# Analysis of Type-II Censored Competing Risks’ Data under Reduced New Modified Weibull Distribution

Authors: Saad J. Almalki; Tahani A. Abushal; M. D. Alsulami; G. A. Abd-Elmougod
Journal: Complexity
 (2021)

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## Abstract

Models with the bathtub-shaped hazard rate function are widely used in lifetime analysis and reliability engineering. In this paper, we adopted the reduced new modified Weibull (RNMW) distribution with a bathtub-shaped hazard rate function. Under consideration that the population units are failing with two independent causes of failure and the failure time is distributed with RNMW distribution, we formulate the model which is known as competing risks model. The model parameters under the type-II censoring scheme are estimated with the maximum likelihood method with the corresponding asymptotic confidence intervals. Also, the Bayes point and credible intervals with the help of MCMC methods are constructed. The real and simulated datasets are analyzed for illustrative purposes. Finally, the estimators are compared with the Monte Carlo simulation study.


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## Body

## 1. Introduction

In the past few years, different modifications of Weibull distribution were constructed with a bathtub-shaped hazard rate function. These modifications were discussed with different authors, see [1–4]. Some modification models of Weibull distribution are discussed with with two or three parameters’ vectors [4–6]. Also, some of the modifications of Weibull distribution were discussed with four parameters (see [7]) and the modifications with five parameters (see [8]). Then, the large sets of distribution are generalized with generalized linear exponential distribution as the bathtub-shaped hazard rate presented by Sarhan et al. [9]. The unimodal, decreasing, increasing, or bathtub-shaped hazard rate beta-Weibull distribution was proposed by Lee et al. [10]. The four parameters’ generalized modified Weibull distribution was proposed by Carrasco et al. [11].Almalki and Yuan [12] have presented new modified Weibull distribution with five parameters and distribution function (CDF) given by(1)Ft=1−e−α1tβ1−α2tβ2eλt,t>0,α1,α2,β1,β2,λ>0,where α1 and α2 are called shape parameters, β1 and β2 are called scale parameters, and λ is called the accelerated parameter. Most effective and flexible lifetime distributions with bathtub-shaped hazard functions have more than three or four parameters but the reduced lifetime distributions have a few number of parameters, two or three, but these distributions are not much (for example, the modified Weibull distribution, the exponentiated Weibull distribution, and the modified Weibull extension). Almalki [13] presented a new version of new modified Weibull distribution with a bathtub-shaped failure rate function (FRF) called reduced new modified Weibull distribution by taking β1 = β2=1/2, with CDF given by(2)Ft=1−e−tα1+α2eλt,t>0,α1,α2,λ>0,where α1 and α2 are called scale parameters and λ is called the accelerated parameter.In this paper, we considered the special case of reduced version (2) called new reduced modified Weibull (NRMW) distribution with parameters α1=α2=α as follows:(3)Ft=1−e−αt1+eλt,t>0,α,λ>0.The density, survival, and hazard failure rate functions are given, respectively, by(4)ft=α2t1+1+2λteλte−αt1+eλt,(5)st=e−αt1+eλt,ht=α2t1+1+2λteλt,where α and λ are nonnegative parameters. The PDF of NRMW distribution has different shape, decreasing, or decreasing-increasing-decreasing, as shown in Figure 1. Also, different shapes of hazard failure rate function, decreasing, and the bathtub-shape are given in Figure 1.Figure 1
The PDF and HFR of RNMW distribution for different parameters’ values.
(a)(b)If the random variableT is distributed with NRMW, then the following two properties are satisfied.(1)
Therth moment of the random variable of NRMW is given by(6)μr=2r∑i=0∞∑j=0∞−αiλiji!j!Γi+2j+rαi+2j+r,whereΓ⋅ is the gamma function.(2)
The moment generating function is given by(7)Mtt=1+2∑i,j,k=0∞−αiiλjtk+1i!j!k!Γi+2j+k+2αi+2j+k+2.The lifetime data of product units are presented in complete or censored data dependent on some consideration of the test time or cost. The word complete is used when the failure time data is obtained from all units under the test. But, the censored data is used when some but not all units under the test are failed through determined period of time. The censoring is available in various types; type-I and type-II censoring schemes are from the oldest censoring scheme. In the type-I censoring scheme, the experimenter removed the test at prefixed time; then, the failure times are random. But, in the type-II censoring scheme, the experimenter removed the test at the prefixed number of failure; then, the test time is random.In life-testing experiments or reliability analysis, tested units fail with different modes of failure which is a common phenomenon in the life testing experiment and known by the competing risks’ problem. In practice, for the life products, the product units fail under different failure modes, one of them causes failure, and the problem of assessment of the risk of any failure mode in the presence of other modes takes the attention of several authors; for more details, see [14-18] and recent [19, 20]. In our population, we record the failure time and the corresponding failure mode under consideration that only two failure modes exist and the unit fails under only one mode.The paper aims to model type-II competing risk samples with NRMW distribution when the failure modes are independent and failure occurs under only one mode. Then, we have a description of the model mechanism and the corresponding likelihood function. Also, the model parameters are estimates for point and interval with maximum likelihood and Bayes methods. The theoretical results are discussed through simulation Monte Carlo study and real data analysis.The paper is organized as follows. Section2 contains some abbreviations and model description. Section 3 contains the classical estimation with the MLE method. Section 4 contains Bayes estimation with the MCMC method. In Section 5, the two lifetime data are analyzed for the illustrating purpose. Section 6 reports the results of the Monte Carlo study.


