# Statistical Inference to the Parameter of the Inverse Power Ishita Distribution under Progressive Type-II Censored Data with Application to COVID-19 Data

Authors: Kariema A. Elnagar; Dina A. Ramadan; Beih S. El-Desouky
Journal: Journal of Mathematics
 (2022)

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## Abstract

The goal of the article is the inference about the parameters of the inverse power ishita distribution (IPID) using progressively type-II censored (Prog–II–C) samples. For IPID parameters, maximum likelihood and Bayesian estimates were obtained. Two bootstrap “confidence intervals” (CIs) are also proposed in addition to “approximate confidence intervals” (ACIs). In addition, Bayesian estimates for “squared error loss” (SEL) and LINEX loss functions are provided. The Gibbs within Metropolis–Hasting samplers process is used to provide Bayes estimators of unknown parameters also “credible intervals” (CRIs) of them by using the “Markov Chain Monte Carlo” (MCMC) technique. Then, an application of the suggested approaches is considered a set of real-life data this data set COVID-19 data from France of 51 days recorded from 1 January to 20 February 2021 formed of mortality rate. To evaluate the quality of the proposed estimators, a simulation study is conducted.


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## Body

## 1. Introduction

Statistical researchers in the fields of industrial and mechanical engineering are spent so much time looking at component and unit failure, which provide the foundation of operational systems. Their research involves watching operating units until they fail, recording their lifetimes, applying statistical inference methods to analyse obtained data and then predicting reliability and risk functions for the entire system using the data. However, because some experimental units are costly and have a high level of reliability. Following a specified censoring scheme, units are excluded from the experiment, and the censored sample is the final sample obtained by Klein et al. [1]. One of the main motivations for filtering is to reduce the total test time and the accompanying test. The life test is considered to be complete when all of the items under test are monitored until failure. In most real conditions, the data available is incomplete. As a result, whether due to test time or budget limits, censoring is critical in lifetime data analysis. The most common censoring techniques are type-I and type-II censored. Following that, a progressive censoring scheme is suggested. As a result, the researchers can release the unused units of various points throughout the test, making it more flexible and realistic.


### 1.1. Progressive Type-II Censored

A progressive type-II censoring strategy is a useful method for removing a particular fraction of at-risk participants from an experiment at each of many ordered failure times. The following is an example of the Prog –II– C scheme: On the life test, the tester allocates independent and identical units. Let us suppose at the start of the experiment, there aren units to test. The lifetime test is ended when the mthm<n unit fails, assuming there are n units to test when the experiment begins. After the first failure, time t1 is recorded, and R1 units were selected randomly as from reminder n−1 survival units. As a result, when the second failure occurs, time t2 is recorded, with the remnants n−R1−2 survival units R2 units were chosen at random. This experiment ends when the mth failure occurs, at time tm, and Rm=n−m−∑i=1m−1Ri as shown in Figure 1.Figure 1
Type-II progressive censoring.As special cases (ifRi=0 and i=1,2,…,m with n=m) and the conventional type-II right censoring scheme if Ri=0 and i=1,2,…,m−1 with Rm=n−m. Many authors have devoted for a Prog–II–C method with various failure time breakdowns. For example, Balakrishnan and Aggarwala [2], Alshenawy [3], Balakrishnan [4], Ahmed [5], and Almetwally et al. [6]. Mann et al. [7] and Meeker et al. [8] looked into the characteristics of progressively censored order statistics and it offered an outline of several inferential approach expansions depend on Prog-I and Prog-II right censored samples, as well as identifying some for future research challenges. Balakrishnan and Cramer [9] give a thorough survey of the literature on progressive censoring, as well as specifics on this progressive censoring technique and its various uses. According to Tse et al. [10], the amount of patients falls out of a clinical trial at each stage is random and not possible predicted. With each failure, the pattern of removal become more random. See Balakrishnan and Sandhu [11] for further details on the gradually censored samples. As lifetime distributions are exponential, log normal, or Weibull, Aggarwala and Balakrishnan [12] has studied inference for Prog–II–C cases. Balakrishnan [11] and Aggarwala [12] have devised a method for simulating general Prog–II–C samples drawn from continuous or uniform distributions. Eryilmaz and Bairamov [13]and Montanari et al. [14] also have been researching parameter estimation from various lifetime distributions using Prog–II–C. Other authors, include Balakrishnan et al. [9], Mousa et al. [15], and Mousa et al. [16], have been researching the estimate of parameters from various lifetime distributions depending on Prog–II–C. In Prog-II right censored order statistics, Salemi et al. [17] recently investigated A-optimal and D-optimal censoring strategies, Qin et al. [18] offer a novel spacing-based test statistic for determining whether general Prog–II–C data are from an exponential distribution and parameters of a log-logistic distribution under the Prog–II–C sample, when Maiti et al. [19] given access to a managed step-wise sample of the second type, there was also a look at the generalized Rayleigh distribution. The IPID is a generalized of the Ishita distribution, which was presented by Elnagar et al. [20] is shown to be a good model for the Covid-19 and glass fiber data failure times. They also started at the maximum likelihood of the unknown parameters estimators, besides their asymptotic confidence intervals, using complete data. The random variable X of the IPI distribution, has a probability density function (PDF):(1)fy=αθ3θ3+2θ+y−2αy−α−1e−θy−α,y>0,θ,α>0,and cumulative distribution function (CDF) is(2)Fy=1+θy−αθy−α+2θ3+2e−θy−αy>0,θ,α>0.The IPID is a generalized of the ishita distribution, which was presented by Shanker and Shukla [21] and proven to be adaptable to a variety of applications.Modeling the COVID-19 is the driving force behind the new distribution. The COVID-19 of France was employed of 51 days that is recorded from 1 January to 20 February 2021 as real data for evaluating model techniques. COVID-19 data on new cases or deaths are discrete data (count). For France, we have been using the COVID-19 daily mortality rate. The daily mortality rate is continuous data.LetX1:m:n,X2:m:n,…,Xm:m:n ; 1≤m≤n be a result of a Prog–II–C sample lifetime test that included n units derived from a IPI α,θ distribution and its censoring scheme is R1,R2,…,Rm. Balakrishnan and Aggarwala [2] gives the joint PDF of a Prog-II.(3)fx1:m:n,x2:m:n,….xm:m:n=C∏i=1mfxi:m:n;α,θ.1−Fxi:m:n;α,θRi,m<n,as C=nn−R1−1…n−∑i=1m−1Ri−1.Balakrishnan et al. [4] have excellent provides a review essay a recent report on progressive censoring schemes. This article is organized as follows: Section 2, introduced MLEs of α and θ. We survey the confidence intervals for the unknown parameters in Section 3. Two parametric bootstrap approaches are used to constructed unknown parameters in Section 4. In Section 5 different loss functions, such as SEL and LINEX loss functions, Bayes estimates for unknown parameters were introduced. In Section 6, a real data set was studied. Section 7 introduces a simulation study to evaluate the accuracy of the various estimators generated in this study. At the end of this paper, conclusions are given in Section 8.


