# Classical and Bayesian Inference under Burr-X Distribution Based on New Unified Progressive Hybrid Censoring Scheme with Engineering Applications

Authors: Saieed F. Ateya; Randa Alharbi; Mutua Kilai; Ramy Aldallal
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, a new unified progressive hybrid censoring schemeUPHCS has been constructed. This unified censoring scheme covers eleven famous censoring schemes. The estimation problem of Burr-X distribution parameters has been studied using the maximum likelihood and Bayes approaches based on the suggested unified progressive hybrid censored samples. Two real data sets have been used as illustrative engineering examples.


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## Body

## 1. Introduction

For reasons of cost and time, life testing studies must be terminated before all failures are observed. Data censoring techniques are extensively employed to reduce test duration and costs. Both conventional type-I and type-II censoring are commonly employed in life testing. The progressive type-II censoring schemePCS is currently the most used. As an illustration, consider the following: assume that r is a predetermined number of failures in a life-time test of n identical units or devices, where r≤n. R1 surviving units are deleted from the test when the first failure t1:r:n occurs. Whenever the second failure t2:r:n takes place, the remaining units are also removed from the test. This goes on until the r-th failure is observed, where the test is terminated and the remaining surviving Rr=n−r−R1+⋯+Rr−1 units are removed. For more details, see [9, 10].Many censoring schemes that are linked to the preceding censoring scheme are established, including type-I progressive hybrid censoring schemeType−IPHCS [13], whenever the life testing experiments are stopped at time T∗=mintr:n,τ, where τ∈0,∞ and 1≤r≤n are predetermined, type-II progressive hybrid censoring scheme Type−IIPHCS [13], that stopes the experiment at time T∗=maxtr:n,τ, generalized Type−IPHCS [20], that stopes the experiment at time T∗=maxtk:n,mintr:n,τ, where 1≤k≤r≤n, and generalized Type−IIPHCS [3, 6, 7], that terminates the experiment at time T∗=maxτ1,mintr:n,τ2, where 0<τ1≤τ2<∞.The contribution of this paper is suggesting a new unified progressively hybrid censoring scheme that covers eleven famous censoring schemes. Also, two real data sets are used as illustrative examples in the engineering field.The suggested censoring scheme can be described as follows.For predetermined,k,r∈1,2,…,n and τ1<τ2∈0,∞, where k<r, in the event that the failure number k happens before time τ1, the experiment should be stopped at time minmaxtr:n,τ1,τ2. In case of the failure number k happens at the interval between τ1 and τ2, the experiment should be stopped at the time mintr:n,τ2, and in case of the failure number k happens after time τ2, then the experiment should be stopped at time tk:n. This censoring scheme allows us to guarantee that the experiment will be completed with at most time τ2 with at least k failures, and if it is not, we can guarantee precisely k fails. Thus, we have the following UPHCS with six cases (according to the relationship between τ1,τ2,tk, and tr).(1)
If0<tk:n<tr:n<τ1<τ2, then the experiment should be stopped at time τ1(2)
If0<tk:n<τ1<tr:n<τ2, then the experiment should be stopped at time tr:n(3)
If0<tk:n<τ1<τ2<tr:n, then the experiment should be stopped at time τ2(4)
If0<τ1<tk:n<tr:n<τ2, then the experiment should be stopped at time tr:n(5)
If0<τ1<tk:n<τ2<tr:n, then the experiment should be stopped at time τ2(6)
If0<τ1<τ2<tk:n<tr:n, then the experiment should be stopped at time tk:nIt is clear that the proposedUPHCS includes a second termination time τ2 in addition to τ1 and the second number k in addition to r in order to provide more flexibility than the generalized type-I PHCS and generalized type-II PHCS, also to have more observations which will develop the inference.Assume that there have beendi failures up to the time ti≡ti:n, i=1,2. Then, given a sample related to this censoring scheme, the likelihood function LF will be in the following form:(1)Lθ|data∝∏i=1dfti;θRti;θRiRτ1;θRd∗,Rd∗=n−d−∑i=1r−1Ri,Rr=0ford1=d2=r,r+1,…,n,∏i=1rfti;θRti;θRi,d1=k,k+1,…,r−1,d2=r,∏i=1d2fti;θRti;θRiRτ2;θRd2∗,Rd2∗=n−d2−∑i=1d2Ri,d1,d2=k,k+1,k+2,…,r−1,d1<d2∏i=1rfti;θRti;θRi,d1=0,1,…,k−1,d2=r,∏i=1d2fti;θRti;θRiRτ2;θRd2∗,Rd2∗=n−d2−∑i=1d2Ri,d1=0,1,…,k−1,d2=k,k+1,2,…,r−1,∏i=1kfti;θRti;θRi,d1,d2=0,2,…,k−1.Special cases:From the suggestedUPHCS, many well-known censoring schemes can be obtained as follows:(1)
Unified HCS [8] when Ri=0,i=1,2,…,r−1,Rr=n−r(2)
Generalized type-I PHCS [20] when τ1⟶0(3)
Generalized type-II PHCS [3, 6, 7] when k=1 and τ1>tk(4)
Generalized type-I HCS [11] when τ1⟶0 and Ri=0,i=1,2,…,r−1,Rr=n−r(5)
Generalized type-II HCS [11] when k=1, τ1>tk and Ri=0,i=1,2,…,r−1,Rr=n−r(6)
Type-I PHCS [13] when τ1⟶0 and k=1(7)
Type-II PHCS [13] when τ2⟶∞ and k=1(8)
Type-I HCS [12, 19] when τ1⟶0 and k=1 and Ri=0,i=1,2,…,r−1,Rr=n−r(9)
Type-II HCS [12] when τ2⟶∞ and k=1 and Ri=0,i=1,2,…,r−1,Rr=n−r(10)
Type-I censoring [18] when τ1⟶0, k=1, r=n and Ri=0,i=1,2,…,r−1,Rr=n−r(11)
Type-II censoring [18] when τ1⟶0, τ2⟶∞ and k=1 and Ri=0,i=1,2,…,r−1,Rr=n−rA random variableT is said to have a Burr-X with vector of parameters θ=η,γ if its PDF is given by(2)ft;η,γ=2ηγxe−γt21−e−γt2η−1,t>0,η>0,γ>0.The corresponding cumulative distribution functionCDF and reliability function RF are given, respectively, as(3)Ft;η,γ=1−e−γt2η,t>0,η>0,γ>0,(4)Rt;η,γ=1−1−e−γt2η,t>0,η>0,γ>0.Estimation and prediction problems are studied under the Burr model by many authors, see [1, 2, 17, 22].The paper is organized as follows: in Section2, the point and interval estimation problems have been studied using the maximum likelihood (classical) approach. In Section 3, the same problems have been studied using the Bayesian approach. The point and interval estimates of the suggested distribution parameters are obtained based on simulated and real data in Section 4. Finally, the paper is concluded in Section 5.