## 2. The Abbreviations and Model

In this section, we give the list of abbreviations that are used in this article as well as the complete description of the model mechanism as follows.


### 2.1. List of Abbreviations

NRMW: Reduced new modified WeibullPDF: Probability density functionCDF: Cumulative distribution functionS.: Survival functionHF: Hazard functionti: The observed time to failuretij: ith time to failure under cause jδj: Indicator denoted to the causeMCMC: Markov chain Monte CarloMH: Metropolis–Hastings algorithmMSE: Mean squared errorMIL: Mean interval lengthCP: Coverage percentageMLE: Maximum likelihood estimateLF: Loss functionACI: Approximate confidence intervalCI: Credible intervals


### 2.2. The Model Description

Letn identical independent (i.i) units are selected from population to test under the life testing experiment. And, the integer number m needing for statistical inference is determined at the beginning of the experiment. When the experiment is running, failure time and the corresponding cause of failure are recorded, say Ti,ηii=1,2,…,m. Under consideration of that only two causes of failure and observed sample t = {t1,η1, t2,η2, ..., tm,ηm}, the joint likelihood function is formulated by(8)Lt¯∣ω¯=AStmi=1n−m∏i=1mh1tiIηi=1h2tiIηi=2Sti,where ω=α1,α2,λ, A=n!/n−m!, S.=s1.s2., and(9)Iηi=j=1,ηi=j,0,ηi≠j.For the given type-II competing risks’ sample drawn from NRMW distributions, sayTij∼ NRMW αj,λ, j=1,2, and the observed variable Ti  = min (Ti1, Ti2), i= 1, 2, …, m, are independent and distributed as NRMW distribution with shape parameter α1+α2 and accelerate parameter λ. Hence, from (8), the likelihood function is reduced to(10)Lα1,α2,λ|t¯=A2−mα1m1α2m2∏i=1m1ti1+1+2λtieλti×e−α1+α2∑i=1mti1+eλti+n−mtm1+eλtm,where the two integers m1=∑i=1mIηi=1 and m2=∑i=1mIηi=2 present numbers of units that fail under first and second causes, respectively.Remarks 1.
(1)
The latent failure time is distributed with NRMW distribution withα1+α2 and λ shape and accelerate parameters, respectively(2)
The valuem1 is distributed with binomial distribution with sample size m1 and probability of success α1/α1+α2 but m1 is binomial distributed with sample size m2 and probability of success α2/α1+α2


## 2.1. List of Abbreviations

NRMW: Reduced new modified WeibullPDF: Probability density functionCDF: Cumulative distribution functionS.: Survival functionHF: Hazard functionti: The observed time to failuretij: ith time to failure under cause jδj: Indicator denoted to the causeMCMC: Markov chain Monte CarloMH: Metropolis–Hastings algorithmMSE: Mean squared errorMIL: Mean interval lengthCP: Coverage percentageMLE: Maximum likelihood estimateLF: Loss functionACI: Approximate confidence intervalCI: Credible intervals