## 1.1. Progressive Type-II Censored

A progressive type-II censoring strategy is a useful method for removing a particular fraction of at-risk participants from an experiment at each of many ordered failure times. The following is an example of the Prog –II– C scheme: On the life test, the tester allocates independent and identical units. Let us suppose at the start of the experiment, there aren units to test. The lifetime test is ended when the mthm<n unit fails, assuming there are n units to test when the experiment begins. After the first failure, time t1 is recorded, and R1 units were selected randomly as from reminder n−1 survival units. As a result, when the second failure occurs, time t2 is recorded, with the remnants n−R1−2 survival units R2 units were chosen at random. This experiment ends when the mth failure occurs, at time tm, and Rm=n−m−∑i=1m−1Ri as shown in Figure 1.Figure 1
Type-II progressive censoring.As special cases (ifRi=0 and i=1,2,…,m with n=m) and the conventional type-II right censoring scheme if Ri=0 and i=1,2,…,m−1 with Rm=n−m. Many authors have devoted for a Prog–II–C method with various failure time breakdowns. For example, Balakrishnan and Aggarwala [2], Alshenawy [3], Balakrishnan [4], Ahmed [5], and Almetwally et al. [6]. Mann et al. [7] and Meeker et al. [8] looked into the characteristics of progressively censored order statistics and it offered an outline of several inferential approach expansions depend on Prog-I and Prog-II right censored samples, as well as identifying some for future research challenges. Balakrishnan and Cramer [9] give a thorough survey of the literature on progressive censoring, as well as specifics on this progressive censoring technique and its various uses. According to Tse et al. [10], the amount of patients falls out of a clinical trial at each stage is random and not possible predicted. With each failure, the pattern of removal become more random. See Balakrishnan and Sandhu [11] for further details on the gradually censored samples. As lifetime distributions are exponential, log normal, or Weibull, Aggarwala and Balakrishnan [12] has studied inference for Prog–II–C cases. Balakrishnan [11] and Aggarwala [12] have devised a method for simulating general Prog–II–C samples drawn from continuous or uniform distributions. Eryilmaz and Bairamov [13]and Montanari et al. [14] also have been researching parameter estimation from various lifetime distributions using Prog–II–C. Other authors, include Balakrishnan et al. [9], Mousa et al. [15], and Mousa et al. [16], have been researching the estimate of parameters from various lifetime distributions depending on Prog–II–C. In Prog-II right censored order statistics, Salemi et al. [17] recently investigated A-optimal and D-optimal censoring strategies, Qin et al. [18] offer a novel spacing-based test statistic for determining whether general Prog–II–C data are from an exponential distribution and parameters of a log-logistic distribution under the Prog–II–C sample, when Maiti et al. [19] given access to a managed step-wise sample of the second type, there was also a look at the generalized Rayleigh distribution. The IPID is a generalized of the Ishita distribution, which was presented by Elnagar et al. [20] is shown to be a good model for the Covid-19 and glass fiber data failure times. They also started at the maximum likelihood of the unknown parameters estimators, besides their asymptotic confidence intervals, using complete data. The random variable X of the IPI distribution, has a probability density function (PDF):(1)fy=αθ3θ3+2θ+y−2αy−α−1e−θy−α,y>0,θ,α>0,and cumulative distribution function (CDF) is(2)Fy=1+θy−αθy−α+2θ3+2e−θy−αy>0,θ,α>0.The IPID is a generalized of the ishita distribution, which was presented by Shanker and Shukla [21] and proven to be adaptable to a variety of applications.Modeling the COVID-19 is the driving force behind the new distribution. The COVID-19 of France was employed of 51 days that is recorded from 1 January to 20 February 2021 as real data for evaluating model techniques. COVID-19 data on new cases or deaths are discrete data (count). For France, we have been using the COVID-19 daily mortality rate. The daily mortality rate is continuous data.LetX1:m:n,X2:m:n,…,Xm:m:n ; 1≤m≤n be a result of a Prog–II–C sample lifetime test that included n units derived from a IPI α,θ distribution and its censoring scheme is R1,R2,…,Rm. Balakrishnan and Aggarwala [2] gives the joint PDF of a Prog-II.(3)fx1:m:n,x2:m:n,….xm:m:n=C∏i=1mfxi:m:n;α,θ.1−Fxi:m:n;α,θRi,m<n,as C=nn−R1−1…n−∑i=1m−1Ri−1.Balakrishnan et al. [4] have excellent provides a review essay a recent report on progressive censoring schemes. This article is organized as follows: Section 2, introduced MLEs of α and θ. We survey the confidence intervals for the unknown parameters in Section 3. Two parametric bootstrap approaches are used to constructed unknown parameters in Section 4. In Section 5 different loss functions, such as SEL and LINEX loss functions, Bayes estimates for unknown parameters were introduced. In Section 6, a real data set was studied. Section 7 introduces a simulation study to evaluate the accuracy of the various estimators generated in this study. At the end of this paper, conclusions are given in Section 8.


