# Assessing the Lifetime Performance Index with Digital Inferences of Power Hazard Function Distribution Using Progressive Type-II Censoring Scheme

Authors: Rashad M. EL-Sagheer; Taghreed M. Jawa; Neveen Sayed-Ahmed
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

This paper deals with estimating the lifetime performance index. The maximum likelihood (ML) and Bayesian estimators for lifetime performance indexCLX where LX is the lower specification limit are derived based on progressive type-II censored (Prog-Type-II-C) sample from two-parameter power hazard function distribution (PHFD). Knowing the lower specification limit, the MLE of CLX is applied to construct a new hypothesis testing procedure. Bayesian estimator of CLX is also utilized to develop a credible interval. Also, the relationship between the CLX and the conforming rate of products is investigated. Moreover, the Bayesian test to evaluate the lifetime performance of units is proposed. A simulation study and illustrative example based on a real dataset are discussed to evaluate the performance of the two tests.


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## Body

## 1. Introduction

Process capability analysis plays an important role in the quality control field to measure the performance of process in the industry. The lifetime performance index ( process capability index orPCI) has been proposed by Montgomery [1] and Kane [2]. The index is used to assess the lifetime performance of electronic units. All process capability indices (PCIs) have been discussed using the assumption that the lifetime of products (units) follows a normal distribution. Nevertheless, the assumption of normality for many processes in industry and business cannot be valid. A lot of papers were studied on the statistical inference for CLX based on various types of censored and progressive censored data for different models of which [3–17] dealt with progressive censoring from various points of view considering several lifetime distributions and its applications. Soliman et al. [18, 19] discussed assessing the life time performance index for exponentiated Frechet distribution using Prog-Type-II-C and progressive first failure censoring scheme. Recently, Wu et al. [20] introduced the reliability sampling design for CLX of Gompertz lifetime distribution under progressive type-I interval censoring, Zhang and Gui [21] studied the statistical inference for CLX with Pareto distribution on the basis of general progressive type-II censored sample, Wu et al. [22] proposed experimental design for CLX of Weibull products based on the progressive type-I interval censored sample, and Hassan and Assar [23] discussed assessing CLX of Burr type-III distribution under progressive type-II censoring.The Prog-Type-II-C scheme is of use in this paper [11–17], which can be characterized as follows: we presume that n units are placed on a life testing experiment. Let Xi, i=1,2,…,n be continuous identically distributed failure times of these units. The following scheme R1,R2,…,Rm are considered where m<n and R1,R2,…,Rm all are previously fixed and ∑i=1mRi+m=n. At the first failure time X1, R1 units can be removed randomly from the remaining n−1 surviving units. Then, immediately at the second failure time X2, R2 surviving units are randomly removed. This technique continues until the time of mth failure Xm. Then, the remaining units Rm are deleted.LetX1:m:nR1,…,Rm,X2:m:nR1,…,Rm,…,Xm:m:nR1,…,Rm be the prog-Type-II-C sample with size m from a sample of size n. One can easily note the following:(i)
IfR1=R2=⋯=Rm−1=0, then type-II right censored order statistics are deduced(ii)
IfR1=R2=⋯=Rm=0, then ordinary order statistics are obtainedIn this paper, the lifetimes of units have the two-parameterPHFDα,β. Statistical inference for the unknown CLX is developed based on Prog-Type-II-C data. Mugdadi [24] proposed the two-parameter PHFDα,β. The hazard rate function (hrf), cumulative distribution function (cdf), probability density function (pdf), and survival function (sf) are given respectively by(1)hx=αxβ,α>0,β>−1,x>0,(2)FXx;α,β=1−exp−αβ+1xβ+1,α>0,β>−1,x>0,(3)fXx;α,β=αxβexp−αβ+1xβ+1,α>0,β>−1,x>0,(4)SXx=exp−αβ+1xβ+1,α>0,β>−1,x>0,where α and β are the scale and shape parameters, respectively. It is clear that(i)
Forβ>0, the PHFD has an increasing hrf(ii)
For−1<β<0, it has decreasing hrfWeibull, Rayleigh, and exponential distributions can be deduced as special cases of the PHFD as follows:(1)
Ifβ=α−1, then PHFD reduces to Weibull α,1(2)
Ifα=1/λ2 and β=1, then PHFD reduces to Rayleigh λ(3)
Ifβ=0, then PHFD reduces to exponential distribution with mean 1/αTherefore, the results in this paper are valid for Weibull, Rayleigh, and exponential distributions. ThePHFDα,β has the following properties:(1)
The expected value ofX is(5)EX=∫0∞xfXxdx=α∫0∞xβ+1exp−αβ+1xβ+1dx=αβ+1−1/β+1Γβ+2β+1,whereΓ. is the complete gamma function.(2)
The expected value ofX2 is(6)EX2=∫0∞x2fXxdx=α∫0∞xβ+2exp−αβ+1xβ+1dx=αβ+1−2/β+1Γβ+3β+1.(3)
The standard deviation(7)σ=EX2−EX2=γ2αβ+1−1/β+1,where(8)γ2=Γβ+3β+1−γ12,γ1=Γβ+2β+1.The main aim of this article is to secure the ML and Bayesian estimators forCLX in view of Prog-Type-II-C sample from PHFDα,β where β is known parameter. The ML estimator of CLX is then used to construct a new hypothesis testing based on known LX. The Bayesian test is also proposed to evaluate CLX of units.The rest of this paper is organized as follows: Section2 contains a derivation of CLX. Section 3 discusses the relationship between CLX and the conforming rate Pr of products. The ML estimate of CLX and some of the corresponding statistical properties are investigated in Section 4. Bayesian approach of CLX in the presence of gamma prior distribution is presented in Section 5, and Section 6 improves testing procedures for CLX. A real dataset has been analyzed to illustrate the use of the testing procedure based on the proposed estimators in Section 7. Simulation studies are given in Section 8. Finally, conclusion appears in Section 9.