## 2.2. The Model Description

Letn identical independent (i.i) units are selected from population to test under the life testing experiment. And, the integer number m needing for statistical inference is determined at the beginning of the experiment. When the experiment is running, failure time and the corresponding cause of failure are recorded, say Ti,ηii=1,2,…,m. Under consideration of that only two causes of failure and observed sample t = {t1,η1, t2,η2, ..., tm,ηm}, the joint likelihood function is formulated by(8)Lt¯∣ω¯=AStmi=1n−m∏i=1mh1tiIηi=1h2tiIηi=2Sti,where ω=α1,α2,λ, A=n!/n−m!, S.=s1.s2., and(9)Iηi=j=1,ηi=j,0,ηi≠j.For the given type-II competing risks’ sample drawn from NRMW distributions, sayTij∼ NRMW αj,λ, j=1,2, and the observed variable Ti  = min (Ti1, Ti2), i= 1, 2, …, m, are independent and distributed as NRMW distribution with shape parameter α1+α2 and accelerate parameter λ. Hence, from (8), the likelihood function is reduced to(10)Lα1,α2,λ|t¯=A2−mα1m1α2m2∏i=1m1ti1+1+2λtieλti×e−α1+α2∑i=1mti1+eλti+n−mtm1+eλtm,where the two integers m1=∑i=1mIηi=1 and m2=∑i=1mIηi=2 present numbers of units that fail under first and second causes, respectively.Remarks 1.
(1)
The latent failure time is distributed with NRMW distribution withα1+α2 and λ shape and accelerate parameters, respectively(2)
The valuem1 is distributed with binomial distribution with sample size m1 and probability of success α1/α1+α2 but m1 is binomial distributed with sample size m2 and probability of success α2/α1+α2


## 3. Maximum Likelihood Estimations

The maximum likelihood estimators of model parameters in this section are formulated for the point and interval estimate with given observed type-II competing risks’ censoring sample as follows.


### 3.1. Point MLE

The natural logarithms of likelihood function (10) is reduced to(11)ℓα1,α2,λ|t¯=Constant+m1logα1+m2logα2+∑i=1mlog1+1+2λtieλti−α1+α2∑i=1mti1+eλti+n−mtm1+eλtm.After taking the partial derivative of the log-likelihood function with respect to parameters’ vectorα1 and α2, we have the likelihood equations which are defined by the following theorem.Theorem 1.
The conditional ML estimators of parametersα1 and α2, given λ> 0 and two numbers m1 and m2>0, can be written by(12)αjλ=mjD1λ,j=1and2,where(13)D1λ=∑i=1mti1+eλti+n−mtm1+eλtm.Remark 2.
Ifmj=0, there are no failures due to cause j. Then, the observed data t = {t1,η1, t2,η2, …, tm,ηm} does not provide information about parameter αj.
Also, the partial derivative of the log-likelihood function with respect to parameterλ(14)∑i=1m2tieλti+1+2λtitieλti1+1+2λtieλti−α1+α2∑i=1mtitieλti+n−mtmtmeλtm=0.
It is clearly from (14) that there is not a closed form for the ML estimator of λ; then, the iteration method can solve this problem as follows.Theorem 2.
The ML estimators of parameterλ presented by the nonlinear equation can be given by(15)λ=D2λ,where(16)D2λ=m1+m2∑i=1mtitieλti+n−mtmtmeλtm/D1λ−∑i=1m3tieλti/1+1+2λtieλti∑i=1m2ti2eλti/1+1+2λtieλti.Remark 3.
Nonlinear equation (16) can be solved with any iteration method such as Newton–Raphson or fixed point but needs to find the initial value of λ which can be obtained from the profile likelihood function presented by(17)gλ|t¯∝∑j=12mjlogmjD1λ+∑i=1mlog1+1+2λtieλti−m1+m2.
Then, from theorems (1) and (2), the ML estimate of the parameters’ vector ω=α1,α2,λ is obtained to be ω^=α^1,α^2,λ^.