## 2. Maximum Likelihood Estimation

In this section, theLF (1) has been maximized to obtain the maximum likelihood estimates MLE′s of the parameters η and γ. Then, the 1−ϵ×100% approximate and bootstrap confidence intervals of the same parameters have been obtained.


### 2.1. Point Estimation

Based on the suggestedUPHCS, the LF of the parameters η and γ given a progressively type-II censored sample t=t1:r:n,t2:r:n,…,tr:r:n, which may be written as t=t1,…,tr, can be obtained by substituting (2) and (4) in (1) as follows:(5)Lη,γ|t∝ηdγde−γ∑i=1dti2∏i=1dti∏i=1d1−1−e−γti2η1−1−e−γτ12ηRd∗,Rd∗=n−d−∑i=1r−1Ri,Rr=0,ford1=d2=r,r+1,…,n,ηrγre−γ∑i=1rti2∏i=1rti∏i=1r1−1−e−γti2ηRi,d1=k,k+1,…,r−1,d2=r,ηd2γd2e−γ∑i=1d2ti2∏i=1d2ti∏i=1d21−1−e−γti2η1−1−e−γτ22ηRd2∗,Rd2∗=n−d2−∑i=1d2Ri,d1,d2=k,k+1,k+2,…,r−1,d1<d2,ηrγre−γ∑i=1rti2∏i=1rti∏i=1r1−1−e−γti2ηRi,d1=0,1,…,k−1,d2=r,ηd2γd2e−γ∑i=1d2ti2∏i=1d2ti∏i=1d21−1−e−γti2η1−1−e−γτ22ηRd2∗,Rd2∗=n−d2−∑i=1d2Ri,d1=0,1,…,k−1,d2=k,k+1,2,…,r−1,ηkγke−γ∑i=1kti2∏i=1kti∏i=1k1−1−e−γti2ηRi,d1,d2=0,2,…,k−1.To obtain theMLE′s of η and γ, denoted by η^M and γ^M, the values of the parameters η and γ which maximize the function LogLη,γ|t are obtained numerically.


### 2.2. Approximate Confidence Interval

Under the assumption thatη^,γ^ is approximately bivariate normal with mean η,γ and covariance matrix I−1η,γ, which can be approximated by I0η^,γ^, where(6)I0η^,γ^=−∂2LogLη,γ|t∂η2−∂2LogLη,γ|t∂η∂γ−∂2LogLη,γ|t∂η∂γ−∂2LogLη,γ|t∂γ2η^,γ^.The approximated confidence intervals forη and γ may be determined, respectively, by using the following formulas:(7)η^∓zϵ/2ν11,γ^∓zϵ/2ν22,where ν11 and ν22 are the main diagonal elements of the covariance matrix I0−1η^,γ^ and zϵ/2 is the standard normal variate such that PZ≤ϵ/2=1−ϵ/2.


### 2.3. Bootstrap Confidence Interval

In this section, based on the idea of [14], bootstrap confidence intervals have been obtained as follows:(1)
Based on the suggested unified progressive hybrid censored sample with censoring schemeR=R1,R2,…,Rr, t=t1,t2,…,tr and selected τ1,τ2,r, and k, the MLEs of the parameters η and γ, η^ and γ^ have been obtained(2)
Using the estimated parameters in step 1, a progressive type-II censored samplet∗, with the same censoring scheme R=R1,R2,…,Rr, is generated(3)
Based on the unified progressive hybrid censored sample related tot∗ and the same values of τ1,τ2,r, and k, the MLEs of the parameters η and γ say η^∗ and γ^∗ have been obtained(4)
Repeat steps 2 and 3N times(5)
After arranging allη^∗’s and γ^∗’s in an ascending order to be η^∗1,η^∗2,…,η^∗N and γ^∗1,γ^∗2,…,γ^∗N, η∗Nτ/2,η∗N1−τ/2 and γ∗Nτ/2,γ∗N1−τ/2 will be the two-sided 1−τ×100%bootstrap−p confidence intervals for η and γ