## 2. The Lifetime Performance IndexCLX

LetX be the lifetime of products which has PHFDα,β with pdf and cdf given in (2) and (3), respectively. It is known to the consumer that the high-quality product is the one that lasts for a longer lifetime, and therefore, the lifetime is the characteristic of the high-quality product. Moreover, in order for the product to be profitable from a financial point of view and satisfactory to customers as well, the lifetime required to exceed LX unit times. Montgomery [1] suggested a capability index CLX to measure the features of a product that has better and higher quality. Then, CLX is defined as(9)CLX=μ−LXσ,where μ=EX denotes the mean lifetime, σ=EX2−EX2 represents the lifetime standard deviation, and LX indicates the lower specification limit. Under the condition that X has PHFDα,β, then from (5), (7), and (9), the lifetime performance index CLX is written as(10)CLX=1γ2γ1−αβ+11/β+1LX,where γ1 and γ2 are given in (8).


## 3. Conforming Rate

If the new lifetime of a product (or item)X exceeds the lower specification limit LX (i.e., X>LX), then the product is labelled as a conforming productA. Otherwise, the product is labelled as a nonconforming product. Therefore, the ratio of the conforming product is known as the conforming probability, or sometimes also called conforming rate Pr, and can be defined as(11)Pr=PX≥LX=exp−αβ+1αβ+1−1/β+1γ1−γ2CLXβ+1.Obviously, a strictly positive relationship exists betweenPr and CLX, for given β>−1. Thus, the higher the index value CLX gives the higher of the conforming rate Pr. Table 1 lists some numerical values of CLX and the corresponding Pr for given β=0.332 and α=1.1835, which can be obtained by using the graphical method, the Pr can be calculated by (11) for given α, β, and CLX.Table 1
The numerical values ofCLX vs. the corresponding Pr.
CLXPr−∞0.000 000−5.000.000 743−4.500.001 572−4.000.003 256−3.500.006 594−3.000.013 039−2.500.025 127−2.000.047 086−1.500.085 566−1.000.150 250−0.500.253 7030.000.409 0620.050.427 8270.100.447 1910.150.467 1530.200.487 7040.250.508 8370.300.530 5390.350.552 7930.400.575 5790.450.598 8700.500.622 6360.550.646 8400.600.671 4370.650.696 3750.700.721 5940.750.747 0220.800.772 5750.850.798 1560.900.823 6490.950.848 9161.000.873 7931.050.898 0781.100.921 5171.150.943 7781.200.964 3941.250.982 6221.300.996 8721.310.998 850