### 3.2. Interval Estimation

The second derivative of the log likelihood function with respect toω=α1,α2,λ is given by(18)∂2ℓα1,α2,λ|t¯∂αj2=−mjαj,i=1and2,∂2ℓα1,α2,λ|t¯∂λ2=∑i=1mD3λd/dλD3λ−d/dλD3λ2D32λ−α1+α2∑i=1mti2tieλti+n−mtm2tmeλtm,(19)∂2ℓα1,α2,λ|t¯∂α1∂α2=∂2ℓα1,α2,λ|t¯∂α2∂α1=0,(20)∂2ℓα1,α2,λ|t¯∂α1∂λ=∂2ℓα1,α2,λ|t¯∂λ∂α1=∂2ℓα1,α2,λ|t¯∂α2∂λ=∂2ℓα1,α2,λ|t¯∂λ∂α2=−∑i=1mtitieλti+n−mtmtmeλtm,where(21)D3λ=1+1+2λtieλti.Hence, from equations (19) to (21), the Fisher information matrix (IM) is defined as minus expectation of equations (19)–(21) to be given by(22)IM=E−∂2ℓα1,α2,λ|t¯∂α12−∂2ℓα1,α2,λ|t¯∂α1∂α2−∂2ℓα1,α2,λ|t¯∂α1∂λ−∂2ℓα1,α2,λ|t¯∂α2∂α1−∂2ℓα1,α2,λ|t¯∂α22−∂2ℓα1,α2,λ|t¯∂α2∂λ−∂2ℓα1,α2,λ|t¯∂λ∂α1−∂2ℓα1,α2,λ|t¯∂λ∂α2−∂2ℓα1,α2,λ|t¯∂λ2.The approximate value of the Fisher information matrix is defined by(23)AIM=IM0=−∂2ℓα1,α2,λ|t¯∂α12−∂2ℓα1,α2,λ|t¯∂α1∂α2−∂2ℓα1,α2,λ|t¯∂α1∂λ−∂2ℓα1,α2,λ|t¯∂α2∂α1−∂2ℓα1,α2,λ|t¯∂α22−∂2ℓα1,α2,λ|t¯∂α2∂λ−∂2ℓα1,α2,λ|t¯∂λ∂α1−∂2ℓα1,α2,λ|t¯∂λ∂α2−∂2ℓα1,α2,λ|t¯∂λ2atα^1,α^2,λ^.Under limiting properties of the MLEs is using it to present the asymptotic distribution ofω^=α^1,α^2,λ^. Then, the asymptotically normally distributed mean ω=α1,α2,λ and variance covariance matrix IM0ω¯^−1 describe the asymptotic distribution of ω=α1,α2,λ Therefore, the interval estimate with confidence level 1001−2γ% is given by(24)α^1∓Zγϵ11,α^2∓Zγϵ22,andλ^∓Zγϵ33,where the vector ϵ11,ϵ22,ϵ33 presents the diagonal of inverse of IM0ω¯^−1 and the value Zγ presents the percentile of the standard normal distribution with right-tail probability γ.


## 3.1. Point MLE

The natural logarithms of likelihood function (10) is reduced to(11)ℓα1,α2,λ|t¯=Constant+m1logα1+m2logα2+∑i=1mlog1+1+2λtieλti−α1+α2∑i=1mti1+eλti+n−mtm1+eλtm.After taking the partial derivative of the log-likelihood function with respect to parameters’ vectorα1 and α2, we have the likelihood equations which are defined by the following theorem.Theorem 1.
The conditional ML estimators of parametersα1 and α2, given λ> 0 and two numbers m1 and m2>0, can be written by(12)αjλ=mjD1λ,j=1and2,where(13)D1λ=∑i=1mti1+eλti+n−mtm1+eλtm.Remark 2.
Ifmj=0, there are no failures due to cause j. Then, the observed data t = {t1,η1, t2,η2, …, tm,ηm} does not provide information about parameter αj.
Also, the partial derivative of the log-likelihood function with respect to parameterλ(14)∑i=1m2tieλti+1+2λtitieλti1+1+2λtieλti−α1+α2∑i=1mtitieλti+n−mtmtmeλtm=0.
It is clearly from (14) that there is not a closed form for the ML estimator of λ; then, the iteration method can solve this problem as follows.Theorem 2.
The ML estimators of parameterλ presented by the nonlinear equation can be given by(15)λ=D2λ,where(16)D2λ=m1+m2∑i=1mtitieλti+n−mtmtmeλtm/D1λ−∑i=1m3tieλti/1+1+2λtieλti∑i=1m2ti2eλti/1+1+2λtieλti.Remark 3.
Nonlinear equation (16) can be solved with any iteration method such as Newton–Raphson or fixed point but needs to find the initial value of λ which can be obtained from the profile likelihood function presented by(17)gλ|t¯∝∑j=12mjlogmjD1λ+∑i=1mlog1+1+2λtieλti−m1+m2.
Then, from theorems (1) and (2), the ML estimate of the parameters’ vector ω=α1,α2,λ is obtained to be ω^=α^1,α^2,λ^.


