# Statistical Inference of Reliability Parameter for the Skew-Normal Distribution under Progressive Type-II Censored Samples

Authors: Salman Babayi; Mohammad Seidpisheh; Gholamhossein Gholami
Journal: Journal of Mathematics
 (2022)

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## Abstract

This study investigates the inference of the reliability parameterR=PY<X for the skew-normal distribution based on progressively Type-II censored samples, where Y and X are two independent random variables. The maximum likelihood estimator and the asymptotic confidence interval for S=2,2,…,2,51×20 are extracted via an iterative algorithm. Moreover, the Bayes estimator and the corresponding credible interval are computed using Gibbs sampling. Also, the MLE, UMVUE, Bayes estimator, and confidence interval are acquired when the scale parameter is known. We compare the performances of the different estimation methods using Monte Carlo simulations. Finally, the real datasets are analyzed for illustrative purposes.


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## Body

## 1. Introduction

The reliability parameter,R=PY<X, has received great attention in the literature. The applications of the stress-strength model had rapid growth in many fields, including quality control, engineering statistics, reliability, medicine, psychology, biostatistics, and probabilistic mechanical design. The strength, X, and the stress, Y, are considered random variables. The system fails whenever X is less than Y, and R is the reliability of the system performance. Another application arises in the lifetime comparison of the two devices represented by X and Y. The stress-strength studies in the reliability context started with the paper of Birnbaum [1]. Since then, several authors have studied the inference of the R by applying different classes of statistical distribution. The estimation of R has been studied by Ali and Woo [2], in which X and Y are Levy random variables. Kotz et al. [3] provided a comprehensive review of the development of these studies until 2003. On the basis of the complete samples, several papers were published on the inference of R, for example, Kundu and Gupta [4], Huang et al. [5], Babayi et al. [6] and Guo and Gui [7].Recently, estimation ofR under progressive censoring samples has been considered by many researchers, for example, the two-parameter Rayleigh distribution discussed by Kohansal and Rezakhah [8]. Extended Chen distribution proposed by Maiti and Kayal [9], Joukar et al. [10] studied power Lindley distribution, Kumar and Kumar [11] investigated inverse Pareto distribution, and Ghanbari et al. [12] studied Marshall–Olkin distributions. The multicomponent stress-strength model for Weibull distribution is considered by Kohansal et al. [13].Ordinarily, the lifetime experiments are costly and time-consuming. There are different types of censoring schemes for reducing the cost and time in life-testing experiments. Type-I and Type-II censoring schemes are the two well-known and extensively applied in the literature. However, none of these schemes allow intermediate removal of the active units from the experiment other than the final termination point. Because of that, the progressive censoring scheme is used, in which some units remove after each failure. This practical scheme was first studied by Cohen [14]. It has become trendy in reliability and life-testing experiments in the last decade, see [15–20].The Type-II progressive censoring scheme can be described as follows. Letn units be placed on a life-testing experiment, and the given scheme S1,S2,…,Sm is prefixed. S1 of the n−1 surviving units are chosen randomly at the time of the first failure and removed from the experiment. Likewise, S2 of the n−S1−2 surviving units are withdrawn randomly at the time of the second failure and so on. Last of all, at the time of the m th failure, all remaining surviving observations, Sm=n−S1−S2−⋯−Sm−1−m, are eliminated. Note that Type-II progressive censoring scheme includes complete sampling scheme by choosing S1=S2=⋯=Sm=0 and Type-II right censoring scheme by choosing S1=S2=⋯=Sm−1=0,Sm=n−m. For more information about progressively censoring, see [21].The first problem in data analysis is to find suitable distribution to fit them. The skew-normal density function, which is an extension of the normal one, has acceptable flexibility to model various datasets. Furthermore, having a closed form of the cumulative distribution function that simplifies calculations is another advantage of this distribution. In Section2, we will provide more details in this regard. Babayi et al. [6] and Babayi and Khorram [22] analyzed the strength data based on complete sample data by applying the generalized logistic (GL) and skew-normal distributions, respectively. In this study, as an extension of their work, we study the inference of stress-strength parameter for skew-normal (SN) distribution based on Type-II progressive censored sample data. Also, in this study, we compute the Bayes estimator and the corresponding credible interval using Gibbs sampling. Moreover, we show that the SN model is better than the GL model to fit the strength data reported by Badr and Priest [23]. The rest of this study is as follows. The skew-normal distribution is introduced in Section 2. We derive classical and Bayesian inference for R in Section 3 when the scale parameter is unknown. In Section 4, the scale parameter of skew-normal distribution is assumed to be known, and Bayesian estimation, MLE, and UMVUE of R are presented. Analysis of the proposed methods is applied to the simulated data in Section 5. Finally, the skew-normal distribution with high flexibility is fitted to real datasets.


## 2. Skew-Normal Distribution

Standard distributions such as the gamma, Weibull, and log-normal distributions are used to analyze the positive real line of skewed data. Azzalini [24] introduced the skew-normal distribution by following the probability density function (pdf) to skewed data on the real line:(1)ϕx;λ=2ϕxΦλx,−∞<x<+∞,where ϕ. and Φ. are the pdf and the cumulative distribution function (cdf) of standard normal distribution, respectively. It is clear that when λ=0, (1) corresponds to standard normal distribution. The generalization of (1) presented by Gupta and Gupta [25] is as follows:(2)fx;λ,k=ΦλxkϕxCkλ,where k∈N and Ckλ=∫−∞∞Φλxkϕxdx. Note that (1) is a special case of (2) when k=1.Now, consider other skew-normal model, suppose thatY be a continuous random variable with cdf FYy; then, FXx=1−1−FYxα, α>0 is a continuous random variable with baseline distribution FY.. These models are called proportional hazard rate models in reliability theory. By choosing N0,σ2 as baseline distribution, the new form of the skew-normal (SN) distribution is resulted in the following cdf and pdf, respectively:(3)Fx;α,σ=1−1−Φxσα,−∞<x<+∞,and(4)fx;α,σ=ασϕxσ1−Φxσα−1,−∞<x<+∞,where α and σ are the shape and scale parameters, respectively. Plots of the pdf for different values of α are shown in Figure 1. If set α=1 and σ=1, F reduces to standard normal distribution function. When α is a positive integer number, (3) is the cdf of the first order statistic of random samples of size α from the standard normal distribution. It is easy to show that if X∼SNα,σ, then −X has power normal distribution with cdf, Fx;α,σ=Φx/σα. Hence, the skew-normal model has properties similar to power normal distribution. Kundu and Gupta [26] surveyed useful applications of the power normal distribution.Figure 1
The pdf of skew-normal distribution for different values ofα.