## 2.1. Point Estimation

Based on the suggestedUPHCS, the LF of the parameters η and γ given a progressively type-II censored sample t=t1:r:n,t2:r:n,…,tr:r:n, which may be written as t=t1,…,tr, can be obtained by substituting (2) and (4) in (1) as follows:(5)Lη,γ|t∝ηdγde−γ∑i=1dti2∏i=1dti∏i=1d1−1−e−γti2η1−1−e−γτ12ηRd∗,Rd∗=n−d−∑i=1r−1Ri,Rr=0,ford1=d2=r,r+1,…,n,ηrγre−γ∑i=1rti2∏i=1rti∏i=1r1−1−e−γti2ηRi,d1=k,k+1,…,r−1,d2=r,ηd2γd2e−γ∑i=1d2ti2∏i=1d2ti∏i=1d21−1−e−γti2η1−1−e−γτ22ηRd2∗,Rd2∗=n−d2−∑i=1d2Ri,d1,d2=k,k+1,k+2,…,r−1,d1<d2,ηrγre−γ∑i=1rti2∏i=1rti∏i=1r1−1−e−γti2ηRi,d1=0,1,…,k−1,d2=r,ηd2γd2e−γ∑i=1d2ti2∏i=1d2ti∏i=1d21−1−e−γti2η1−1−e−γτ22ηRd2∗,Rd2∗=n−d2−∑i=1d2Ri,d1=0,1,…,k−1,d2=k,k+1,2,…,r−1,ηkγke−γ∑i=1kti2∏i=1kti∏i=1k1−1−e−γti2ηRi,d1,d2=0,2,…,k−1.To obtain theMLE′s of η and γ, denoted by η^M and γ^M, the values of the parameters η and γ which maximize the function LogLη,γ|t are obtained numerically.


## 2.2. Approximate Confidence Interval

Under the assumption thatη^,γ^ is approximately bivariate normal with mean η,γ and covariance matrix I−1η,γ, which can be approximated by I0η^,γ^, where(6)I0η^,γ^=−∂2LogLη,γ|t∂η2−∂2LogLη,γ|t∂η∂γ−∂2LogLη,γ|t∂η∂γ−∂2LogLη,γ|t∂γ2η^,γ^.The approximated confidence intervals forη and γ may be determined, respectively, by using the following formulas:(7)η^∓zϵ/2ν11,γ^∓zϵ/2ν22,where ν11 and ν22 are the main diagonal elements of the covariance matrix I0−1η^,γ^ and zϵ/2 is the standard normal variate such that PZ≤ϵ/2=1−ϵ/2.


## 2.3. Bootstrap Confidence Interval

In this section, based on the idea of [14], bootstrap confidence intervals have been obtained as follows:(1)
Based on the suggested unified progressive hybrid censored sample with censoring schemeR=R1,R2,…,Rr, t=t1,t2,…,tr and selected τ1,τ2,r, and k, the MLEs of the parameters η and γ, η^ and γ^ have been obtained(2)
Using the estimated parameters in step 1, a progressive type-II censored samplet∗, with the same censoring scheme R=R1,R2,…,Rr, is generated(3)
Based on the unified progressive hybrid censored sample related tot∗ and the same values of τ1,τ2,r, and k, the MLEs of the parameters η and γ say η^∗ and γ^∗ have been obtained(4)
Repeat steps 2 and 3N times(5)
After arranging allη^∗’s and γ^∗’s in an ascending order to be η^∗1,η^∗2,…,η^∗N and γ^∗1,γ^∗2,…,γ^∗N, η∗Nτ/2,η∗N1−τ/2 and γ∗Nτ/2,γ∗N1−τ/2 will be the two-sided 1−τ×100%bootstrap−p confidence intervals for η and γ


## 3. Bayes Estimation

In this section, the Bayes estimatesBEs (point and interval) have been obtained using the squared error loss function and the bivariate prior used in [4–7] which of the following form:(8)πη,γ∝ηc1+c3−1γc3−1e−ηγ+c2,η>0,γ>0,c1>0,c2>0,c3>0,where c1,c2, and c3 are the prior parameters.


### 3.1. Point Estimation

Using the squared error loss function, theBE of a general continuous function uθ of the vector of parameters θ=θ1,θ2,…,θK is given by(9)u^BSθ=Euθ|t=∫…∫uθπ∗θ|tdθ1,…,dθK,which can be approximated by(10)u^BSθ=∑i=1KuθiK,where θi=θ1i,…,θmi,i=1,2,…,K is a sample generated from the posterior PDFπ∗θ|t∝πθLθ|t using Gibbs sampler and Metropolis-Hastings techniques, see [16, 21, 23, 24].For the studied distribution,θ=η,γ and the posterior PDF, using the prior PDF (8) and the LF (5), can be written in the following form:(11)π∗η,γ|t=A1ηd+c1+c3−1γd+c3−1e−ηγ+c2e−γ∑i=1dti2∏i=1dti∏i=1d1−1−e−γti2η1−1−e−γτ12ηRd∗,Rd∗=n−d−∑i=1r−1Ri,Rr=0,ford1=d2=r,r+1,…,n,A2ηr+c1+c3−1γr+c3−1e−ηγ+c2e−γ∑i=1dti2∏i=1rti∏i=1r1−1−e−γti2ηRi,d1=k,k+1,…,r−1,d2=r,A3ηd2+c1+c3−1γd2+c3−1e−ηγ+c2e−γ∑i=1dti2∏i=1d2ti∏i=1d21−1−e−γti2η1−1−e−γτ22ηRd2∗,Rd2∗=n−d2−∑i=1d2Ri,d1,d2=k,k+1,k+2,…,r−1,d1<d2,A4ηr+c1+c3−1γr+c3−1e−ηγ+c2e−γ∑i=1dti2∏i=1rti∏i=1r1−1−e−γti2ηRi,d1=0,1,…,k−1,d2=r,A5ηd2+c1+c3−1γd2+c3−1e−ηγ+c2e−γ∑i=1dti2∏i=1d2ti∏i=1d21−1−e−γti2η1−1−e−γτ22ηRd2∗,Rd2∗=n−d2−∑i=1d2Ri,d1=0,1,…,k−1,d2=k,k+1,2,…,r−1,A6ηk+c1+c3−1γk+c3−1e−ηγ+c2e−γ∑i=1dti2∏i=1kti∏i=1k1−1−e−γti2ηRi,d1,d2=0,2,…,k−1,where Ai,i=1,2,…,6 are normalizing constants.Using Gibbs sampler and Metropolis-Hastings techniques to generate the sampleηi,γi,i=1,2,…,K and then using formula (10), the BEs of the parameters η and γ will be of the following forms:(12)η^BSθ=∑i=1KηiK,and(13)γ^BSθ=∑i=1KγiK,