## 3.2. Interval Estimation

The second derivative of the log likelihood function with respect toω=α1,α2,λ is given by(18)∂2ℓα1,α2,λ|t¯∂αj2=−mjαj,i=1and2,∂2ℓα1,α2,λ|t¯∂λ2=∑i=1mD3λd/dλD3λ−d/dλD3λ2D32λ−α1+α2∑i=1mti2tieλti+n−mtm2tmeλtm,(19)∂2ℓα1,α2,λ|t¯∂α1∂α2=∂2ℓα1,α2,λ|t¯∂α2∂α1=0,(20)∂2ℓα1,α2,λ|t¯∂α1∂λ=∂2ℓα1,α2,λ|t¯∂λ∂α1=∂2ℓα1,α2,λ|t¯∂α2∂λ=∂2ℓα1,α2,λ|t¯∂λ∂α2=−∑i=1mtitieλti+n−mtmtmeλtm,where(21)D3λ=1+1+2λtieλti.Hence, from equations (19) to (21), the Fisher information matrix (IM) is defined as minus expectation of equations (19)–(21) to be given by(22)IM=E−∂2ℓα1,α2,λ|t¯∂α12−∂2ℓα1,α2,λ|t¯∂α1∂α2−∂2ℓα1,α2,λ|t¯∂α1∂λ−∂2ℓα1,α2,λ|t¯∂α2∂α1−∂2ℓα1,α2,λ|t¯∂α22−∂2ℓα1,α2,λ|t¯∂α2∂λ−∂2ℓα1,α2,λ|t¯∂λ∂α1−∂2ℓα1,α2,λ|t¯∂λ∂α2−∂2ℓα1,α2,λ|t¯∂λ2.The approximate value of the Fisher information matrix is defined by(23)AIM=IM0=−∂2ℓα1,α2,λ|t¯∂α12−∂2ℓα1,α2,λ|t¯∂α1∂α2−∂2ℓα1,α2,λ|t¯∂α1∂λ−∂2ℓα1,α2,λ|t¯∂α2∂α1−∂2ℓα1,α2,λ|t¯∂α22−∂2ℓα1,α2,λ|t¯∂α2∂λ−∂2ℓα1,α2,λ|t¯∂λ∂α1−∂2ℓα1,α2,λ|t¯∂λ∂α2−∂2ℓα1,α2,λ|t¯∂λ2atα^1,α^2,λ^.Under limiting properties of the MLEs is using it to present the asymptotic distribution ofω^=α^1,α^2,λ^. Then, the asymptotically normally distributed mean ω=α1,α2,λ and variance covariance matrix IM0ω¯^−1 describe the asymptotic distribution of ω=α1,α2,λ Therefore, the interval estimate with confidence level 1001−2γ% is given by(24)α^1∓Zγϵ11,α^2∓Zγϵ22,andλ^∓Zγϵ33,where the vector ϵ11,ϵ22,ϵ33 presents the diagonal of inverse of IM0ω¯^−1 and the value Zγ presents the percentile of the standard normal distribution with right-tail probability γ.


## 4. MCMC Bayes Estimation

Bayesian estimation depends on the amount of the past information about the model parameters which is known by prior distribution. And, the amount of the information existing in data is known by the likelihood function. Then, we aim to formulate these two data in the form of probability distribution, known by posterior distribution, as follows.Suppose that the parameters’ vectorω=α1,α2,λ is distributed with independent gamma priors. Therefore, the joint prior distribution can be presented by(25)π∗α1,α2,λ∝α1a1−1α2a2−1λa3−1exp−b1α1−b2α2−b3λ,aiandbi>0andi=1,2,and3.Generally, the posterior distribution can be formulated by(26)πα1,α2,λ|t¯=π∗α1,α2,λLα1,α2,λ|t¯∫∫ω¯∫π∗α1,α2,λLα1,α2,λ|t¯dα1dα2dλ.Also, generally the ratio of two integrals (26), especially, in the high-dimensional case, cannot obtain analytical. Then, different methods are available to approximate this ratio such as Lindley approximation and numerical integration. But, one of the most applied methods known by MCMC method is discussed in this article. Hence, without loss of the generality, the Bayes estimate of the function Ωω¯ under squared error loss (SEL) function is given by(27)Ω^B=∫∫ω¯∫Ωω¯πω¯|t¯dα1dα2dλ.