## 3. The MLE ofR

We want to estimate ofR in which X∼SNα,σ and Y∼SNβ,σ, and X and Y are independent random variables. Hence,(5)R=PY<X=∫−∞+∞∫−∞xασϕxσ1−Φxσα−1.βσϕyσ1−Φyσβ−1dydx,=1−∫−∞+∞ασϕxσ1−Φxσα+β−1dx=βα+β.Therefore, for estimating ofR, we first evaluate the MLE of σ, α, and β based on progressively Type-II censored data. Let X1:m1:n1,X2:m1:n1,…,Xm1:m1:n1 and Y1:m2:n2,Y2:m2:n2,…,Ym2:m2:n2 be two progressively Type-II censored samples from SNα,σ and SNβ,σ with censoring schemes S1,S2,…,Sm1 and S1′,S2′,…,Sm2′, respectively. So, the likelihood function of the observed sample is given by [21](6)Lα,β,σ=c1∏i=1m1fxi:m1:n11−Fxi:m1:n1Si×c2∏i=1m2fyi:m2:n21−Fyi:m2:n2Si′,where(7)c1=n1n1−S1−1⋯n1−S1−S2−⋯−Sm1−1−m1+1,c2=n2n2−S1′−1⋯n2−S1′−S2′−⋯−Sm2−1′−m2+1.Considering1−Φa=Φ−a, the log-likelihood function is given by(8)lα,β,σ=c−m1+m2lnσ+m1lnα+m2lnβ+∑i=1m1lnϕxi:m1:n1σ+∑i=1m2lnϕyi:m2:n2σ+∑i=1m1αSi+α−1lnΦ−xi:m1:n1σ+∑i=1m2βSi′+β−1lnΦ−yi:m2:n2σ.Hence, the MLEs ofα and β, denotes by α^σ^ and β^σ^, are(9)α^σ^=−m1∑i=1m1Si+1lnΦ−Xi:m1:n1/σ^,β^σ^=−m2∑i=1m2Si′+1lnΦ−Xi:m2:n2/σ^.and σ^ could be computed using the following nonlinear equation via an iterative scheme(10)gσ=σ,where(11)gσ=m1+m2−11σ∑i=1m1xi:m1:n12+∑i=1m1α^σSi+α^σ−1xi:m1:n1×hxi:m1:n1σ+1σ∑i=1m2yi:m2:n22+∑i=1m2β^σSi′+β^σ−1yi:m2:n2×hyi:m2:n2σ,and ht=ϕt/1−Φt is the hazard rate function of the standard normal distribution.Thus, using the invariant property of the ML estimators, the MLE of theR is evaluated as(12)R^=β^σ^α^σ^+β^σ^.Considering Figure2 and the definition of fσ=gσ−σ, we observe that fσ is a strictly decreasing function of σ for σ>0. Therefore, equation (10) has just one positive root, and the MLE of R is unique.Figure 2
The graph offσ=gσ−σ with respect to σ.


### 3.1. Asymptotic Distribution

Here, we extract the asymptotic distribution ofR^ applying the observed Fisher information matrix, I=Iij=−∂2l/∂θi∂θjθ=θ^, where i,j=1,2,3 and θ=θ1,θ2,θ3T=α,β,σT is used to acquire the asymptotic distribution of θ^=α^,β^,σ^T. The elements of the observed Fisher information matrix can be derived as follows:(13)I11=−∂2l∂α2=m1α2,I12=I21=−∂2l∂α∂β=0,I22=−∂2l∂β2=m2β2,I13=I31=−∂2l∂α∂σ=−∑i=1m1Si+1xi:m1:n1σ2hxi:m1:n1σ,I23=I32=−∂2l∂β∂σ=−∑i=1m2Si′+1yi:m2:n2σ2hyi:m2:n2σ,and using ϕ′t=−tϕt, the last element of the matrix will be as(14)I33=−∂2l∂σ2=−m1+m2σ2−∑i=1m1−3xi:m1:n12σ4+xi:m1:n14σ8+xi:m1:n13σ5ϕxi:m1:n1/σ−∑i=1m2−3yi:m2:n22σ4+yi:m2:n24σ8+yi:m2:n23σ5ϕyi:m2:n2/σ+∑i=1m1αSi+Si−1−2xi:m1:n1σ3+xi:m1:n13σ5+xi:m1:n12σ4Φxi:m1:n1/σ×h−xi:m1:n1σ+∑i=1m2βSi′+Si′−1−2yi:m2:n2σ3+yi:m2:n23σ5+yi:m2:n22σ4Φyi:m2:n2/σh−yi:m2:n2σ.The confidence interval (CI) ofR based on the asymptotic distribution of R^ is determined. In the regularity conditions, the distribution of vector θ^=α^,β^,σ^T converges to the normal with μ=α,β,σT, and Σ=I−1α,β,σ.Theorem 1.
Asm1⟶∞ and m2⟶∞, then(15)m1α^−α,m2β^−β,m1σ^−σ⟶N30,A−1α,β,σ,where A−1α,β,σ is the symmetric matrix as(16)Aα,β,σ=a110a13a22a23a33,a11=I11m1,a13=I13m1,a22=I22m2,a23=I23m1m2,a33=I33m1.Proof.
The proof is resulted by applying the asymptotic normality of the MLE.Theorem 2.
Asm1⟶∞ and m2⟶∞, then(17)m1R^−R⟶N0,V,where(18)V=1cα+β4β2a22a33−a232−2αβa23a13+α2a11a33−a132,c=a11a22a33−a232a11−a132a22.Proof.
Here,R^=fα^,β^,σ^, where(19)fα,β,σ=βα+β.
Proof, using Theorem1 and the delta method, is achieved as follows:(20)V=eTA−1e,e=∂g∂α∂g∂β∂g∂σ=−βα+β2αα+β20.
□By Theorem2, the 1001−γ% asymptotic CI of R is given by(21)R^−z1−γ/2V^m1,R^+z1−γ/2V^m1,where z1−γ/2 is 1001−γ/2th percentile of standard normal distribution.