## 4. Maximum Likelihood Estimator ofCLx

LetX1:m::n<X2:m::n<⋯<Xm:m::n be a Prog-Type-II-C sample from PHFDα,β, with pdf and cdf as defined in equations (2) and (3), respectively. We denote the observed values of such Prog-Type-II-C sample by xi , i=1,2,…,m. According to Balakrishnan and Sandhu [12], the likelihood function of Prog-Type-II-C with scheme Ri≥0, i=1,2,…,m is(12)Lα,β;x¯=C∏i=1mfXx;α,β1−FXx;α,βRi,where(13)C=nn−R1−1n−R1−R2−2…n−∑i=1m−1Ri−m+1.Substituting from (2) and (3) into (12), the likelihood function for x¯ is given by(14)Lα,β;x¯=Cαm∏i=1mxiβexp−αβ+1∑i=1mRi+1xiβ+1.The log-likelihood function can be obtained from (14) as(15)ℓα,β;x¯=logC+mlogα+β∑i=1mlogxi−αβ+1∑i=1mRi+1xiβ+1.Consequently, for knownβ, the likelihood equation of α is obtained as(16)∂ℓα,β;x¯∂α=mα−1β+1∑i=1mRi+1xiβ+1=0.Therefore, the ML estimator ofα is(17)α^ML=m1β+1∑i=1mRi+1xiβ+1−1.Thus, according to Zehna [25], the MLE of CLX becomes(18)C^LXML=1γ2γ1−mW1/β+1LX,where W=∑i=1mRi+1xiβ+1 and γ1 and γ2 are given in (8).


## 5. Bayes Estimation ofCLX

In this section, based on Prog-Type-II-C sample underPHFDα,β with known β, the Bayesian method for deriving estimates of both α and CLX is discussed. In lifetime data analysis, such prior knowledge is usually summarized into a prior density, denoted by πα|a,b. We consider the conjugate prior distribution to be gamma distribution with the pdf as(19)πα|a,b=baΓaαa−1exp−bα,a>0,b>0,α>0,where the hyperparameters a and b are chosen to reflect prior knowledge about the unknown parameter α. Based on (14) and (19), we can obtain the posterior density function of α, given the data as(20)π∗α|x¯=W′m+aΓm+aαm+a−1exp−αW′,for α>0 and zero elsewhere, where(21)W′=b+1β+1∑i=1mRi+1xiβ+1.Based on a SELFℓα,α^=α^−α2 and (20), the Bayesian estimation for the parameter α becomes(22)α^BS=Eα|x¯=m+aW′.Hence, the Bayes estimatorC^LXBS of CLX can be written as(23)C^LXBS=ECLX|x¯=1γ2γ1−Γm+a+1/β+1Γm+a1β+11/β+1LXW′−1/β+1,where W′ is given in (21).Lemma 1.
IfW′=b+1/β+1∑i=1mRi+1xiβ+1, then 2αW′ follows the chi-square distribution with 2m+a degrees of freedom, denoted by 2αW′∼χ2m+a2.Proof.
Letα=y/2W′, then Jy=1/2W′ (see Casella and Berger [26], pp.); we obtain the density function of y as(24)fYy=π∗y2W′|x¯Jy=W′m+aΓm+ay2W′m+a−1exp−y212W′=122m+a/2Γ2m+a/2y2m+a/2−1exp−y2=12m+aΓm+aym+a−1exp−y2.
Therefore,Y=2αW′∼χ2m+a2.