### 3.2. Credibility Interval

For a certain0<ϵ<1, 1−ϵ×100% credibility intervals CI′s for the parameters η and γ are two intervals Lη,Uη and Lγ,Uγ, respectively, satisfy the following conditions:(14)∫Lη∞π1∗η|tdη=1−ϵ2,∫Uη∞π1∗η|tdη=ϵ2,∫Lγ∞π2∗γ|tdγ=1−ϵ2,∫Uγ∞π2∗|tdγ=ϵ2,where π1∗η|t and π2∗γ|t can be obtained using the previously generated sample ηi,γi,i=1,2,…,K from π∗η,γ|t as follows:(15)π1∗η|t=1K∑i=1Kπ∗η,γi|t,π2∗γ|t=1K∑i=1Kπ∗γ,ηi|t.Simple formulas for computing the credibility intervals forη and γ have been found by substituting from (15) into (14), and they are as follows:(16)1K∑i=1K∫Lη∞π∗η,γi|tdη=1−ϵ2,1K∑i=1K∫Uη∞π∗η,γi|tdη=ϵ2,1K∑i=1K∫Lγ∞π∗γ,ηi|tdγ=1−ϵ2,1K∑i=1K∫Uγ∞π∗γ,ηi|tdγ=ϵ2.Solving the set of nonlinear (16), the CIs for η and γ can be obtained (see [5, 6]).An alternative simple technique can be used to calculate theCIs for the parameters η and γ. This technique can be summarized in the following steps (see [5, 6]):(1)
Generateηi,γi,i=1,2,…,K from π∗η,γ|t(2)
Arrange allηi’s and γi’s in an ascending order to be η∗1,η∗2,…,η∗K and γ∗1,γ∗2,…,γ∗N(3)
A two-sided1−ϵ×100%CI for η, say Lη,Uη is then given by η∗Kϵ/2,η∗K1−ϵ/2(4)
Also, a two-sided1−ϵ×100%CI for γ, say Lγ,Uγ is given by γ∗Kϵ/2,γ∗K1−ϵ/2


### 3.3. Highest Posterior Density Interval

To calculate the1−ϵ×100% highest posterior density HPD interval for the parameter η, Lη,Uη, we should solve the following two nonlinear equations (see [5, 6]):(17)1K∑i=1K∫LηUηπ∗η,γi|tdη=1−ϵ,∑i=1Kπ∗Lη,γi|t=∑i=1Kπ∗Uη,γi|t.Similarly, the 1−ϵ×100%HPD interval for γ, Lγ,Uγ, can be calculated by solving the following two nonlinear equations (see [5, 6]):(18)1K∑i=1K∫LγUγπ∗γ,ηi|,tdγ=1−ϵ,∑i=1Kπ∗Lγ,ηi|,t=∑i=1Kπ∗Uγ,ηi|,t.


## 3.1. Point Estimation

Using the squared error loss function, theBE of a general continuous function uθ of the vector of parameters θ=θ1,θ2,…,θK is given by(9)u^BSθ=Euθ|t=∫…∫uθπ∗θ|tdθ1,…,dθK,which can be approximated by(10)u^BSθ=∑i=1KuθiK,where θi=θ1i,…,θmi,i=1,2,…,K is a sample generated from the posterior PDFπ∗θ|t∝πθLθ|t using Gibbs sampler and Metropolis-Hastings techniques, see [16, 21, 23, 24].For the studied distribution,θ=η,γ and the posterior PDF, using the prior PDF (8) and the LF (5), can be written in the following form:(11)π∗η,γ|t=A1ηd+c1+c3−1γd+c3−1e−ηγ+c2e−γ∑i=1dti2∏i=1dti∏i=1d1−1−e−γti2η1−1−e−γτ12ηRd∗,Rd∗=n−d−∑i=1r−1Ri,Rr=0,ford1=d2=r,r+1,…,n,A2ηr+c1+c3−1γr+c3−1e−ηγ+c2e−γ∑i=1dti2∏i=1rti∏i=1r1−1−e−γti2ηRi,d1=k,k+1,…,r−1,d2=r,A3ηd2+c1+c3−1γd2+c3−1e−ηγ+c2e−γ∑i=1dti2∏i=1d2ti∏i=1d21−1−e−γti2η1−1−e−γτ22ηRd2∗,Rd2∗=n−d2−∑i=1d2Ri,d1,d2=k,k+1,k+2,…,r−1,d1<d2,A4ηr+c1+c3−1γr+c3−1e−ηγ+c2e−γ∑i=1dti2∏i=1rti∏i=1r1−1−e−γti2ηRi,d1=0,1,…,k−1,d2=r,A5ηd2+c1+c3−1γd2+c3−1e−ηγ+c2e−γ∑i=1dti2∏i=1d2ti∏i=1d21−1−e−γti2η1−1−e−γτ22ηRd2∗,Rd2∗=n−d2−∑i=1d2Ri,d1=0,1,…,k−1,d2=k,k+1,2,…,r−1,A6ηk+c1+c3−1γk+c3−1e−ηγ+c2e−γ∑i=1dti2∏i=1kti∏i=1k1−1−e−γti2ηRi,d1,d2=0,2,…,k−1,where Ai,i=1,2,…,6 are normalizing constants.Using Gibbs sampler and Metropolis-Hastings techniques to generate the sampleηi,γi,i=1,2,…,K and then using formula (10), the BEs of the parameters η and γ will be of the following forms:(12)η^BSθ=∑i=1KηiK,and(13)γ^BSθ=∑i=1KγiK,