### 3.2. Bayesian Estimation ofR

We develop the Bayesian inference ofR based on the Bayesian estimation of the parameters α, β, and σ. A starting point of any Bayesian analysis is the choice of the prior. We have assumed the priors follow gamma distribution because, firstly, the supports of α, β, and σ are positive. Secondly, gamma priors are conjugate ones. The advantage of using conjugate priors is that the posterior belongs to the same class as the prior, so updating the prior reduces updating its parameters. As a rule, conjugate priors lead to straightforward mathematical calculations. Therefore, we assume α, β, and σ follow the conjugate independent gamma priors. Therefore,(22)πiθi=biaiΓaiθiai−1e−biθi,where θi>0,ai>0,bi>0,i=1,2,3, and θ1,θ2,θ3=α,β,σ.The joint density of the data,α, β, and σ, is(23)κdata,α,β,σ=Ldata|α,β,σ×∏i=13πiθi.Therefore, the joint posterior density ofα, β, and σ given data can be written as(24)ηα,β,σ|data=κdata,α,β,σ∫0∞∫0∞∫0∞κdata,α,β,σdαdβdσ.Since (24) cannot be determined in a closed form, therefore, for inference of R, the MCMC methods are used. The posterior PDFs of α, β, and σ are given in the following way:(25)α|β,σ,data∼Γa1+m1,b1−∑i=1m1Si+1lnΦ−xi:m1:n1σ,β|α,σ,data∼Γa2+m2,b2−∑i=1m2Si′+1lnΦ−yi:m2:n2σ,ησ|α,β,data∝σa3+m1+m2−1e−b3σ×∏i=1m1ϕxi:m1:n1σ∏i=1m1Φ−xi:m1:n1σαSi+α−1×∏i=1m2ϕyi:m2:n2σ∏i=1m2Φ−yi:m2:n2σβSi′+β−1,where Γ.,. stands for Gamma distribution.Since the posterior PDF ofσ is unknown, the normal distribution is a good proposal based on Figure 3. Hence, we apply the normal proposal distribution to simulate σ using the Metropolis–Hasting method. Therefore, the Gibbs sampling algorithm is the following.(1)
Step 1: take the initial valuesα0,β0,σ0 for α,β,σ.(2)
Step 2: setj=1.(3)
Step 3: simulateαj from Γa1+m1,b1−∑i=1m1Si+1lnΦ−xi:m1:n1/σj−1.(4)
Step 4: simulateβj from Γa2+m2,b2−∑i=1m2Si′+1lnΦ−yi:m2:n2/σj−1.(5)
Step 5: simulateσj from(26)ησ|αj−1,βj−1,data,with the Nσj−1,1 as a proposal distribution using the Metropolis–Hastings method.(6)
Step 6: determineRj using (5).(7)
Step 7: setj=j+1.(8)
Step 8:T times repeat steps 3–7.Figure 3
Proposal and posterior density functions of the scale parameter.Figure4 illustrates the convergence of the Metropolis–Hasting algorithm based on the normal proposal. In addition, the cumulative average graph (center) shows that the algorithm converges after 1000 iteration, and the autocorrelation graph demonstrates no correlation.Figure 4
Graphical diagnoses of Metropolis–Hasting algorithm. (a) A sequence of 1000 generations from posterior density functions ofσ; (b) cumulative average graph of σ sequence; (c) autocorrelation graph of σ sequence.The posterior mean and variance ofR can be evaluated as(27)E^R|data=1T∑j=1TRj,V^R|data=1T∑j=1TRj−E^R|data2.By choosing the shortest length fromR1,R1−γT,R2,R1−γT+1,…,RγT,RT, the HPD credible interval is determined in which Ri is the ith-order statistics from sample size T.


## 3.1. Asymptotic Distribution

Here, we extract the asymptotic distribution ofR^ applying the observed Fisher information matrix, I=Iij=−∂2l/∂θi∂θjθ=θ^, where i,j=1,2,3 and θ=θ1,θ2,θ3T=α,β,σT is used to acquire the asymptotic distribution of θ^=α^,β^,σ^T. The elements of the observed Fisher information matrix can be derived as follows:(13)I11=−∂2l∂α2=m1α2,I12=I21=−∂2l∂α∂β=0,I22=−∂2l∂β2=m2β2,I13=I31=−∂2l∂α∂σ=−∑i=1m1Si+1xi:m1:n1σ2hxi:m1:n1σ,I23=I32=−∂2l∂β∂σ=−∑i=1m2Si′+1yi:m2:n2σ2hyi:m2:n2σ,and using ϕ′t=−tϕt, the last element of the matrix will be as(14)I33=−∂2l∂σ2=−m1+m2σ2−∑i=1m1−3xi:m1:n12σ4+xi:m1:n14σ8+xi:m1:n13σ5ϕxi:m1:n1/σ−∑i=1m2−3yi:m2:n22σ4+yi:m2:n24σ8+yi:m2:n23σ5ϕyi:m2:n2/σ+∑i=1m1αSi+Si−1−2xi:m1:n1σ3+xi:m1:n13σ5+xi:m1:n12σ4Φxi:m1:n1/σ×h−xi:m1:n1σ+∑i=1m2βSi′+Si′−1−2yi:m2:n2σ3+yi:m2:n23σ5+yi:m2:n22σ4Φyi:m2:n2/σh−yi:m2:n2σ.The confidence interval (CI) ofR based on the asymptotic distribution of R^ is determined. In the regularity conditions, the distribution of vector θ^=α^,β^,σ^T converges to the normal with μ=α,β,σT, and Σ=I−1α,β,σ.Theorem 1.
Asm1⟶∞ and m2⟶∞, then(15)m1α^−α,m2β^−β,m1σ^−σ⟶N30,A−1α,β,σ,where A−1α,β,σ is the symmetric matrix as(16)Aα,β,σ=a110a13a22a23a33,a11=I11m1,a13=I13m1,a22=I22m2,a23=I23m1m2,a33=I33m1.Proof.
The proof is resulted by applying the asymptotic normality of the MLE.Theorem 2.
Asm1⟶∞ and m2⟶∞, then(17)m1R^−R⟶N0,V,where(18)V=1cα+β4β2a22a33−a232−2αβa23a13+α2a11a33−a132,c=a11a22a33−a232a11−a132a22.Proof.
Here,R^=fα^,β^,σ^, where(19)fα,β,σ=βα+β.
Proof, using Theorem1 and the delta method, is achieved as follows:(20)V=eTA−1e,e=∂g∂α∂g∂β∂g∂σ=−βα+β2αα+β20.
□By Theorem2, the 1001−γ% asymptotic CI of R is given by(21)R^−z1−γ/2V^m1,R^+z1−γ/2V^m1,where z1−γ/2 is 1001−γ/2th percentile of standard normal distribution.