## 2. Maximum-Likelihood Estimation

The maximum likelihood estimation method (MLE) is used to study the problem of estimating IPID parameters under Prog–II–C data.Letx1:m:n,x2:m:n,….,xm:m:n be a Prog–II–C sample from the IPI distribution with PDF (1) and parameters α and θ. The likelihood function has the following:(4)Lα,θ=Cαmθ3mθ3+2me−θ∑i=1myi−α∏i=1mθ+yi−2αyi−α−1∏i=1m1−1+θyi−αθyi−α+2θ3+2e−θyi−αRi.On Prog–II–C samples, the log-likelihood function ofα and θ parameters as a result of(5)l=lnLα,θ=lnc+mlnα+3mlnθ−mlnθ3+2−θ∑i=1myi−α+∑i=1mlnθ+yi−2α−α+1∑i=1mlnyi+∑i=1mRiln1−1+θyi−αθyi−α+2θ3+2e−θyi−α.Computing the first partial derivatives of (5) in relation to α and θ and solving this nonlinear system of equations ∂/∂αlnLα,θ=0 and ∂/∂θlnLα,θ=0, the maximum likelihood estimates (α^ and θ^) are obtained. Accordingly, results in(6)mα+∑i=1mθyi−α−1lnyi+∑i=1mRiθyi−αlnyie−θyi−α1−1+θyi−αθyi−α+2/θ3+2e−θyi−α=0,and(7)3mθ−m3θ2θ3+2−∑i=1myi−α+∑i=1m1θ+yi−2α,+∑i=1mRie−θyi−αyi−α1+θyi−αθyi−α+2θ3+2−yi−α2θ3+2θyi−α+1−3θ2θyi−α+2θ3+221−1+θyi−αθyi−α+2/θ3+2e−θyi−α=0,Since Equations (6) and (7) cannot be solved analytically, so to obtain the estimations, a numerical method like Newton–Raphson method should be employed. In Ahmed [5], the algorithm is given in detail.


## 3. Asymptotic Confidence Intervals

The elements of the Fisher information matrix’s inverse give us the approximate variances and covariances of the MLEs,α^ and θ^; Iij=E−∂2lΨ/∂ψi∂ψj, As i,j=1,2 also Ψ=ψ1,ψ2=α,θ. So, precise asymptotic forms for these equations are tough to comprehend get. The Fisher information matrix Iij^=−∂2lΨ/∂ψi∂ψjΨ=Ψ^ is then used, we will use the expectation to establish confidence intervals (CIs) for the parameters, which is obtained through inference. Consequently, the information matrix that has been seen is(8)I^α^,θ^=−∂2l∂α2−∂2l∂α∂θ−∂2l∂θ∂α−∂2l∂θ2α=α^,θ=θ^.As a result, the asymptotic variance-covariance matrixV^ for MLEs derived through converting observed information matrix I^α,θ or similar to the asymptotic variance-covariance matrix(9)V^=I^α^,θ^−1=varα^covα,θcovα,θvarθ^α^,θ^.It is evident thatα^,θ^ is approximately distributed as multivariate normal with mean α,θ and covariance matrix I−1α,θ given specific symmetry requirements, see Bebbington et al. [22]. As a result. The 1−γ100% convergent confidence intervals (ACIs) for α and θ may be calculated as follows:(10)α^±zγ/2varα^,θ^±zγ/2varθ^,as zγ/2 is the right-tail probability percentile standard normal distribution for γ/2 See Lawless [23].


## 4. Bootstrap Confidence Intervals

Different bootstrap confidence intervals are suggested here. The percentile bootstrap (Bp) and bootstrap-t (Bt) confidence intervals are used in this study. Efron [24] and Hall [25] have more information on bootstrap confidence intervals.


### 4.1. Parametric Percentile Bootstrap (Boot-P)

The formula for calculating the parametric percentile bootstrap confidence interval for model parameters is as follows:(1)
Calculate the MLE ofα and θ by maximizing Equations (6) and (7).(2)
Create bootstrap samples withα and θ to get the bootstrap estimate of α and θ, say ψ^⋆ where ψ be α or θ, from the bootstrap sample.(3)
To getψ1^⋆,ψ2^⋆,…,ψNboot^⋆, repeat step 2 N times, where ψi^⋆=αi^⋆,θ^i⋆, i=1,2,3,..Nboot.(4)
Assignψ^i⋆=1,2,3,..,Nboot to ψ^1⋆,ψ^1⋆,…,ψ^Nboot⋆ in ascending order.(11)ψ^boot−pη2,ψ^boot−p1−η2,Gives a two-sided1001−η% bootstrap-p for the unknown parameters α and θ.