## 3.2. Credibility Interval

For a certain0<ϵ<1, 1−ϵ×100% credibility intervals CI′s for the parameters η and γ are two intervals Lη,Uη and Lγ,Uγ, respectively, satisfy the following conditions:(14)∫Lη∞π1∗η|tdη=1−ϵ2,∫Uη∞π1∗η|tdη=ϵ2,∫Lγ∞π2∗γ|tdγ=1−ϵ2,∫Uγ∞π2∗|tdγ=ϵ2,where π1∗η|t and π2∗γ|t can be obtained using the previously generated sample ηi,γi,i=1,2,…,K from π∗η,γ|t as follows:(15)π1∗η|t=1K∑i=1Kπ∗η,γi|t,π2∗γ|t=1K∑i=1Kπ∗γ,ηi|t.Simple formulas for computing the credibility intervals forη and γ have been found by substituting from (15) into (14), and they are as follows:(16)1K∑i=1K∫Lη∞π∗η,γi|tdη=1−ϵ2,1K∑i=1K∫Uη∞π∗η,γi|tdη=ϵ2,1K∑i=1K∫Lγ∞π∗γ,ηi|tdγ=1−ϵ2,1K∑i=1K∫Uγ∞π∗γ,ηi|tdγ=ϵ2.Solving the set of nonlinear (16), the CIs for η and γ can be obtained (see [5, 6]).An alternative simple technique can be used to calculate theCIs for the parameters η and γ. This technique can be summarized in the following steps (see [5, 6]):(1)
Generateηi,γi,i=1,2,…,K from π∗η,γ|t(2)
Arrange allηi’s and γi’s in an ascending order to be η∗1,η∗2,…,η∗K and γ∗1,γ∗2,…,γ∗N(3)
A two-sided1−ϵ×100%CI for η, say Lη,Uη is then given by η∗Kϵ/2,η∗K1−ϵ/2(4)
Also, a two-sided1−ϵ×100%CI for γ, say Lγ,Uγ is given by γ∗Kϵ/2,γ∗K1−ϵ/2


## 3.3. Highest Posterior Density Interval

To calculate the1−ϵ×100% highest posterior density HPD interval for the parameter η, Lη,Uη, we should solve the following two nonlinear equations (see [5, 6]):(17)1K∑i=1K∫LηUηπ∗η,γi|tdη=1−ϵ,∑i=1Kπ∗Lη,γi|t=∑i=1Kπ∗Uη,γi|t.Similarly, the 1−ϵ×100%HPD interval for γ, Lγ,Uγ, can be calculated by solving the following two nonlinear equations (see [5, 6]):(18)1K∑i=1K∫LγUγπ∗γ,ηi|,tdγ=1−ϵ,∑i=1Kπ∗Lγ,ηi|,t=∑i=1Kπ∗Uγ,ηi|,t.