## 3.2. Bayesian Estimation ofR

We develop the Bayesian inference ofR based on the Bayesian estimation of the parameters α, β, and σ. A starting point of any Bayesian analysis is the choice of the prior. We have assumed the priors follow gamma distribution because, firstly, the supports of α, β, and σ are positive. Secondly, gamma priors are conjugate ones. The advantage of using conjugate priors is that the posterior belongs to the same class as the prior, so updating the prior reduces updating its parameters. As a rule, conjugate priors lead to straightforward mathematical calculations. Therefore, we assume α, β, and σ follow the conjugate independent gamma priors. Therefore,(22)πiθi=biaiΓaiθiai−1e−biθi,where θi>0,ai>0,bi>0,i=1,2,3, and θ1,θ2,θ3=α,β,σ.The joint density of the data,α, β, and σ, is(23)κdata,α,β,σ=Ldata|α,β,σ×∏i=13πiθi.Therefore, the joint posterior density ofα, β, and σ given data can be written as(24)ηα,β,σ|data=κdata,α,β,σ∫0∞∫0∞∫0∞κdata,α,β,σdαdβdσ.Since (24) cannot be determined in a closed form, therefore, for inference of R, the MCMC methods are used. The posterior PDFs of α, β, and σ are given in the following way:(25)α|β,σ,data∼Γa1+m1,b1−∑i=1m1Si+1lnΦ−xi:m1:n1σ,β|α,σ,data∼Γa2+m2,b2−∑i=1m2Si′+1lnΦ−yi:m2:n2σ,ησ|α,β,data∝σa3+m1+m2−1e−b3σ×∏i=1m1ϕxi:m1:n1σ∏i=1m1Φ−xi:m1:n1σαSi+α−1×∏i=1m2ϕyi:m2:n2σ∏i=1m2Φ−yi:m2:n2σβSi′+β−1,where Γ.,. stands for Gamma distribution.Since the posterior PDF ofσ is unknown, the normal distribution is a good proposal based on Figure 3. Hence, we apply the normal proposal distribution to simulate σ using the Metropolis–Hasting method. Therefore, the Gibbs sampling algorithm is the following.(1)
Step 1: take the initial valuesα0,β0,σ0 for α,β,σ.(2)
Step 2: setj=1.(3)
Step 3: simulateαj from Γa1+m1,b1−∑i=1m1Si+1lnΦ−xi:m1:n1/σj−1.(4)
Step 4: simulateβj from Γa2+m2,b2−∑i=1m2Si′+1lnΦ−yi:m2:n2/σj−1.(5)
Step 5: simulateσj from(26)ησ|αj−1,βj−1,data,with the Nσj−1,1 as a proposal distribution using the Metropolis–Hastings method.(6)
Step 6: determineRj using (5).(7)
Step 7: setj=j+1.(8)
Step 8:T times repeat steps 3–7.Figure 3
Proposal and posterior density functions of the scale parameter.Figure4 illustrates the convergence of the Metropolis–Hasting algorithm based on the normal proposal. In addition, the cumulative average graph (center) shows that the algorithm converges after 1000 iteration, and the autocorrelation graph demonstrates no correlation.Figure 4
Graphical diagnoses of Metropolis–Hasting algorithm. (a) A sequence of 1000 generations from posterior density functions ofσ; (b) cumulative average graph of σ sequence; (c) autocorrelation graph of σ sequence.The posterior mean and variance ofR can be evaluated as(27)E^R|data=1T∑j=1TRj,V^R|data=1T∑j=1TRj−E^R|data2.By choosing the shortest length fromR1,R1−γT,R2,R1−γT+1,…,RγT,RT, the HPD credible interval is determined in which Ri is the ith-order statistics from sample size T.


## 4. Estimation ofR with Known Scale Parameter

The estimation ofR is considered when the common scale parameter is known and equals 1.


### 4.1. The MLE ofR

Considering (12), the MLE of R will be(28)R^=β^α^+β^=11+m1W2/m2W1,where W1=−∑i=1m1Si+1lnΦ−Xi:m1:n1 and W2=−∑i=1m2Si′+1lnΦ−Yi:m2:n2.Lemma 3.
LetX1:m1:n1,X2:m1:n1,…,Xm1:m1:n1 be progressively Type-II censored samples from SNα,σ with censoring schemes S1,S2,…,Sm1. By defining(29)Z1=−n1lnΦ−X1:m1:n1,Zi=n1−∑k=1i−1Sk−i+1−lnΦ−Xi:m1:n1−−lnΦ−Xi−1:m1:n1,i=1,2,…,m1,then distribution of Zi, i=1,…,m1, is exponential with mean 1/α.Proof.
See the work of Balakrishnan and Aggarwala [21].Theorem 4.
Suppose thatX1:m1:n1,X2:m1:n1,…,Xm1:m1:n1 and Y1:m2:n2,Y2:m2:n2,…,Ym2:m2:n2 be two progressively Type-II censored samples from SNα,σ and SNβ,σ with censoring schemes S1,S2,…,Sm1 and S1′,S2′,…,Sm2′, respectively. Then, 1001−γ% CI for R is(30)11+1/R^−1F2m1,2m2;1−γ/2,11+1/R^−1F2m1,2m2;γ/2,where F2m1,2m1;γ/2 and F2m1,2m2;1−γ/2 are the lower and upper γ/2th percentile points of the F distribution with 2m1 and 2m2 degrees of freedom.Proof.
From Lemma3, it is evident that(31)W1=−∑i=1m1Si+1lnΦ−Xi:m1:n1,=∑i=1m1Zi∼Γm1,α,and 2αW1 is the chi-square random variable with 2m1 df, and likewise, 2βW2 is the chi-square random variable with 2m2 df. Thus, by using(32)R^∼11+α/βF,orR1−R×1−R^R^∼F2m1,2m2,PF2m1,2m2;γ/2<R1−R×1−R^R^<F2m1,2m2;1−γ/2=1−γ,
The proof is completed.From Theorem4, the PDF of R^ is resulted as(33)fR^r=1Bm2,m1r2m2βm1αm21−r/rm2−11+m2β/m1α1−r/rm1+m2,0<r<1.