### 4.1. MCMC Approach

The posterior distribution presented by (26) with prior distribution (24) and likelihood function (10) is reduced to(28)πω¯|t¯=Cα1m1+a1−1α2m2+a2−1λa3−1α1m1α2m2∏i=1m1+1+2λtieλti×e−α1+α2∑i=1mti1+eλti+n−mtm1+eλtm−b1α1−b2α2−b3λ,where C is the normalized constant. From the posterior distribution the full-conditional PDFs of the vector ω, ω=α1,α2,λ is given by(29)π1α1|α2,λ,t¯∝Gammam1+a1,b1+∑i=1mti1+eλti+n−mtm1+eλtm,(30)π2α2|α1,λ,t¯∝Gammam2+a2,b2+∑i=1mti1+eλti+n−mtm1+eλtm,(31)π3λ|α1,α2,t¯∝λa3−1e∑i=1mlog1/ti1+1+2λtieλti−α1+α2ti1+eλti+n−mtm1+eλtm−b3λ.The full-conditional PDFs have shown that generation from posterior distribution depend on two-conditional gamma density and more with general conditional function which is similar to normal distribution. Hence, the MH algorithm under Gibbs is a more suitable algorithm to construct the empirical posterior distribution (see [24]) Also, for the recent review of MH algorithms under Gibbs, see [25] and [26].


### 4.2. Gibbs with MH Algorithms

Step 1: set the indicatorκ=1 and ω0=α^1,α^2,λ^.Step 2: from (28) and (29), generate α1κ and α2κ.Step 3: with normal proposal distribution with meanλκ−1 and variance V(λ) = ϵ33, generate the value λκ under proposal normal distribution with MH algorithms as follows.(i)
Generate the proposal,λ∗, from Nλκ−1,ϵ33.(ii)
Calculate the acceptance probability:(32)σλκ−1,λ∗=min1,π3λ∗λκ−1.(iii)
Generate uniform (0, 1) valueU.(iv)
IfU≤σλκ−1,λ∗, accept the proposal λ∗. Otherwise, reject the proposal and repeat the value λκ−1.Step 4: reportωκ=α1κ,α2κ,λκ and set κ=κ+1.Step 5: steps from (2) to (4) are repeated N times.Step 6: the point Bayes estimators for any functionΩω¯ are defined by(33)Ω^B=1N−Mi=M+1NΩi,and integerM is the number needed to achieve the stationary distribution (burn-in). The corresponding posterior variance of Ωω¯ is given by(34)V^Ω=1N−Mi=M+1NΩi−Ω^B2.Step 7: for interval estimators with confidence level1001−2γ% of Ωω¯ is given by(35)ΩγN−M,Ω1−γN−M.


## 4.1. MCMC Approach

The posterior distribution presented by (26) with prior distribution (24) and likelihood function (10) is reduced to(28)πω¯|t¯=Cα1m1+a1−1α2m2+a2−1λa3−1α1m1α2m2∏i=1m1+1+2λtieλti×e−α1+α2∑i=1mti1+eλti+n−mtm1+eλtm−b1α1−b2α2−b3λ,where C is the normalized constant. From the posterior distribution the full-conditional PDFs of the vector ω, ω=α1,α2,λ is given by(29)π1α1|α2,λ,t¯∝Gammam1+a1,b1+∑i=1mti1+eλti+n−mtm1+eλtm,(30)π2α2|α1,λ,t¯∝Gammam2+a2,b2+∑i=1mti1+eλti+n−mtm1+eλtm,(31)π3λ|α1,α2,t¯∝λa3−1e∑i=1mlog1/ti1+1+2λtieλti−α1+α2ti1+eλti+n−mtm1+eλtm−b3λ.The full-conditional PDFs have shown that generation from posterior distribution depend on two-conditional gamma density and more with general conditional function which is similar to normal distribution. Hence, the MH algorithm under Gibbs is a more suitable algorithm to construct the empirical posterior distribution (see [24]) Also, for the recent review of MH algorithms under Gibbs, see [25] and [26].


## 4.2. Gibbs with MH Algorithms

Step 1: set the indicatorκ=1 and ω0=α^1,α^2,λ^.Step 2: from (28) and (29), generate α1κ and α2κ.Step 3: with normal proposal distribution with meanλκ−1 and variance V(λ) = ϵ33, generate the value λκ under proposal normal distribution with MH algorithms as follows.(i)
Generate the proposal,λ∗, from Nλκ−1,ϵ33.(ii)
Calculate the acceptance probability:(32)σλκ−1,λ∗=min1,π3λ∗λκ−1.(iii)
Generate uniform (0, 1) valueU.(iv)
IfU≤σλκ−1,λ∗, accept the proposal λ∗. Otherwise, reject the proposal and repeat the value λκ−1.Step 4: reportωκ=α1κ,α2κ,λκ and set κ=κ+1.Step 5: steps from (2) to (4) are repeated N times.Step 6: the point Bayes estimators for any functionΩω¯ are defined by(33)Ω^B=1N−Mi=M+1NΩi,and integerM is the number needed to achieve the stationary distribution (burn-in). The corresponding posterior variance of Ωω¯ is given by(34)V^Ω=1N−Mi=M+1NΩi−Ω^B2.Step 7: for interval estimators with confidence level1001−2γ% of Ωω¯ is given by(35)ΩγN−M,Ω1−γN−M.