### 4.2. Parametric Bootstrap-t Boot-t

The percentile bootstrap-t confidence interval for model parameters can be calculated using the following algorithm:(1)
Steps 1 and 2 in boot-p are the same(2)
Using the asymptotic variance-covariance matrix (9), the variance-covariance matrix I−1⋆∂2l/∂ψi∂ψj,i,j=1,2 and get the t-statistic of ψ as T⋆ψ=ψ^⋆−ψ^/varψ^⋆^(3)
NBoot times repeat steps 2 and 3 to getT1⋆ψ,T2⋆ψ,…,TNboot⋆ψ(4)
T1⋆ψ,T2⋆ψ,…,TNboot⋆ψ the ordered sequences are sorted in ascending order as T1⋆ψ,T2⋆ψ,…,TNboot⋆ψ(12)ψ^boot−tη2,ψ^boot−t1−η2,Calculates a two-sided bootstrap-t1001−η% CI for the unknown parameters α and θ.


## 4.1. Parametric Percentile Bootstrap (Boot-P)

The formula for calculating the parametric percentile bootstrap confidence interval for model parameters is as follows:(1)
Calculate the MLE ofα and θ by maximizing Equations (6) and (7).(2)
Create bootstrap samples withα and θ to get the bootstrap estimate of α and θ, say ψ^⋆ where ψ be α or θ, from the bootstrap sample.(3)
To getψ1^⋆,ψ2^⋆,…,ψNboot^⋆, repeat step 2 N times, where ψi^⋆=αi^⋆,θ^i⋆, i=1,2,3,..Nboot.(4)
Assignψ^i⋆=1,2,3,..,Nboot to ψ^1⋆,ψ^1⋆,…,ψ^Nboot⋆ in ascending order.(11)ψ^boot−pη2,ψ^boot−p1−η2,Gives a two-sided1001−η% bootstrap-p for the unknown parameters α and θ.


## 4.2. Parametric Bootstrap-t Boot-t

The percentile bootstrap-t confidence interval for model parameters can be calculated using the following algorithm:(1)
Steps 1 and 2 in boot-p are the same(2)
Using the asymptotic variance-covariance matrix (9), the variance-covariance matrix I−1⋆∂2l/∂ψi∂ψj,i,j=1,2 and get the t-statistic of ψ as T⋆ψ=ψ^⋆−ψ^/varψ^⋆^(3)
NBoot times repeat steps 2 and 3 to getT1⋆ψ,T2⋆ψ,…,TNboot⋆ψ(4)
T1⋆ψ,T2⋆ψ,…,TNboot⋆ψ the ordered sequences are sorted in ascending order as T1⋆ψ,T2⋆ψ,…,TNboot⋆ψ(12)ψ^boot−tη2,ψ^boot−t1−η2,Calculates a two-sided bootstrap-t1001−η% CI for the unknown parameters α and θ.


## 5. Bayes Estimation

The Bayesian study of the IPID under Prog–II–C is discussed in this section. Under the SEL function and LINEX loss functions, we obtain Bayes estimators. When the parametersα and θ are independent and follow the following gamma prior distributions according to Zellner [26].(13)π1α∝αk1−1exp−h1α,α>0,π2θ∝θk2−1exp−h2θ,θ>0,where the hyper-parameters ki and hi, i=1,2 are assumed to be known and chosen to reflect the prior belief about the unknown parameters. The posterior distribution is derived from the likelihood function equation (4) and the prior distribution (13), and with the outcomes posterior distribution of α and θ denoted as π∗α,θ|x¯.(14)π∗α,θ|x¯=Lα,θ;xπ1απ2θ∫0∞∫0∞Lα,θ;xπ1απ2θdαdθ.


### 5.1. Loss Function

We need to design an asymmetric loss function to make statistical Bayesian inference more practical and relevant. The loss function, as defined by Press and James [27], is a real-valued function that satisfies all feasible estimates and parameters. We must adopt an asymmetric loss function as well as create statistical Bayesian inference extra sensible and relevant. According to Press and James [27], the loss function is a real-valued function that satisfy all possible estimations and parameters.


#### 5.1.1. Squared Error Loss Function (SEL)

The SEL function is(15)Lψ,ψ^=ψ^−ψ2,Then, for any function ofα and θ, the Bayes estimate is gα,θ under the SEL function is(16)g^BSα,θ|x=Eα,θ|xgα,θ,where(17)Eα,θ|xgα,θ=∫0∞∫0∞gα,θπ1απ2θLα,θ|xdαdθ∫0∞∫0∞π1απ2θLα,θ|xdαdθ.