### 4.2. The UMVUE ofR

The UMVUE ofR is obtained using the following lemma. Because of the convenience of proof, we ignore it.Lemma 5.
IfT1,T2,…,Tn have i.i.d Exp λ, then conditional density function of T1 given Y=∑i=1nTi is(34)fT1|Yt1|y=n−1y1−t1yn−2,0<t1<y.Theorem 6.
The UMVUE ofR is(35)R˜=1−∑k=0m2−1−1km2−1km1+k−1kW1W2k,ifW1≤W2,∑k=0m1−1−1km1−1km2+k−1kW2W1k,ifW2≤W1,where W1,W2 is a CS statistic for α,β.Proof.
From Lemma3, −n1lnΦ−X1:m1:n1∼Expα and −n2lnΦ−Y1:m2:n2∼Expβ, so by defining(36)IX1:m1:n1,Y1:m2:n2=1,if−n2lnΦ−Y1:m2:n2<−n1lnΦ−X1:m1:n1,0,if−n2lnΦ−Y1:m2:n2>−n1lnΦ−X1:m1:n1.
It is clear thatIX1:m1:n1,Y1:m2:n2 is the unbiased estimator of R. Then, using Lehmann and Scheffe theorem and the fact that W1,W2 is a CS statistic [27], the UMVUE is given by(37)R˜=EIX1:m1:n1,Y1:m2:n2|W1=w1,W2=w2,=P−n2lnΦ−Y1:m2:n2<−n1lnΦ−X1:m1:n1|W1=w1,W2=w2=∫B∫fZ1|W1=w1z1|w1fV1|W2=w2v1|w2dz1dv1,where(38)B=z1,v1|0<z1<w1n1,0<v1<w2n2,n1z1>n2v1,and Z1=−lnΦ−X1:m1:n1 and V1=−lnΦ−Y1:m2:n2. Equation (37) using Lemma 5 and for w1<w2 is obtained in the following way:(39)R˜=∫0w1/n1∫0n1z1/n2n1m1−1w1−n1z1m1−2w1m1−1n2m2−1×w2−n2v1m2−2w2m2−1dv1dz1=1−n1m1−1w1m1−1w2m2−1∫0w1/n1w1−n1z1m1−2w2−n1z1m2−1dz1=1−m1−1∫011−tm1−21−w1tw2m2−1dt.
Considering the binomial expansion1−w1t/w2m2−1=∑k=0m2−1−1km2−1kw1t/w2k, for w1<w2, R˜ is simplified as(40)R˜=1−∑k=0m2−1−1km2−1km1+k−1kw1w2k.
By the same method, one can show that, forw2<w1,(41)R˜=∑k=0m1−1−1km1−1km2+k−1kw2w1k.
□


### 4.3. The Bayesian Estimation ofR

Since we assumed that the parameters are a priori independent with gamma density, the posterior density ofα and β are independent Γa1+m1,b1+W1 and Γa2+m2,b2+W2, receptively. Therefore, the posterior distribution of R will be(42)ηR|data=c.ra1+m1−11−ra2+m2−1,1−zr−a1+a2+m1+m2,where 0<r<1,(43)c=Γa1+a2+m1+m2Γa1+m1Γa2+m2.b2+W2b1+W1a2+m2,and z=1−b2+W2/b1+W1.By assuming the quadratic loss function, the Bayesian estimation will be the posterior mean which could be computed by considering the following well-known equation:(44)Bb,c−bF21a,b;c;z=∫01xb−11−xc−b−11−zx−adx,c>b>0,where Bb,c−b and F21a,b;c;z are beta and hypergeometric functions, respectively.Therefore, the Bayesian estimation ofR is(45)R^Bayes=ER|data=Γa1+a2+m1+m2Γa1+m1Γa2+m2b2+W2b1+W1a2+m2Ba1+m1+1,a2+m×F21q,a1+m1+1;a1+a2+m1+m2+1;1−b2+W2b1+W1,where q=a1+a2+m1+m2. The variance of the Bayesian estimator could be achieved by using(46)ER2|data=Γa1+a2+m1+m2Γa1+m1Γa2+m2.b2+W2b1+W1a2+m2Ba1+m1+2,a2+m×2F1q,a1+m1+2;a1+a2+m1+m2+2;1−b2+W2b1+W1,where q=a1+a2+m1+m2. As the posterior is not tractable, we can generate a sample from the posterior using an indirect sampling algorithm, such as the accept-reject method for constructing the HPD intervals.


## 4.1. The MLE ofR

Considering (12), the MLE of R will be(28)R^=β^α^+β^=11+m1W2/m2W1,where W1=−∑i=1m1Si+1lnΦ−Xi:m1:n1 and W2=−∑i=1m2Si′+1lnΦ−Yi:m2:n2.Lemma 3.
LetX1:m1:n1,X2:m1:n1,…,Xm1:m1:n1 be progressively Type-II censored samples from SNα,σ with censoring schemes S1,S2,…,Sm1. By defining(29)Z1=−n1lnΦ−X1:m1:n1,Zi=n1−∑k=1i−1Sk−i+1−lnΦ−Xi:m1:n1−−lnΦ−Xi−1:m1:n1,i=1,2,…,m1,then distribution of Zi, i=1,…,m1, is exponential with mean 1/α.Proof.
See the work of Balakrishnan and Aggarwala [21].Theorem 4.
Suppose thatX1:m1:n1,X2:m1:n1,…,Xm1:m1:n1 and Y1:m2:n2,Y2:m2:n2,…,Ym2:m2:n2 be two progressively Type-II censored samples from SNα,σ and SNβ,σ with censoring schemes S1,S2,…,Sm1 and S1′,S2′,…,Sm2′, respectively. Then, 1001−γ% CI for R is(30)11+1/R^−1F2m1,2m2;1−γ/2,11+1/R^−1F2m1,2m2;γ/2,where F2m1,2m1;γ/2 and F2m1,2m2;1−γ/2 are the lower and upper γ/2th percentile points of the F distribution with 2m1 and 2m2 degrees of freedom.Proof.
From Lemma3, it is evident that(31)W1=−∑i=1m1Si+1lnΦ−Xi:m1:n1,=∑i=1m1Zi∼Γm1,α,and 2αW1 is the chi-square random variable with 2m1 df, and likewise, 2βW2 is the chi-square random variable with 2m2 df. Thus, by using(32)R^∼11+α/βF,orR1−R×1−R^R^∼F2m1,2m2,PF2m1,2m2;γ/2<R1−R×1−R^R^<F2m1,2m2;1−γ/2=1−γ,
The proof is completed.From Theorem4, the PDF of R^ is resulted as(33)fR^r=1Bm2,m1r2m2βm1αm21−r/rm2−11+m2β/m1α1−r/rm1+m2,0<r<1.


