π-1.246074685)×Γ(5+1.001-1/2)Γ(5+1.001)(χ2(5+1.001),0.9522)1/2=0.96984.
So, the95% one-sided credible interval for CL is [LB_BS,∞)=[0.96984,∞). 							(52)LB_BL=π4-π-(π4-π-C^L,BL)×[12c(1-e-c/(n+b+1))χ2(n+b),1-α2]1/2=π4-π-(π4-π-1.129561059)×[12(0.5)(1-e-0.5/(5+1.001+1))χ2(5+1.001),0.952]1/2=0.96984.
So, the95% one-sided credible interval for CL is [LB_BL,∞)=[0.96984,∞). 							(53)LB_BG=π4-π-(π4-π-C^L,BG)×[Γ(n+b+q/2)Τ(n+b)]-1/q(χ2(n+b),1-α22)1/2=π4-π-(π4-π-1.200432998)×[Γ(5+1.001+2/2)Γ(5+1.001)]-1/2(χ2(5+1.001),0.9522)1/2=0.96984.
So, the95% one-sided credible interval for CL is [LB_BG,∞)=[0.96984,∞).Step 5.
Because of the performance index valuec0=0.90∉[LB_BS,∞)=[LB_BL,∞)=[LB_BG,∞), we reject the null hypothesis H0:CL≤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:CL≤0.90. Hence, the lifetime performance index of data meets the required level.


## 8. Conclusions

Montgomery [1] proposed a process capability index CL 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 ofCL 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 CL 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 correspondingCL can be obtained.Further, the Bayesian estimator and posterior distribution ofCL are also utilized to construct the testing procedure of CL 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.



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*Source: 547209-2013-03-07.xml*