## 4. Results

This section contains two subsections. In the first, the estimation problem has been studied based on simulated samples, and in the second, the same problem has been studied based on real data sets as an illustrative example.Table 1
Parameters point estimates and its relatedMSEs using η=1.2951,γ=3.1627,c1=3.0,c2=1.8,c3=0.6 based on UPHCS.
τ1,τ20.2,2.50.8,3.5n,r,kMethodη^¯,MSEη^γ^¯,MSEγ^η^¯,MSEη^γ^¯,MSEγ^50,30,10ML(1.1468, 0.0363)(2.4065, 0.6508)(1.3203, 0.0284)(3.1111, 0.4369)B(1.2691, 0.0031)(3.0065, 0.0501)(1.3131, 0.0027)(3.2049, 0.0407)50,40,10ML(1.2191, 0.0245)(2.6747, 0.3372)(1.3546, 0.0215)(3.1903, 0.3231)B(1.2852, 0.0029)(3.0583, 0.0256)(1.3218, 0.0021)(3.1796, 0.0142)50,50,10ML(1.2997, 0.0177)(3.0359, 0.1136)(1.3137, 0.0146)(3.1903, 0.1025)B(1.3091, 0.0017)(3.1225, 0.0215)(1.3125, 0.0012)(3.1465, 0.0202)τ1,τ20.3,0.40.5,0.850,40,10ML(1.3526, 0.0224)(3.2601, 0.7257)(1.3477, 0.0217)(3.0654, 0.2733)B(1.3021, 0.0205)(3.2702, 0.6403)(1.2908, 0.0191)(3.1107, 0.2629)50,40,20ML(1.2719, 0.0162)(3.0461, 0.3811)(1.3078, 0.0121)(3.1603, 0.2296)B(1.2814, 0.0133)(3.2782, 0.3315)(1.2883, 0.0104)(3.1221, 0.1709)50,40,30ML(1.3103, 0.0093)(3.1016, 0.1413)(1.3178, 0.0081)(3.1713, 0.1035)B(1.2873, 0.0052)(3.1414, 0.1017)(1.3032, 0.0043)(3.1521, 0.0804)τ1,τ20.8,1.50.5,0.730,25,10ML(1.2941, 0.0247)(3.0931, 0.3404)(1.3063, 0.0436)(2.9565, 0.3666)B(1.2814, 0.0202)(3.1105, 0.3309)(1.3118 0.0321)(3.0621, 0.3476)40,25,10ML(1.2002, 0.0353)(2.6083, 0.3608)(1.3224, 0.0487)(3.1683, 0.3738)B(1.2797, 0.0253)(3.1208, 0.3554)(1.2916, 0.0331)(3.0442, 0.3602)50,25,10ML(1.0676, 0.0818)(2.5086, 0.8373)(1.2424, 0.0933)(2.7589, 0.9498)B(1.3208, 0.0338)(3.4271, 0.7175)(1.2288, 0.0552)(3.4108, 0.8208)Table 2
Parameters interval estimates.
τ1,τ20.8,3.50.2,2.5n,r,kMethodLη,UηLγ,UγLη,UηLγ,UγLengthLengthLengthLength50,30,10Approximate CI(0.8889,1.3665)(1.7267,3.0762)(0.9501,1.6489)(1.8879,4.3712)0.47761.34950.69882.4833Bootstrap−pCI(0.8657,1.3184)(2.0009,3.2093)(0.9467,1.594)(2.0178,4.1566)0.45271.20850.64732.1388Credible CI(1.0778,1.4113)(2.505,3.4028)(1.1341,1.4943)(2.7255,3.6655)0.33340.89780.36030.9399HPD CI(1.1021,1.3819)(2.6107,3.3109)(1.1006,1.4001)(2.8108,3.6004)0.27980.70020.29950.789650,40,10Approximate CI(1.0737,1.4954)(1.5826,2.7100)(1.0520,1.5018)(1.9831,4.0045)0.42171.12740.44982.0214Bootstrap−pCI(1.1219,1.5414)(2.0952,3.2940)(1.1942,1.6317)(2.2807,3.8635)0.41951.19880.43751.5828Credible CI(1.1804,1.4939)(2.7527,3.4271)(1.1644,1.5871)(2.7127,3.5526)0.31340.77440.32270.8400HPD CI(1.2016,1.4332)(2.8417,3.4935)(1.1104,1.4032)(2.8103,3.5206)0.23160.65180.29280.710650,50,10Approximate CI(1.0557,1.4682)(2.3427,3.4182)(1.0487,1.4972)(1.8415,3.8109)0.41251.07550.44851.9695Bootstrap−pCI(1.0952,1.4096)(2.3131,3.3303)(1.1113,1.625)(2.2669,3.7300)0.31441.01720.51361.4631Credible CI(1.1503,1.3757)(2.7547,3.5071)(1.2535,1.4906)(2.7578,3.5226)0.22540.75250.23710.7648HPD CI(1.1502,1.3519)(2.9017,3.4415)(1.3019,1.5535)(2.9071,3.4619)0.20170.53980.21160.5548τ1,τ20.3,0.40.5,0.8n,r,kMethodLη,UηLγ,UγLη,UηLγ,UγLengthLengthLengthLength50,40,10Approximate CI(1.0724,1.6351)(1.5121,5.0415)(1.1515,1.4836)(2.0265,4.093)0.56263.52930.33212.0664Bootstrap−pCIb2−4ac(1.095,1.5682)(2.0553,4.7915)(1.153,1.4463)(2.0694,4.1432)0.47312.73620.29332.0738Credible CI(1.1745,1.4950)(2.7433,3.6639)(1.2095,1.4343)(2.7404,3.5848)0.32050.92060.22480.8444HPD CI(1.2018,1.4224)(2.6819,3.4417)(1.2091,1.4106)(2.6614,3.4727)0.22060.75980.20150.811350,40,20Approximate CI(1.179,1.4917)(2.0176,4.0466)(1.2125,1.5138)(2.1051,3.9666)0.31272.0290.30121.8614Bootstrap−pCI(1.1924,1.4498)(2.1724,3.9639)(1.203,1.4498)(2.2694,3.9639)0.25741.79150.24671.6945Credible CI(1.2251,1.3324)(2.5630,3.3530)(1.1109,1.3168)(2.7234,3.4374)0.22730.79010.20590.714HPD CI(1.2508,1.4611)(2.6602,3.3918)(1.2071,1.3876)(2.8091,3.4705)0.21030.73160.18050.661450,40,30Approximate CI(1.2159,1.5298)(2.0767,4.0633)(0.9715,1.275)(1.7222,2.7712)0.31381.98660.30341.049Bootstrap−pCI(1.2124,1.4911)(2.7031,3.8219)(1.0496,1.2633)(1.8027,2.8455)0.22871.11880.21361.0428Credible CI(1.2144,1.4317)(2.4819,3.2151)(1.1743,1.3671)(2.7878,3.5)0.21730.73320.19280.7122HPD CI(1.3082,1.4987)(2.7209,3.3922)(1.2153,1.3778)(2.8831,3.4824)0.19050.67130.16250.5993τ1,τ20.8,1.50.5,0.7n,r,kMethodLη,UηLγ,UγLη,UηLγ,UγLengthLengthLengthLength30,25,10Approximate CI(0.9843,1.6086)(1.9840,4.2101)(0.8839,1.7684)(1.9389,4.4165)0.62422.22610.88452.4776Bootstrap−pCI(1.0325,1.5609)(2.172,4.1193)(0.9965,1.7586)(2.1277,4.2044)0.52831.94720.76212.0767Credible CI(1.1168,1.5105)(2.6342, 3.501)(1.1165,1.4993)(2.5825,3.6066)0.39370.86680.38281.0241HPD CI(1.0192,1.3320)(2.7103,3.4720)(1.1093,1.4418)(2.4417,3.3523)0.31280.76170.33250.910640,25,10Approximate CI(0.6742,1.4293)(1.8723,4.3159)(0.7154,1.5825)(1.2186,4.102)0.7552.44360.86712.7834Bootstrap−pCI(0.7367,1.4707)(1.9075,4.2957)(0.9937,1.8522)(1.3056,3.9789)0.7342.38820.85852.6733Credible CI(1.066,1.5437)(2.4646,3.3834)(1.1254,1.577)(2.3588,3.5071)0.47770.91880.45161.1482HPD CI(1.1106,1.4624)(2.6154,3.4870)(1.2109,1.6246)(2.4414,3.4328)0.35180.87160.41370.991450,25,10Approximate CI(0.6716,1.4703)(1.6141,4.2879)(0.5807,1.4623)(1.0651,3.939)0.79872.67390.88162.8739Bootstrap−pCI(0.7223,1.4845)(1.6421,4.0944)(0.6213,1.4832)(1.2694,3.9734)0.76222.45230.86192.704Credible CI(1.0475,1.5928)(2.5555,3.5541)(1.1163,1.689)(2.6321,3.6052)0.54530.99860.57271.2731HPD CI(1.1329,1.6340)(2.6172,3.5088)(1.1114,1.7018)(2.8815,3.9868)0.50110.89160.59041.1053