## 4.2. The UMVUE ofR

The UMVUE ofR is obtained using the following lemma. Because of the convenience of proof, we ignore it.Lemma 5.
IfT1,T2,…,Tn have i.i.d Exp λ, then conditional density function of T1 given Y=∑i=1nTi is(34)fT1|Yt1|y=n−1y1−t1yn−2,0<t1<y.Theorem 6.
The UMVUE ofR is(35)R˜=1−∑k=0m2−1−1km2−1km1+k−1kW1W2k,ifW1≤W2,∑k=0m1−1−1km1−1km2+k−1kW2W1k,ifW2≤W1,where W1,W2 is a CS statistic for α,β.Proof.
From Lemma3, −n1lnΦ−X1:m1:n1∼Expα and −n2lnΦ−Y1:m2:n2∼Expβ, so by defining(36)IX1:m1:n1,Y1:m2:n2=1,if−n2lnΦ−Y1:m2:n2<−n1lnΦ−X1:m1:n1,0,if−n2lnΦ−Y1:m2:n2>−n1lnΦ−X1:m1:n1.
It is clear thatIX1:m1:n1,Y1:m2:n2 is the unbiased estimator of R. Then, using Lehmann and Scheffe theorem and the fact that W1,W2 is a CS statistic [27], the UMVUE is given by(37)R˜=EIX1:m1:n1,Y1:m2:n2|W1=w1,W2=w2,=P−n2lnΦ−Y1:m2:n2<−n1lnΦ−X1:m1:n1|W1=w1,W2=w2=∫B∫fZ1|W1=w1z1|w1fV1|W2=w2v1|w2dz1dv1,where(38)B=z1,v1|0<z1<w1n1,0<v1<w2n2,n1z1>n2v1,and Z1=−lnΦ−X1:m1:n1 and V1=−lnΦ−Y1:m2:n2. Equation (37) using Lemma 5 and for w1<w2 is obtained in the following way:(39)R˜=∫0w1/n1∫0n1z1/n2n1m1−1w1−n1z1m1−2w1m1−1n2m2−1×w2−n2v1m2−2w2m2−1dv1dz1=1−n1m1−1w1m1−1w2m2−1∫0w1/n1w1−n1z1m1−2w2−n1z1m2−1dz1=1−m1−1∫011−tm1−21−w1tw2m2−1dt.
Considering the binomial expansion1−w1t/w2m2−1=∑k=0m2−1−1km2−1kw1t/w2k, for w1<w2, R˜ is simplified as(40)R˜=1−∑k=0m2−1−1km2−1km1+k−1kw1w2k.
By the same method, one can show that, forw2<w1,(41)R˜=∑k=0m1−1−1km1−1km2+k−1kw2w1k.
□


## 4.3. The Bayesian Estimation ofR

Since we assumed that the parameters are a priori independent with gamma density, the posterior density ofα and β are independent Γa1+m1,b1+W1 and Γa2+m2,b2+W2, receptively. Therefore, the posterior distribution of R will be(42)ηR|data=c.ra1+m1−11−ra2+m2−1,1−zr−a1+a2+m1+m2,where 0<r<1,(43)c=Γa1+a2+m1+m2Γa1+m1Γa2+m2.b2+W2b1+W1a2+m2,and z=1−b2+W2/b1+W1.By assuming the quadratic loss function, the Bayesian estimation will be the posterior mean which could be computed by considering the following well-known equation:(44)Bb,c−bF21a,b;c;z=∫01xb−11−xc−b−11−zx−adx,c>b>0,where Bb,c−b and F21a,b;c;z are beta and hypergeometric functions, respectively.Therefore, the Bayesian estimation ofR is(45)R^Bayes=ER|data=Γa1+a2+m1+m2Γa1+m1Γa2+m2b2+W2b1+W1a2+m2Ba1+m1+1,a2+m×F21q,a1+m1+1;a1+a2+m1+m2+1;1−b2+W2b1+W1,where q=a1+a2+m1+m2. The variance of the Bayesian estimator could be achieved by using(46)ER2|data=Γa1+a2+m1+m2Γa1+m1Γa2+m2.b2+W2b1+W1a2+m2Ba1+m1+2,a2+m×2F1q,a1+m1+2;a1+a2+m1+m2+2;1−b2+W2b1+W1,where q=a1+a2+m1+m2. As the posterior is not tractable, we can generate a sample from the posterior using an indirect sampling algorithm, such as the accept-reject method for constructing the HPD intervals.