#### 5.1.2. Linear Exponential (LINEX) Loss Function

The LINEX loss functionLΔ for a parameter ψ is proposed according to Varian and Hall [28] given by the following equation:(18)LΔ=ecΔ−cΔ−1,c≠0,Δ=ψ^−ψ,Hence, the bayes estimate of a functiongα,θ under LINEX loss function illustrated by the following equation:(19)g^BLα,θ|x=−1clogEe−gα,θ|x,c≠0,Ee−gα,θ=∫0∞∫0∞e−gα,θπ1απ2θLα,θ|xdαdθ∫0∞∫0∞π1απ2θLα,θ|xdαdθ.It should be noted that the ratio of multiple integrals in (17) and (19) cannot be expressed explicitly. To produce samples from the joint posterior density function in (14), the MCMC approach is used. We use the Gibbs within Metropolis–Hasting samplers process to add the MCMC approach. The joint posterior distribution expressed as follows:(20)π∗α,θ|x∝mαm+k1−1θ3m+k2−1θ3+2me−h1α+h2θ∑i=1myi−α∏i=1mθ+yi−2αyi−α−1∏i=1m1−1+θyi−αθyi−α+2θ3+2e−θyi−αRi.The conditional posterior densities ofα and θ represented as follows:(21)π1∗αθ,x∝αm+k1−1e−h1α+θ∑i=1myi−α∏i=1mθ+yi−2αyi−α−1∏i=1m1−1+θyi−αθyi−α+2θ3+2e−θyi−αRi,and(22)π2∗θ|α,x∝θ3m+k2−1θ3+2me−θh2+∑i=1myi−α∏i=1mθ+yi−2α∏i=1m1−1+θθyi−α+2θ3+2e−θyi−αRi.Because the conditional posteriors ofα and θ in the previous equations do not qualify for a specified distribution, the Metropolis–Hasting sampler must be used to apply the MCMC approach. Tierney and Luke [29] proposed the Metropolis–Hastings algorithm within Gibbs sampling, which generates the posterior samples as follows:(1)
Begin with the first suggestionα0,θ0(2)
Specifyj=1(3)
Produceαj and θj by using M-H algorithm below using Equations (23) and (24) with the normal indicated distribution Nαj−1,varα and the inverse Fisher information matrix is used to calculate Nθj−1,varθ, where varα and varθ(i)
Create aNαj−1,varα is suggestion to α∗ and θ∗ by Nθj−1,varθ(ii)
Calculate the probabilities of approval(23)ρα=min1,π1∗α∗|θj−1,xπ1∗αj−1|θj−1,x,and(24)ρθ=min1,π2∗θ∗|αj,xπ2∗θj−1∣αj,x.(iii)
Create au1 and u2 from a Uniform 0,1 distribution(iv)
Ifu1=ρα, confess the proposition and Specify αj=α∗ else Specify αj=αj−1(v)
Ifu1=ρθ, confess the proposition and Specify θj=θ∗ else Specify θj=θj−1(4)
Calculateαj and θj(5)
Specifyj=j+1(6)
Steps from 3 to 6 should be reiterated N times(7)
To evaluate the CRIs ofα and θ of ψki, i=1,2,..,N, k=1,2 and ψ1,ψ2=α,θ as ψk1<ψk2<…<ψkN then the 1−γ100% CRIs of ψk is(25)ψkγ2N−M,ψk1−γ2N−M.For sufficiently largeN, the samples chosen are ψki,j=M+1,…,N. SEL function is accustomed to get the approximate Bayes estimates of α^ and θ^ as follows:(26)α^BS=1N−M∑j=M+1Nαj,and(27)θ^BS=1N−M∑j=M+1Nθj.And, the estimates for the mentioned parameters under LINEX loss function are(28)αBL=−1clog1N−M∑i=M+1Ne−cαi,and(29)θBL=−1clog1N−M∑i=M+1Ne−cθi.


## 5.1. Loss Function

We need to design an asymmetric loss function to make statistical Bayesian inference more practical and relevant. The loss function, as defined by Press and James [27], is a real-valued function that satisfies all feasible estimates and parameters. We must adopt an asymmetric loss function as well as create statistical Bayesian inference extra sensible and relevant. According to Press and James [27], the loss function is a real-valued function that satisfy all possible estimations and parameters.


### 5.1.1. Squared Error Loss Function (SEL)

The SEL function is(15)Lψ,ψ^=ψ^−ψ2,Then, for any function ofα and θ, the Bayes estimate is gα,θ under the SEL function is(16)g^BSα,θ|x=Eα,θ|xgα,θ,where(17)Eα,θ|xgα,θ=∫0∞∫0∞gα,θπ1απ2θLα,θ|xdαdθ∫0∞∫0∞π1απ2θLα,θ|xdαdθ.