## 5. Data Analysis

In this section, we discussed two lifetime datasets; real and simulated data to illustrate the developed results in this paper.


### 5.1. Example 1: Real Dataset

Under a laboratory experiment with a conventional laboratory environment, Hoel [23] tested a male mice exposed to radiation dose of 300 roentgens at age of 5-6 weeks. The lifetime data under two causes of failure, thymic lymphoma as a first cause and other causes as a second cause of failure, are reported in Table 1. The causes of failure are obtained for each mouse by autopsy. Restricting the analysis to two causes of death, for the purpose of analysis, we consider thymic lymphoma as cause 1 and reticulum cell sarcoma as cause 2. Different combinations of Hoel data are used by several authors [24–26]. Hence, under sample size n = 61 and effected sample size m = 23, the type-II competing risks’ data after divided by 100 for simplicity are given by t¯={(0.40, 2), (0.42, 2), (0.51, 2), (0.62, 2), (1.59, 1), (1.63, 2), (1.79, 2), (1.89, 1), (1.91, 1), (1.98. 1), (2.00, 1), (2.06, 2), (2.07, 1), (2.20, 1), (2.22, 2), (2.28, 2), (2.35, 1), (2.45, 1), (2.49, 2), (2.50, 1), (2.52, 2), (2.56, 1), (2.61, 1)}.Table 1
The failure data of 61 male mice under radiation dose or 300 r at age 5-6 weeks.
Thymic159189191198200207220235245250256lymphoma261265266280343356383403414428432Other causes40425162163179206222228252249282324333341366385407420431441461462482517517524564567586619620621622647651686761763For givent¯ dataset, the efficiency of the theoretical results of NRMW distribution for analysis of this data is constructed. Under type-II competing risks’ data, the profile log-likelihood function of λ given in Figure 2 is a unimodal function, and the ML estimate is computed with iteration with initial guess of λ as 1.0. The noninformative prior is used for prior information as ai=bi=0.0001, i=1,2,3,and4. The results of MLE and Bayes estimate are reported in Table 2 for point and 95% interval estimate. For the MCMC approach in the Bayes method. We run the chan (Gibbs with MH algorithm) 11000 times and the first 1000 times which are needed to reach the stationary distribution are deleted as known by “brun-in.” The empirical posterior distribution under the MCMC approach is described with Figures 3 and 4. The MCMC convergence to the posterior distribution is described by Figures 3 and 4.Figure 2
Profile log-likelihood function ofλ.Table 2
Point and 95% interval ML and Bayes estimate.
Pa.⋅ML⋅B−MCMC95% ACILength95% CILengthα10.005120.0068(0.00000, 0.01472)0.0147(0.0012, 0.0183)0.0172α20.004690.0062(0.00000, 0.01512)0.0151(0.0011, 0.0171)0.0161λ1.283891.2411(0.54999, 2.01780)1.4678(0.7868, 1.8240)1.0372Figure 3
Simulation number generated by the MCMC method forα1, α2, and λ, respectively.
(a)(b)(c)Figure 4
Histogram generated by the MCMC method forα1, α2, and λ, respectively.
(a)(b)(c)