## 5. Simulation Study

For evaluating the proposed methods, we implement the methods on artificial datasets. For inference ofR, two cases are considered: (i) unknown σ and (ii) known σ.In the first case, the performances of the MLE and the Bayes estimation in terms of biases and mean squares errors (MSE) are compared. Furthermore, the asymptotic confidence intervals (CIs) and the HPD credible intervals are compared with average confidence lengths and coverage percentages. We use MATLAB software for all the computations. For comparing the performance of the MLE and Bayes estimation, we do the following steps (Table1).(1)
Step 1: we consider different parameter values,θ1=α=1,β=1,σ=0.5, θ2=α=1,β=2,σ=1, and θ3=α=2,β=0.75,σ=1.5, and we assign n=n1=n2=30 and m=m1=m2=10 for all schemes.(2)
Step 2: we consider three progressively censored schemes in Table1.(3)
Step 3: for generating two independent progressive Type-II censored samplesX1:m1:n1,X2:m1:n1,…,Xm1:m1:n1 from SNα,σ and Y1:m2:n2,Y2:m2:n2,…,Ym2:m2:n2 from SNβ,σ based on the algorithm of Balakrishnan and Sandhu [28].(4)
Step 4: to evaluate the Bayes estimation and HPD credible intervals, noninformative prior (P1) and two informative gamma priors (P2 and P3) are assumed as(47)P1:aj=0.0001,bj=0.0001,j=1,2,3,P2:aj=1,bj=1,j=1,2,3,P3:aj=2,bj=3,j=1,2,3.(5)
Step 5: for extracting MLE ofR, the estimation of σ is computed from (10). The iterative process stops when the difference between the two consecutive iterates becomes less than 10−6. The 95% asymptotic confidence intervals are obtained by applying Theorem 2.(6)
Step 6: in the algorithm of Gibbs sampling, we takeT = 10000 and assume 20% of T as burn-in-period experimentally for eliciting the Bayes estimators and HPD credible intervals. Therefore, the results will not be affected by the initial values of the parameters α0,β0,σ0.(7)
Step 7: the average biases and MSE for the MLE and Bayesian estimations are reported in Table2 for 1000 replications.(8)
Step 8: average lengths and coverage percentages of the 95% CIs and HPD credible intervals forR are reported in Table 3 for various sets of parameters.Table 1
Censoring schemes.
Schemem,nP.C.S.psc110,300,0,0,0,0,0,0,0,0,20psc210,3020,0,0,0,0,0,0,0,0,0psc310,302,2,2,2,2,2,2,2,2,2As seen in Table2, both estimators are quite competitive. The results of Table 2 indicate that the MLE outperforms the Bayes estimation in terms of bias and MSE. Also, Table 2 demonstrates that almost all the Bayes estimators are not affected by the values of hyperparameters. From Table 3, we observe that the average lengths of credible intervals are less than the average lengths of the CIs. Furthermore, the coverage percentage of the MLE is less than the Bayes estimator.Table 2
The MSE (bias) for the estimators ofR with unknown σ.
θpcsi,pcsjMLEP1P2P3θ11,10.01930.02140.02230.0209(0.0602)(−0.0631)(−0.0609)(−0.0568)1,20.01250.01790.01710.0152(0.0454)(−0.0608)(0.0529)(−0.0527)1,30.03440.04760.04310.0401(0.0211)(0.0278)(0.0266)(0.0239)2,20.01250.01610.01620.0148(0.0149)(0.0241)(0.0232)(0.0214)2,30.01320.01750.01670.0166(0.0135)(0.0217)(0.0193)(−0.0191)3,30.01240.01820.01740.0163(0.0324)(−0.0467)(−0.0441)(−0.0384)θ21,10.02180.02570.02430.0237(−0.0453)(0.0606)(0.0566)(0.0535)1,20.01750.02350.02190.0209(−0.0346)(0.0511)(0.0474)(0.0460)1,30.01300.01870.01610.0156(−0.0269)(−0.0342)(0.0316)(-0.0308)2,20.01100.01800.01610.0151(0.0171)(0.0338)(0.0320)(−0.0293)2,30.00900.01500.01220.0113(0.0130)(0.0312)(0.0259)(−0.0209)3,30.02140.02430.02450.0219(0.0348)(−0.0501)(0.0472)(0.0439)θ31,10.02140.02640.02610.0230(−0.0291)(0.0370)(−0.0352)(−0.0326)1,20.01110.01820.01870.0162(0.0225)(−0.0402)(0.0380)(−0.0344)1,30.00800.01020.01040.0099(−0.0174)(−0.0245)(0.0248)(0.0217)2,20.00850.01220.01150.0118(−0.0172)(0.0246)(0.0239)(0.0230)2,30.01260.02090.01770.0179(0.0144)(0.0252)(−0.0218)(0.0215)3,30.01690.02510.02490.0230(−0.0353)(−0.0510)(0.0481)(−0.0459)Table 3
Average of confidence/credible length (coverage percentage).
θpcsi,pcsjMLEP1P2P3θ11,10.3510.3340.3370.333(0.883)(0.931)(0.930)(0.933)1,20.3400.3320.3240.334(0.883)(0.943)(0.937)(0.941)1,30.3390.3240.3320.325(0.888)(0.945)(0.942)(0.940)2,20.3440.3210.3190.322(0.895)(0.951)(0.939)(0.943)2,30.3420.3190.3210.321(0.889)(0.953)(0.953)(0.949)3,30.3520.3310.3310.324(0.909)(0.955)(0.948)(0.949)θ21,10.3520.3370.3380.339(0.886)(0.934)(0.944)(0.949)1,20.3410.3200.3230.321(0.882)(0.943)(0.951)(0.949)1,30.3460.3240.3300.321(0.896)(0.947)(0.955)(0.951)2,20.3500.3220.3190.313(0.897)(0.951)(0.941)(0.939)2,30.3440.3150.3130.329(0.874)(0.943)(0.945)(0.955)3,30.3520.3210.3230.314(0.915)(0.954)(0.941)(0.952)θ31,10.3420.3150.3180.316(0.893)(0.957)(0.950)(0.951)1,20.3400.3160.3160.313(0.906)(0.962)(0.960)(0.957)1,30.3340.3110.3150.312(0.897)(0.955)(0.962)(0.951)2,20.3320.3040.3090.303(0.912)(0.958)(0.952)(0.960)2,30.3320.3060.3130.304(0.898)(0.950)(0.953)(0.953)3,30.3410.3360.3330.322(0.904)(0.951)(0.949)(0.960)We study the second case in which the scale parameterσ is known. In this case, the MLE and the 95% exact CI of R are constructed using (28) and (30), respectively. Also, the UMVUE and Bayesian estimation of R can be computed using (35) and (45), respectively. All results are presented in Table 4, demonstrating the Bayesian estimator outperforms others and has the smallest MSE of other estimators. As expected, the coverage probabilities for the 95% exact CI of R are close to 95%. Also, the results show that the scheme 2,2 has the smallest bias and MSE of other schemes.Table 4
Compute MSE (bias) of the estimations and the average confidence length (coverage percentage), setσ=1.
θpsci,pscjMLEUMVUEBayesCIθ11,10.01100.01180.01020.3942(0.0029)(0.0019)(−0.0070)(0.9432)1,20.01110.01140.01000.3961(0.0024)(−0.0025)(−0.0059)(0.9523)1,30.01120.01210.01040.3935(−0.0044)(0.0006)(−0.0089)(0.9486)2,20.01070.01170.00980.3932(−0.0015)(0.0035)(-0.0062)(0.9510)2,30.01080.01200.01010.3940(−0.0033)(0.0017)(−0.0077)(0.9426)3,30.01140.01200.01050.3927(0.0002)(0.0050)(−0.0049)(0.9421)θ21,10.00920.01010.00920.3731(0.0110)(0.0119)(0.0148)(0.9558)1,20.00960.01060.00890.3723(0.0073)(0.0002)(0.0144)(0.9526)1,30.00920.01020.00910.3729(0.0090)(0.0020)(0.0163)(0.9571)2,20.00940.01070.00920.3727(0.0072)(0.0004)(0.0147)(0.9514)2,30.00940.01020.00880.3722(0.0084)(0.0015)(0.0154)(0.9556)3,30.01030.01090.00990.3717(0.0077)(0.0002)(0.0147)(0.9421)θ31,10.01230.01340.01110.4012(−0.0030)(−0.0003)(−0.0053)(0.9401)1,20.01310.01430.01190.4014(−0.0054)(−0.0024)(0.0075)(0.9330)1,30.01090.01200.01060.4018(0.0004)(−0.0021)(−0.0035)(0.9492)2,20.01130.01250.01050.4017(0.0025)(−0.0004)(0.0050)(0.9517)2,30.01230.01350.01150.4012(−0.0027)(0.0003)(−0.0058)(0.9403)3,30.01160.01310.01090.4020(−0.0043)(0.0023)(−0.0070)(0.9602)