### 5.1.2. Linear Exponential (LINEX) Loss Function

The LINEX loss functionLΔ for a parameter ψ is proposed according to Varian and Hall [28] given by the following equation:(18)LΔ=ecΔ−cΔ−1,c≠0,Δ=ψ^−ψ,Hence, the bayes estimate of a functiongα,θ under LINEX loss function illustrated by the following equation:(19)g^BLα,θ|x=−1clogEe−gα,θ|x,c≠0,Ee−gα,θ=∫0∞∫0∞e−gα,θπ1απ2θLα,θ|xdαdθ∫0∞∫0∞π1απ2θLα,θ|xdαdθ.It should be noted that the ratio of multiple integrals in (17) and (19) cannot be expressed explicitly. To produce samples from the joint posterior density function in (14), the MCMC approach is used. We use the Gibbs within Metropolis–Hasting samplers process to add the MCMC approach. The joint posterior distribution expressed as follows:(20)π∗α,θ|x∝mαm+k1−1θ3m+k2−1θ3+2me−h1α+h2θ∑i=1myi−α∏i=1mθ+yi−2αyi−α−1∏i=1m1−1+θyi−αθyi−α+2θ3+2e−θyi−αRi.The conditional posterior densities ofα and θ represented as follows:(21)π1∗αθ,x∝αm+k1−1e−h1α+θ∑i=1myi−α∏i=1mθ+yi−2αyi−α−1∏i=1m1−1+θyi−αθyi−α+2θ3+2e−θyi−αRi,and(22)π2∗θ|α,x∝θ3m+k2−1θ3+2me−θh2+∑i=1myi−α∏i=1mθ+yi−2α∏i=1m1−1+θθyi−α+2θ3+2e−θyi−αRi.Because the conditional posteriors ofα and θ in the previous equations do not qualify for a specified distribution, the Metropolis–Hasting sampler must be used to apply the MCMC approach. Tierney and Luke [29] proposed the Metropolis–Hastings algorithm within Gibbs sampling, which generates the posterior samples as follows:(1)
Begin with the first suggestionα0,θ0(2)
Specifyj=1(3)
Produceαj and θj by using M-H algorithm below using Equations (23) and (24) with the normal indicated distribution Nαj−1,varα and the inverse Fisher information matrix is used to calculate Nθj−1,varθ, where varα and varθ(i)
Create aNαj−1,varα is suggestion to α∗ and θ∗ by Nθj−1,varθ(ii)
Calculate the probabilities of approval(23)ρα=min1,π1∗α∗|θj−1,xπ1∗αj−1|θj−1,x,and(24)ρθ=min1,π2∗θ∗|αj,xπ2∗θj−1∣αj,x.(iii)
Create au1 and u2 from a Uniform 0,1 distribution(iv)
Ifu1=ρα, confess the proposition and Specify αj=α∗ else Specify αj=αj−1(v)
Ifu1=ρθ, confess the proposition and Specify θj=θ∗ else Specify θj=θj−1(4)
Calculateαj and θj(5)
Specifyj=j+1(6)
Steps from 3 to 6 should be reiterated N times(7)
To evaluate the CRIs ofα and θ of ψki, i=1,2,..,N, k=1,2 and ψ1,ψ2=α,θ as ψk1<ψk2<…<ψkN then the 1−γ100% CRIs of ψk is(25)ψkγ2N−M,ψk1−γ2N−M.For sufficiently largeN, the samples chosen are ψki,j=M+1,…,N. SEL function is accustomed to get the approximate Bayes estimates of α^ and θ^ as follows:(26)α^BS=1N−M∑j=M+1Nαj,and(27)θ^BS=1N−M∑j=M+1Nθj.And, the estimates for the mentioned parameters under LINEX loss function are(28)αBL=−1clog1N−M∑i=M+1Ne−cαi,and(29)θBL=−1clog1N−M∑i=M+1Ne−cθi.


## 5.1.1. Squared Error Loss Function (SEL)

The SEL function is(15)Lψ,ψ^=ψ^−ψ2,Then, for any function ofα and θ, the Bayes estimate is gα,θ under the SEL function is(16)g^BSα,θ|x=Eα,θ|xgα,θ,where(17)Eα,θ|xgα,θ=∫0∞∫0∞gα,θπ1απ2θLα,θ|xdαdθ∫0∞∫0∞π1απ2θLα,θ|xdαdθ.


## 5.1.2. Linear Exponential (LINEX) Loss Function

The LINEX loss functionLΔ for a parameter ψ is proposed according to Varian and Hall [28] given by the following equation:(18)LΔ=ecΔ−cΔ−1,c≠0,Δ=ψ^−ψ,Hence, the bayes estimate of a functiongα,θ under LINEX loss function illustrated by the following equation:(19)g^BLα,θ|x=−1clogEe−gα,θ|x,c≠0,Ee−gα,θ=∫0∞∫0∞e−gα,θπ1απ2θLα,θ|xdαdθ∫0∞∫0∞π1απ2θLα,θ|xdαdθ.It should be noted that the ratio of multiple integrals in (17) and (19) cannot be expressed explicitly. To produce samples from the joint posterior density function in (14), the MCMC approach is used. We use the Gibbs within Metropolis–Hasting samplers process to add the MCMC approach. The joint posterior distribution expressed as follows:(20)π∗α,θ|x∝mαm+k1−1θ3m+k2−1θ3+2me−h1α+h2θ∑i=1myi−α∏i=1mθ+yi−2αyi−α−1∏i=1m1−1+θyi−αθyi−α+2θ3+2e−θyi−αRi.The conditional posterior densities ofα and θ represented as follows:(21)π1∗αθ,x∝αm+k1−1e−h1α+θ∑i=1myi−α∏i=1mθ+yi−2αyi−α−1∏i=1m1−1+θyi−αθyi−α+2θ3+2e−θyi−αRi,and(22)π2∗θ|α,x∝θ3m+k2−1θ3+2me−θh2+∑i=1myi−α∏i=1mθ+yi−2α∏i=1m1−1+θθyi−α+2θ3+2e−θyi−αRi.Because the conditional posteriors ofα and θ in the previous equations do not qualify for a specified distribution, the Metropolis–Hasting sampler must be used to apply the MCMC approach. Tierney and Luke [29] proposed the Metropolis–Hastings algorithm within Gibbs sampling, which generates the posterior samples as follows:(1)
Begin with the first suggestionα0,θ0(2)
Specifyj=1(3)
Produceαj and θj by using M-H algorithm below using Equations (23) and (24) with the normal indicated distribution Nαj−1,varα and the inverse Fisher information matrix is used to calculate Nθj−1,varθ, where varα and varθ(i)
Create aNαj−1,varα is suggestion to α∗ and θ∗ by Nθj−1,varθ(ii)
Calculate the probabilities of approval(23)ρα=min1,π1∗α∗|θj−1,xπ1∗αj−1|θj−1,x,and(24)ρθ=min1,π2∗θ∗|αj,xπ2∗θj−1∣αj,x.(iii)
Create au1 and u2 from a Uniform 0,1 distribution(iv)
Ifu1=ρα, confess the proposition and Specify αj=α∗ else Specify αj=αj−1(v)
Ifu1=ρθ, confess the proposition and Specify θj=θ∗ else Specify θj=θj−1(4)
Calculateαj and θj(5)
Specifyj=j+1(6)
Steps from 3 to 6 should be reiterated N times(7)
To evaluate the CRIs ofα and θ of ψki, i=1,2,..,N, k=1,2 and ψ1,ψ2=α,θ as ψk1<ψk2<…<ψkN then the 1−γ100% CRIs of ψk is(25)ψkγ2N−M,ψk1−γ2N−M.For sufficiently largeN, the samples chosen are ψki,j=M+1,…,N. SEL function is accustomed to get the approximate Bayes estimates of α^ and θ^ as follows:(26)α^BS=1N−M∑j=M+1Nαj,and(27)θ^BS=1N−M∑j=M+1Nθj.And, the estimates for the mentioned parameters under LINEX loss function are(28)αBL=−1clog1N−M∑i=M+1Ne−cαi,and(29)θBL=−1clog1N−M∑i=M+1Ne−cθi.