### 4.1. Simulated Results

In this subsection, point and intervalML and B estimates based on generated samples under UPHCS have been calculated for different values of r,k,τ1, and τ2 as shown in the following steps:(1)
Forc1,c2, and c3, the parameters η and γ have been generated from the PDF (8), see [15](2)
Using the generatedη and γ, a progressive type-II censored sample of size n and censoring size r from Burr-X has been generated, see [9](3)
For different values ofn,r,k,τ1 and τ2, the point and interval ML and B estimates of the parameters η and γ have been computed(4)
Repeating steps1−3, M times, the mean squared errors MSEs of all estimates have been calculated to study the behaviors of the MSEs of the parameters with respect to the values n,r,k,τ1, and τ2(5)
A1−ϵ×100%ML and B interval estimates of the parameters η and γ have been obtained for different values of r,k,τ1, and τ2(6)
From a high number of runs, some of the obtained results have been summarized in Tables1 and 2From Tables1 and 2, the following remarks can be observed:(1)
The Bayes technique is better than the maximum likelihood technique because it gives smallerMSEs and narrower confidence intervals for the estimates of the parameters(2)
In the maximum likelihood interval estimation problem, the confidence intervals given by the bootstrap method are narrower than that given by the approximate method(3)
In the Bayesian interval estimation problem, the highest posterior confidence intervals are narrower than the credibility confidence intervals(4)
In each technique of estimation, for any three fixed values fromr,k,τ1, and τ2, the MSEs have been decreased, and the confidence intervals have been become narrower by increasing the fourth value(5)
In each technique of estimation, for fixedr,k,τ1, and τ2, the MSEs have been increased, and the confidence intervals have become wider by increasing n.


### 4.2. Data Analysis

In this section, two real data sets, from [25], have been introduced. In [25], the validity of Burr-XII distribution for the given real data sets is discussed, and it is found that Burr-XII distribution fits quite well for the data sets.In this section, the same real data sets have been analyzed using the Burr-X distribution, and a comparison between the two distributions has been made using the Kolmogorov–SmirnovK−S test statistic and it’s P−Value to select the best from the two distributions. These real data sets are as follows:Data 1.
0.19, 0.78, 0.96, 1.31, 2.78, 3.16, 4.15, 4.76, 4.85, 6.5, 7.35, 8.01, 8.27, 12.06, and 31.75 which represent the failure times in hours of 15 devices.Data 2.
0.9, 1.5, 2.3, 3.2, 3.9, 5.0, 6.2, 7.5, 8.3, 10.4, 11.1, 12.6, 15.0, 16.3, 19.3, 22.6, 24.8, 31.5, 38.1, and 53.0, which represent the first failure times in months of 20 electronic cards.
In Table3, the maximum likelihood estimates, the corresponding Kolmogorov–Smirnov K−S test statistic, and its P−values have been computed under Burr-X and Burr-XII models.
Under significance level (0.05), the computedP−values for Burr-X and Burr-XII distributions are greater than the chosen significance level which means that the two distributions fit the two real data sets well. But, the P−Value corresponding to Burr-X distribution is greater than that computed for Burr-XII which means that the Burr-X distribution fits the data better than the Burr-XII distribution.
Point and interval estimates of the parametersη and γ based on the introduced UPHCS from the given real data sets have been obtained and summarized in Table 4 using τ1=1.2,τ2=11.5,r=10, k=7 and R1=1,0,0,0,1,0,0,0,0,3 (for sample I) and R2=2,0,0,3,1,0,0,0,0,4 (for sample II).Table 3
-MLEs of the parameters and the associated K−S and P−Value based on the real data set I and II.
Sample numberModelMLEsK−SP−ValueIBurr−Xη,γη^=0.51104,γ^=1.31870.14960.8881Burr−XIIα,βα^=0.9012,β^=3.22330.18720.5851IIBurr−Xη,γη^=0.7001,γ^=0.20190.16220.7768Burr−XIIα,βα^=1.4105,β^=0.48110.19480.5335Table 4
Parameters estimates based onUPHCS from real data set I and II.
Data set numberValues ofτ1,τ21.2,11.5r,kMethodLη,UηLγ,UγLengthLength0.482191.21093I10,7Approximate CI(0.2106, 0.5618)(0.7163, 1.5358)0.35120.81950.482191.21093Bootstrap−pCI(2208, 0.5425)(0.7618, 1.5535)0.32170.79170.482191.21093Credible CI(0.2417, 0.5191))(0.8015, 1.5567)0.27740.75520.482191.21093HPD CI(0.2557, 0.5250)(0.9173, 1.3988)0.26930.48150.70010.2019II10,7Approximate CI(0.5223, 0.9397)(0.0582,0.2467)0.41740.18850.70010.2019Bootstrap−pCI(0.5568, 0.9185)(0.1069, 0.2395)0.36170.13260.70010.2019Credible CI(0.6021, 0.9339)(0.1506, 0.2661)0.33180.11550.70010.2019HPD CI(0.6418, 0.9392)(0.1937, 0.2970)0.29740.1033