## 6. Data Analysis

In this section, two real strength data reported by Badar and Priest [23] are analyzed (Table 5). The data represent the strength measured in GPA (GigaPascal, GPA = KN/mm2, Kilonewten/squared millimeter) for single carbon fibers and impregnated 1000 carbon fiber tows. Single fibers were tested under tension at gauge lengths of 10 mm (dataset I) and 20 mm (dataset II). The fibers were randomly selected; therefore, the observations are iid. Data in Table 5 were reported in the ascending order. Babayi and Khorram [29] used Type-II generalized Logistic distribution (GL) to analyze these data sets. We try to show the SN distribution models for fitting these datasets are better than GL models. For this purpose, we apply the maximized value of the log-likelihood function (log L), the Kolmogorov–Smirnov (K–S) test statistic, the corresponding p value, and the Akaike information criterion AIC=2k−2logL, where k is the number of parameters in the model. The results are presented in Table 6. According to the results, for data set II, AIC and log-likelihood display that the GL model is better than the SN model. Still, the K–S statistic and p value indicate that the SN model is better than the GL model. For dataset I, all of the log-likelihood, AIC, K–S statistics, and p value indicate that the SN model is better than the GL model. Figure 5 shows the empirical and theoretical cdf graphs. Also, Figure 6 displays the histograms with estimated models for the two data sets. We can observe that the SN distribution fits very well for these datasets.Table 5
The real datasets.
Dataset I (x)Dataset II (y)1.9012.1322.2032.2282.2572.3502.3611.3121.3141.4791.5521.7001.8031.8612.3962.3972.4452.4542.4742.5182.5221.8651.9441.9581.9661.9972.0062.0212.5252.5322.5752.6142.6162.6182.6242.0272.0552.0632.0982.1402.1792.2242.6592.6752.7382.7402.8562.9172.9282.2402.2532.2702.2722.2742.3012.3012.9372.9372.9772.9963.0303.1253.1392.3592.3822.3822.4262.4342.4352.4783.1453.2203.2233.2353.2433.2643.2722.4902.5112.5142.5352.5542.5662.5703.2943.3323.3463.3773.4083.4353.4932.5862.6292.6332.6422.6482.6842.6973.5013.5373.5543.5623.6283.8523.8712.7262.7702.7732.8002.8092.8182.8213.8863.9714.0244.0274.2254.3955.0202.8482.8802.9543.0123.0673.0843.0903.0963.1283.2333.4333.5853.585Table 6
The goodness of fit test for SN and GL distributions on the real datasets.
Data setModelLog-likelihoodAICK–Sp valueIGL distribution−59.1378122.27560.11510.3471SN distribution−58.7108121.42060.10130.5056IIGL distribution−91.1909186.38170.07910.7508SN distribution−105.7333215.46650.04120.9995Figure 5
Empirical CDF plots of SN distributions for datasets. (a) Dataset I. (b) Dataset II.
(a)(b)Figure 6
Fitting the SN distribution to datasets.Based on the four different progressive censoring schemes, we compute the MLE and corresponding %95 asymptotic confidence intervals ofR from (12) and Theorem 2, respectively. Also, we extract the Bayes estimates and HPD credible intervals using the Gibbs sampling algorithm. According to Congdon [30] and Kundu and Gupta [31], we set noninformative prior (P1) to compute the Bayesian estimation. All the results are presented in Table 7. In the first censoring scheme in Table 7, we consider inference of R based on completed datasets. Comparing the results show that both the ML and Bayes estimators perform well, and there is no significant difference between them. Also, we observe that the length of HPD credible intervals is less than the length of the asymptotic confidence intervals.Table 7
The ML, Bayes estimations, and %95 asymptotic confidence/HPD interval ofR using real dataset.
n1,m1n2,m2SchemesMLECIMLEBayesHPD(63, 63)S=0,0,…,01×630.5766(0.4964, 0.6568)0.5748(0.5314, 0.6109)(69, 69)S′=0,0,…,01×69(63, 20)S=2,2,…,2,51×200.5273(0.3763, 0.6784)0.5261(0.4758, 0.5753)(69, 20)S′=2,2,…,2,111×20(63, 20)S=43,0,…,0,01×200.5999(0.5251, 0.6748)0.5982(0.5445, 0.6471)(69, 20)S′=49,0,…,0,01×20(63, 30)S=0,0,…,0,331×300.5260(0.4003, 0.6516)0.5235(0.4802, 0.5637)(69, 30)S′=0,0,…,0,391×30


## 7. Conclusions

In this study, we have studied the inference of reliability parameterR=PY<X for skew-normal distribution in the presence of progressive Type-II censored samples. First, we have considered the scale parameter to be unknown. We have extracted MLE of R and corresponding asymptotic confidence interval using an iterative procedure. Also, we have derived Bayes estimation and HPD credible interval of R applying Gibbs sampling technique. In continuation of our discussion, assuming that the scale parameter is known, we have obtained MLE, UMVUE, and Bayes estimators and the exact confidence interval of R. Furthermore, a simulation study is implemented to compare the performances of different proposed methods. It has been observed that the average lengths of credible intervals are less than the average lengths of the confidence intervals and the coverage percentage of the MLE is less than the Bayes estimator. Finally, the real dataset analysis is presented to illustrate the applicability of the model. In future work, the inference of stress-strength based on ranked set sampling or record values for skew-normal distribution can be made. Also, the neutrosophic inference of stress-strength for skew-normal distribution can be extended [32, 33].



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*Source: 7173293-2022-06-16.xml*