## 6. Real-Life Data Application

In this section, the proposed estimation methods are applied to the parameters of the IPI model under a Prog–II–C scheme using a real data set from the COVID-19 data from France, which was analyses by Almetwally [6]. The considered COVID-19 data belong to France of 51 days that is recorded from 1 January to 20 February 2021. This data formed of mortality rate. A Prog–II–C sample of size m=10 simulated randomly from the sample of size n=20 with censoring scheme 5,5,0,0,0,0,0,0,0,0. For the data, we calculated the Kolmogorov–Smirnov (K–S) distance (D) between the fitted and empirical distribution functions, with K–S = 0.0999069 and p-value = 0.688698.The MLEs of parameters depend on Prog-II failure data in Table 1 are α^ and θ^ respectively, and are shown in Table 2. The mean of 1000 Boot-p and Boot-t samples of the lifetime parameters also given in Table 2 through using the algorithm defined in Section 4 of the bootstrap methods. For various values of the LINEX loss function’s shape parameter c, the parameters α and θ, Table 3 calculated and shewed the Bayes estimates related to both SEL and LINEX functions; the 95% ACIs and CRIs of the parameters α and θ have calculated. The LINEX loss function becomes symmetric as c going to zero, as is well known. In Table 2, the outputs of the SEL and LINEX loss functions are identical at c=0.0001, confirming accuracy of presented s. Figure 2 indicate simulation numbers of the parameters α and θ for COVID-19 data.Table 1
The Progressively Type-II failure COVID-19 data.
xi:0.09950.05250.06150.04550.1474Ri55000xi:0.33730.10870.10550.22350.0633Ri00000xi:0.05650.25770.13450.08430.1023Ri00000xi:0.22960.06910.05050.14340.2326Ri00000Table 2
Point estimates for the parametersα, θ, for COVID-19 data.
ParameterMLEBoot-pBoot-tSELLINEXc1=−2c2=2c3=0.0001α1.008711.32511.11350.9827310.9830720.9823890.982731θ0.3078310.52180.42310.273710.2737410.273710.27371Table 3
95% ACIs and CRI α and θ COVID-19 data.
ParameterMLEBoot-pBoot-tMCMCα(0.603973, 1.41345)(0.8751, 1.6324)(0.5891, 1.3271)(0.951173,1.01104)θ(-0.0186961, 0.634358)(0.6782, 0.8792)(0.4351, 0.7452)(0.264976,0.285227)Figure 2
Simulation numbers of the parameters for COVID-19 data.
(a)(b)