## 6. Testing Procedure forCLX

This section is devoted to construct a statistical testing procedure to evaluate whetherCLX reaches the required level. Credible and confidence intervals for CLX are calculated to objectively evaluate whether CLX adheres to the required level. The null and the alternative hypotheses H0 (the product is unreliable) and H1 (the product is reliable) respectively can be written as(25)H0:CLX≤c,H1:CLX>c,where c denotes the lower bound of CLX.In the Bayesian approach, for given specified significance levelδ, a 1001−δ% one-sided credible interval (CRI) for CLX is derived as follows. Since β is known, by using the pivotal quantity 2αW′∼χ2m+a2, and the lower 1−δ percentile of χ2m+a2 denoted by χ1−δ,2m+a2, we have(26)P2αW′≤χ1−δ,2m+a2=1−δ,⇒P1γ2γ1−αβ+11/β+1LX≥1γ2γ1−χ1−δ,2m+a22W′β+11/β+1LX=1−δ,⇒PCLX≥1γ2γ1−χ1−δ,2m+a22W′β+11/β+1LX=1−δ,⇒PCLX≥1γ2γ1−χ1−δ,2m+a22m+aβ+11/β+1γ1−γ2C^LXBS=1−δ,Here, γ1, γ2, W′, and C^LXBS are given by (8), (21), and (23), respectively. Therefore, the level 1001−δ% lower credible bound for CLX can be written as(27)LB¯BS=1γ2γ1−χ1−δ,2m+a22m+aβ+11/β+1γ1−γ2C^LXBS.At the same time, we derive the maximum likelihood approach by using 2αW∼χ2m2, where W=∑i=1mRi+1xiβ+1. The 1001−δ% one-sided confidence interval (CI) for CLX is then given by(28)CLX≥1γ2γ1−χ1−δ,2m22mβ+11/β+1γ1−γ2C^LXML,where C^LXML is given in (18). Hence, the level 1001−δ% lower confidence bound for CLX can be derived as(29)LB¯ML=1γ2γ1−χ1−δ,2m22mβ+11/β+1γ1−γ2C^LXML.The proposed testing procedure aboutCLX in the Bayesian approach can be organized as follows.Step 1.
Specify the lower lifetime limitLX for products and performance index value c; then, the testing null hypothesis H0: CLX≤c and the alternative hypothesis H1: CLX>c are created.Step 2.
Determine a significance levelδStep 3.
Compute the level1001−δ% one-sided CRI LB¯BS,∞ for CLXStep 4.
The decision rule of the statistical test is provided as follows: if the performance index valuec∉LB¯BS,∞, it is concluded that the lifetime performance index of the product meets the required level
The ML approach uses1001−δ% one-sided CI LB¯ML,∞ for CLX instead of 1001−δ% one-sided CRI LB¯BS,∞ to see if the product performance meets the required level.


## 7. Application to Real-Life Data

The combination between theoretical and application methods plays an important role in modern statistical problems. In this section, a theoretical technique is applied to a set of real data for illustration of the proposed procedures. We consider a real dataset given by Leiblein and Zelen [27], which was used recently by Cho et al. [28]. For the purpose of the goodness-of-fit test, the Kolmogorov–Smirnov distance between the empirical and the fitted distribution functions has been computed. It is 0.150 81, and the associated p-value is 0.672 3. Hence, the p-value for Kolmogorov–Smirnov test has the highest value for the dataset. This leads us to conclude that PHFD is the best fit for the real dataset. Empirical, Q−Q, and P−P plots are shown in Figure 1, which clear that the PHFD fits the data very well. This set of real data represents 23 observed failure times. The Prog-Type-II-C scheme was conducted with n=23, m=18, and the censoring scheme Ri, i=1,…,18 is generated from the original data. The observed data and removed numbers are reported in Table 2, where α^=1.1835 and β^=0.332, which can be obtained by using the graphical method introduced by Balakrishnan and Kateri [29].Figure 1
Empirical, Q-Q, and P–P plots of PHFD for the real dataset.Table 2
Prog-Type-II-C sample from Leiblein and Zelen [19].
i123456789xi,18,230.178 80.330 00.415 20.456 00.484 80.518 60.519 60.541 20.555 6Ri000000000i101112131415161718xi,18,230.678 00.686 40.841 20.931 20.986 41.058 41.279 21.280 41.734 0Ri001010102In the Bayesian approach, we assumed that the values of the hyperparametersa=0.001 and b=0.001. Under the Prog-Type–II–C sample, the proposed testing procedure for CLX can be performed in the following steps:(i) Step 1
:LX is assumed to be 0.441 1. To meet the product purchasers’ concerns regarding operational performance, the Pr of products is required to exceed 82%. Referring to Table 1, the CLX value is required to exceed 0.90. Thus, c=0.90 and H0: CLX≤0.90 vs. H1 : CLX>0.90 is constructed.Step 2
: let the significance level be given byδ=0.05.(ii) Step 3
: the lower boundLB¯BS of the 95% one-sided CRI for CLX is(30)LB¯BS=10.69700.9192−χ1−0.05,218+12218+10.332+11/0.332+10.9192−0.69700.9427=0.9274,whereγ1=0.9192 and γ2=0.6970 are calculated according to (8).Step 4
: becausec=0.90∉0.9274,∞, the null hypothesis H0: CLX≤0.90 is rejected.In the ML approach, the lower boundLB¯ML of the 95% one-sided CI for CLX is(31)LB¯ML=10.69700.9192−χ1−0.05,2×1822×180.332+11/0.332+10.9192−0.69700.9382=0.9201.Because of the performance index valuec=0.90∉0.9201,∞, we reject H0: CLX≤0.90. To sum up, CLX of products meets the required quality level for the Bayesian and ML approaches.