## 4.1. Simulated Results

In this subsection, point and intervalML and B estimates based on generated samples under UPHCS have been calculated for different values of r,k,τ1, and τ2 as shown in the following steps:(1)
Forc1,c2, and c3, the parameters η and γ have been generated from the PDF (8), see [15](2)
Using the generatedη and γ, a progressive type-II censored sample of size n and censoring size r from Burr-X has been generated, see [9](3)
For different values ofn,r,k,τ1 and τ2, the point and interval ML and B estimates of the parameters η and γ have been computed(4)
Repeating steps1−3, M times, the mean squared errors MSEs of all estimates have been calculated to study the behaviors of the MSEs of the parameters with respect to the values n,r,k,τ1, and τ2(5)
A1−ϵ×100%ML and B interval estimates of the parameters η and γ have been obtained for different values of r,k,τ1, and τ2(6)
From a high number of runs, some of the obtained results have been summarized in Tables1 and 2From Tables1 and 2, the following remarks can be observed:(1)
The Bayes technique is better than the maximum likelihood technique because it gives smallerMSEs and narrower confidence intervals for the estimates of the parameters(2)
In the maximum likelihood interval estimation problem, the confidence intervals given by the bootstrap method are narrower than that given by the approximate method(3)
In the Bayesian interval estimation problem, the highest posterior confidence intervals are narrower than the credibility confidence intervals(4)
In each technique of estimation, for any three fixed values fromr,k,τ1, and τ2, the MSEs have been decreased, and the confidence intervals have been become narrower by increasing the fourth value(5)
In each technique of estimation, for fixedr,k,τ1, and τ2, the MSEs have been increased, and the confidence intervals have become wider by increasing n.


## 4.2. Data Analysis

In this section, two real data sets, from [25], have been introduced. In [25], the validity of Burr-XII distribution for the given real data sets is discussed, and it is found that Burr-XII distribution fits quite well for the data sets.In this section, the same real data sets have been analyzed using the Burr-X distribution, and a comparison between the two distributions has been made using the Kolmogorov–SmirnovK−S test statistic and it’s P−Value to select the best from the two distributions. These real data sets are as follows:Data 1.
0.19, 0.78, 0.96, 1.31, 2.78, 3.16, 4.15, 4.76, 4.85, 6.5, 7.35, 8.01, 8.27, 12.06, and 31.75 which represent the failure times in hours of 15 devices.Data 2.
0.9, 1.5, 2.3, 3.2, 3.9, 5.0, 6.2, 7.5, 8.3, 10.4, 11.1, 12.6, 15.0, 16.3, 19.3, 22.6, 24.8, 31.5, 38.1, and 53.0, which represent the first failure times in months of 20 electronic cards.
In Table3, the maximum likelihood estimates, the corresponding Kolmogorov–Smirnov K−S test statistic, and its P−values have been computed under Burr-X and Burr-XII models.
Under significance level (0.05), the computedP−values for Burr-X and Burr-XII distributions are greater than the chosen significance level which means that the two distributions fit the two real data sets well. But, the P−Value corresponding to Burr-X distribution is greater than that computed for Burr-XII which means that the Burr-X distribution fits the data better than the Burr-XII distribution.
Point and interval estimates of the parametersη and γ based on the introduced UPHCS from the given real data sets have been obtained and summarized in Table 4 using τ1=1.2,τ2=11.5,r=10, k=7 and R1=1,0,0,0,1,0,0,0,0,3 (for sample I) and R2=2,0,0,3,1,0,0,0,0,4 (for sample II).Table 3
-MLEs of the parameters and the associated K−S and P−Value based on the real data set I and II.
Sample numberModelMLEsK−SP−ValueIBurr−Xη,γη^=0.51104,γ^=1.31870.14960.8881Burr−XIIα,βα^=0.9012,β^=3.22330.18720.5851IIBurr−Xη,γη^=0.7001,γ^=0.20190.16220.7768Burr−XIIα,βα^=1.4105,β^=0.48110.19480.5335Table 4
Parameters estimates based onUPHCS from real data set I and II.
Data set numberValues ofτ1,τ21.2,11.5r,kMethodLη,UηLγ,UγLengthLength0.482191.21093I10,7Approximate CI(0.2106, 0.5618)(0.7163, 1.5358)0.35120.81950.482191.21093Bootstrap−pCI(2208, 0.5425)(0.7618, 1.5535)0.32170.79170.482191.21093Credible CI(0.2417, 0.5191))(0.8015, 1.5567)0.27740.75520.482191.21093HPD CI(0.2557, 0.5250)(0.9173, 1.3988)0.26930.48150.70010.2019II10,7Approximate CI(0.5223, 0.9397)(0.0582,0.2467)0.41740.18850.70010.2019Bootstrap−pCI(0.5568, 0.9185)(0.1069, 0.2395)0.36170.13260.70010.2019Credible CI(0.6021, 0.9339)(0.1506, 0.2661)0.33180.11550.70010.2019HPD CI(0.6418, 0.9392)(0.1937, 0.2970)0.29740.1033


## 5. Conclusions

In this paper, the point and interval estimates of the Burr-X parameters have been obtained based on a suggestedUPHCS with complete description for the behaviors of the MSE′s and the confidence intervals lengthes with respect to the values n,r,k,τ1, and τ2. From the obtained results, we concluded that the Bayes approach of estimation gives more accurate point estimates and narrower interval estimates. Moreover, in the two approaches of estimation, by increasing one of the values r,k,τ1, and τ2 and fixing the other values, the results will be more accurate, also, by increasing n with fixed r,k,τ1, and τ2, and we will get a lower accuracy. Two engineering real data sets have been introduced and analyzed using Burr-XII and Burr-X distribution, and we have found that the studied distribution fits the given real data sets well. Based on the two real data sets and using the suggested UPHCS, the point and interval estimates of the used distribution parameters have been obtained using different techniques.For future work, we can study the prediction problem based on theUPHCS which suggested in this paper and also extend the results of this paper to other distributions. Furthermore, we can study the estimation problem under the joint UPHCS.



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*Source: 3746821-2022-07-06.xml*