## 7. Simulated Data

Simulation experiments are carried out utilizing 1000 (Prog –II–C) samples per one simulation in order to compare parameter estimators for the IPI distribution. IPID is used to create (Prog –II– C) samples, with initial valuesα=1 and θ=0.6. The comparison of the different approaches of the producing α and θ estimators, has been taken into account in the computation of their mean square error (MSE), for k=1,2ψ1=α,ψ2=θ, as MSE ψk=1/Mψk^−ψk2, as M=1000 simulated samples numbers. The 95% CIs generated through utilizing asymptotic distributions of the MLEs and CRIs are compared using another criterion. The average confidence interval lengths (ACLs) and coverage probability (CP) are used to make comparisons between them. In this study the progressive schemes that have considered:Scheme I:R1=n−m, Ri=0 for i≠1Scheme II:Rm/2=Rm/2+1=n−m/2, Ri=0 for i≠m/2 and i≠m/2+1Scheme III:Rm=n−m, Ri=0 for i≠mTables4 and 5 illustrate the estimated parameters’ results and their MSE and Table 6 shows outcomes of ACL and CP of 95% CIs. From the results, the following notices can be observed:(1)
Tables4 and 5 show that as sample size increase, MSEs decrease, with Bayes estimates having the smallest MSEs for α and θ. As a result, Bayes estimates performance MLE approaches in all situations studied.(2)
In the sense that MSEs are smaller, Bayes estimates under LINEX withc=−2 better results estimates.(3)
Scheme I outperforms schemes II and III in terms of MSEs when the sample size is fixed atn and the effective size is fixed at m.(4)
Adapted from Table6. It can be seen that the CRIs produce more accurate results. Then, the ACIs for different sample sizes, observed failures, and schemes.Table 4
MSE of MLE and BE parameters estimatorsα with α0=1.
MLESELLINEXn,mSchemeC=−2C=2C=0.0001(15, 10)I0.37030.68551.8780.59040.6854(40.24)(9.0013)(1.24439)(10.3129)(9.001)II0.37840.80112.7381.11060.8007(39.37)(23.2342)(6.55607)(22.5079)(23.2216)III0.42040.72392.82291.55480.7236(35.29)(20.2274)(3.03739)(42.8603)(20.2287)(20, 15)I0.27420.49371.1590.11830.4936(5.294)(0.80339)(0.60741)(1.18222)(0.80343)II0.27740.28520.8970.33660.2851(5.268)(3.8652)(3.8086)(5.715)(3.8653)III0.32210.85232.56060.74910.8519(4.651)(1.068194)(1.07274)(1.20591)(1.68009)(30, 20)I0.20880.0970.25140.05840.097(0.6282)(0.13922)(0.11393)(0.17185)(0.13923)II0.2020.23920.47570.01260.2392(0.6389)(0.14607)(0.13187)(0.1949)(0.14608)III0.21240.23770.45320.020.2377(0.6242)(0.15856)(0.14083)(0.20695)(0.15856)(40, 20)I0.15960.22310.31510.13020.2231(0.7071)(0.10083)(0.009619)(0.01739)(0.010083)II0.18710.33820.08910.08910.3382(0.06631)(0.16286)(0.007436)(0.017436)(0.016284)III0.28880.33951.59870.9690.3394(0.5133)(0.096754)(0.02266)(0.0758)(0.096828)(50, 20)I0.1580.18790.28880.08280.1879(0.007097)(0.0011341)(0.001328)(0.01085)(0.001879)II0.22540.43930.89910.01060.4393(0.06032)(0.0018168)(0.0026371)(0.023192)(0.0018168)III0.39132.94937.86261.97992.9482(0.379)(0.0823423)(0.0212)(0.05568)(0.0823)Table 5
MSE of MLE and BE estimates for the parameterβ with β0=0.6.
MLESELLINEXn,mSchemeC=−2C=2C=0.0001(15, 10)I0.95850.60842.442.440.6082(14.85)(12.6991)(1.80526)(14.96)(12.6988)II0.93060.68082.82842.82840.6806(12.8)(11.3128)(1.5829)(2.31639.9981)(11.3137)III0.98450.99953.21663.21660.9992(17.54)(2.3791)(0.31721)(4.35105)(2.379)(20, 15)I0.84520.66752.31460.11830.6672(0.812)(0.38233)(0.2418)(0.8796)(0.38221)II0.82610.69132.00492.00492.0049(6.98)(5.751)(0.83298)(0.99981)(0.99981)III0.91560.83013.1033.1030.8294(12.71)(0.0325622)(0.0356730)(0.9974)(0.325915)(30, 20)I0.87981.12821.75771.12811.2181(0.1006)(0.030879)(0.020327)(0.02195)(0.0309)II0.83070.82661.66631.6630.8265(0.725)(0.39378)(0.047928)(0.047928)(0.39377)III1.03911.07081 .97441.97441.0707(2.074)(0.058934)(0.0087964)(0.049418)(0.058932)(40, 20)I1.15670.83931.7391.7390.8372(0.03138)(0.00437)(0.00571)(0.00571)(0.0636)II1.06610.90432.93071.74361.1098(0.2308)(0.04472)(0.05844)(0.059018)(0.06479)III0.99371.112.59912.59911.1098(1.643)(0.0080091)(0.001266)(0.01403)(0.0901)(50, 20)I1.17121.28842.0352.0351.2884(0.03312)(0.00392)(0.000067)(0.00319)(0.003914)II1.02731.0942.11572.11571.094(0.1902)(0.00744)(0.00894)(0.00813)(0.0074286)III0.96470.5343.45523.45520.5336(0.1407)(0.002475)(0.000545)(0.00472)(0.02475)Table 6
ACL and CB of95% CIs for the parameters α and β.
αβn,mSchemeMLEMCMCMLEMCMC(15, 10)I1.97293.77152.02064.868(0.953)(0.9706)(0.9278)(0.9447)II2.462344.481073.09675.1826(0.9405)(0.9421)(0.9353)(0.9615)III3.17535.18753.8495.6644(0.9491)(0.968)(0.9297)(0.9499)(20, 15)I1.280252.34281.911553.8025(0.9459)(0.9342)(0.926)(0.9305)II1.286372.2441.335693.7107(0.9384)(0.9354)(0.9711)(0.9261)III2.388254.04153.257215.7014(0.9644)(0.9652)(0.9296)(0.9371)(30, 20)I0.6362351.05410.5925252.5754(0.9403)(0.9408)(0.9565)(0.9527)II0.6853031.25930.7381472.8534(0.9506)(0.9431)(0.9636)(0.9464)III0.7125481.21230.7811553.1536(0.9444)(0.9611)(0.9387)(0.9536)(40, 20)I0.4281340.8340.6348943.0317(0.9737)(0.9743)(0.9337)(0.9586)II0.6247131.13380.7264052.8997(0.9254)(0.9715)(0.935)(0.9481)III1.977173.48962.100264.0023(0.964)(0.947)(0.9397)(0.9454)(50, 20)I0.4781480.80910.723612.7625(0.9329)(0.9553)(0.9526)(0.9689)II1.08.731.8231.144553.229(0.9358)(0.9649)(0.9473)(0.946)III5.380979.19655.684296.3552(0.9444)(0.9699)(0.9337)(0.9521)


## 8. Conclusion

We discuss an iterative technique for deriving MLEs from an inverse power ishita lifetime distribution using progressively Type-II censored samples. Another estimation approach, based on Bayes estimates, is also considered. The Bayes estimates were produced using loss functions; however, they are not available in explicit form. Propose that the Bayes estimators and associated IPI distribution CIs be computed using the MCMC technique. Derive the posterior summaries of interest, such as credible intervals for the parameters, in a simple way using typical MCMC simulation methods for a Bayesian study of the model. The findings of this study used to the estimate problem of the IPID using complete data, which are usually progressive type-II censoring samples. The proposed methods are demonstrated with numerical examples [30, 31].



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