## 8. Monte Carlo Simulation Study

To compare the ML and Bayes estimators forCLX which are proposed in previous sections, Monte Carlo simulations were performed utilizing 1000 Prog-Type-II-C samples for each simulation. All obtained simulation study results were performed using Mathematica Ver. 13. The mean square error (MSE) is used to compare the estimators. The samples were generated from PHFD using α,β=1.092,0.332 , with different n, m, and hyperparameters a,b. In this study, we used different censoring schemes as follows.Scheme 1.
R1=n−m,Ri=0 for i≠1.Scheme 2.
Rm+1/2=n−m,Ri=0 for i≠m+1/2, if m is odd, and Rm/2=n−m,Ri=0 for i≠m/2, if m is even.Scheme 3.
Rm=n−m,Ri=0 for i≠m.
Based on the lower lifetime limitLX=0.022, the results of MSEs of the ML, Bayes estimates, and coverage probabilities (CPs) of the 95% CRI and CI for CLX are presented in Table 3.Table 3
MSEs and CPs of the MLEs and Bayes estimates forCLX.
NmSc.MLBayesBayesMLBayesBayesMean squared errors (MSEs)Coverage probabilities (CPs)a = 1, b = 2a = 2, b = 3a = 1, b = 2a = 2, b = 32515I0.000 088 90.000 080 40.000 072 90.9470.9490.956II0.000 091 90.000 084 90.000 075 00.9390.9480.953III0.000 097 20.000 089 70.000 081 10.9450.9390.9483020I0.000 083 70.000 077 40.000 068 70.9410.9430.939II0.000 086 60.000 079 40.000 071 30.9520.9480.947III0.000 092 20.000 084 20.000 079 20.9540.9530.9553025I0.000 076 70.000 069 90.000 061 80.9620.9540.963II0.000 082 30.000 074 20.000 066 60.9410.9510.947III0.000 089 10.000 078 50.000 072 90.9390.9400.9515030I0.000 071 60.000 062 60.000 059 30.9440.9590.945II0.000 075 70.000 068 40.000 063 10.9400.9360.942III0.000 081 40.000 073 20.000 067 50.9450.9410.9395040I0.000 067 30.000 058 40.000 055 80.9600.9520.962II0.000 071 50.000 062 60.000 058 50.9540.9580.957III0.000 076 80.000 066 90.000 062 80.9380.9400.9487050I0.000 059 90.000 049 80.000 046 70.9510.9490.947II0.000 063 30.000 054 30.000 051 60.9440.9450.952III0.000 067 70.000 058 50.000 055 90.9380.9410.9499060I0.000 038 40.000 029 90.000 025 80.9550.9610.954II0.000 043 60.000 036 30.000 033 40.9480.9390.945III0.000 047 10.000 038 20.000 035 10.9540.9400.9409070I0.000 022 80.000 019 60.000 014 90.9560.9530.958II0.000 028 70.000 023 40.000 018 50.9450.9430.951III0.000 034 10.000 028 50.000 023 20.9420.9410.939


## 9. Conclusions

This paper aims to construct Bayesian and non-Bayesian approaches, with various estimators forCLX with the Prog-Type-II-C sample from PHFDα,β. Under the condition of known LX, the MLEs and Bayesian estimator of CLX are then used to develop the new hypotheses testing procedure. From the application of real-life data, it turns out that the suggested test can be carried out easily so that we can assess whether the product quality (lifetime of products) meets the requirements of customers and at the same time brings abundant profit. Furthermore, it indicates that these assessing methods are practical and feasible. An elaborate simulation study was conducted for different sample sizes n, m and different censoring schemes (I, II, III) to evaluate the performance of these proposed procedures. For the point estimations, MSE was compared. For the interval estimations, the coverage rates were obtained. According to the tabulated results of the estimates in Table 3, the following concluding remarks can be drawn:(1)
For the censoring schemeI, the MSE values of all estimates decrease as m/n increases which is consistent with the statistical theory that the larger the sample size, the more accurate of the estimate(2)
Generally, the performance of the Bayes estimators is better than MLE for the all considered cases because it has the smallest MSE(3)
The CPs of both ACIs and CRI are sensibly satisfactory and in most cases are near to the nominal confidence level of95%



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