### 5.2. Example 2: Simulated Dataset

The algorithms used to generate and analyze the type-II competing risks sample from NRMW distribution is described as follows:(1)
Suppose a sample sizen = 50 and effected sample size m=30.(2)
From gamma prior distributions withai,bi|i=1,2,3 = 1,5,1,15,5,2, generate a sample of size 10. Then, the true parameters’ value is computed to be the sample mean ω=α1,α2,λ = 0.2,0.05,2.2.(3)
Generate type-II competing risks’ sample from NRMW with (α1+α2) shape and λ scale parameters to be t¯={0.0273, 0.0349, 0.0604, 0.0779, 0.0834, 0.0902, 0.1604, 0.167, 0.1674, 0.1682, 0.1892, 0.2384, 0.2431, 0.2786, 0.2786, 0.2958, 0.3827, 0.452, 0.5049, 0.5735, 0.5845, 0.5996, 0.6032, 0.6201, 0.6279, 0.6376, 0.6461, 0.6529, 0.6668, 0.6726}; m1 and m2 are generated from binomial distribution to be m1=25 and m2=5.(4)
The graphs of profile log-likelihood function and empirical posterior distribution under MCMC are given in Figures5–7, respectively.(5)
The results of ML and Bayes estimate are reported in Table3 for point and 95% interval estimate.Figure 5
Profile log-likelihood function ofλ.Figure 6
Simulation number generated by the MCMC method forα1, α2, and λ, respectively.
(a)(b)(c)Figure 7
Histogram generated by the MCMC method forα1, α2, and λ, respectively.
(a)(b)(c)Table 3
Point and 95% interval ML and Bayes estimate.
Pa.⋅ML⋅B−MCMC95% ACILength95% CILengthα1=0.200.12740.1467(0.0334, 0.2215)0.1881(0.0742, 0.2469)0.1727α2=0.050.02550.0320(0.0000, 0.0529)0.0529(0.0104, 0.0690)0.0586λ=2.201.28392.4580(1.4654, 3.8878)2.4224(1.5523, 3.4745)1.9222


## 5.1. Example 1: Real Dataset

Under a laboratory experiment with a conventional laboratory environment, Hoel [23] tested a male mice exposed to radiation dose of 300 roentgens at age of 5-6 weeks. The lifetime data under two causes of failure, thymic lymphoma as a first cause and other causes as a second cause of failure, are reported in Table 1. The causes of failure are obtained for each mouse by autopsy. Restricting the analysis to two causes of death, for the purpose of analysis, we consider thymic lymphoma as cause 1 and reticulum cell sarcoma as cause 2. Different combinations of Hoel data are used by several authors [24–26]. Hence, under sample size n = 61 and effected sample size m = 23, the type-II competing risks’ data after divided by 100 for simplicity are given by t¯={(0.40, 2), (0.42, 2), (0.51, 2), (0.62, 2), (1.59, 1), (1.63, 2), (1.79, 2), (1.89, 1), (1.91, 1), (1.98. 1), (2.00, 1), (2.06, 2), (2.07, 1), (2.20, 1), (2.22, 2), (2.28, 2), (2.35, 1), (2.45, 1), (2.49, 2), (2.50, 1), (2.52, 2), (2.56, 1), (2.61, 1)}.Table 1
The failure data of 61 male mice under radiation dose or 300 r at age 5-6 weeks.
Thymic159189191198200207220235245250256lymphoma261265266280343356383403414428432Other causes40425162163179206222228252249282324333341366385407420431441461462482517517524564567586619620621622647651686761763For givent¯ dataset, the efficiency of the theoretical results of NRMW distribution for analysis of this data is constructed. Under type-II competing risks’ data, the profile log-likelihood function of λ given in Figure 2 is a unimodal function, and the ML estimate is computed with iteration with initial guess of λ as 1.0. The noninformative prior is used for prior information as ai=bi=0.0001, i=1,2,3,and4. The results of MLE and Bayes estimate are reported in Table 2 for point and 95% interval estimate. For the MCMC approach in the Bayes method. We run the chan (Gibbs with MH algorithm) 11000 times and the first 1000 times which are needed to reach the stationary distribution are deleted as known by “brun-in.” The empirical posterior distribution under the MCMC approach is described with Figures 3 and 4. The MCMC convergence to the posterior distribution is described by Figures 3 and 4.Figure 2
Profile log-likelihood function ofλ.Table 2
Point and 95% interval ML and Bayes estimate.
Pa.⋅ML⋅B−MCMC95% ACILength95% CILengthα10.005120.0068(0.00000, 0.01472)0.0147(0.0012, 0.0183)0.0172α20.004690.0062(0.00000, 0.01512)0.0151(0.0011, 0.0171)0.0161λ1.283891.2411(0.54999, 2.01780)1.4678(0.7868, 1.8240)1.0372Figure 3
Simulation number generated by the MCMC method forα1, α2, and λ, respectively.
(a)(b)(c)Figure 4
Histogram generated by the MCMC method forα1, α2, and λ, respectively.
(a)(b)(c)


## 5.2. Example 2: Simulated Dataset

The algorithms used to generate and analyze the type-II competing risks sample from NRMW distribution is described as follows:(1)
Suppose a sample sizen = 50 and effected sample size m=30.(2)
From gamma prior distributions withai,bi|i=1,2,3 = 1,5,1,15,5,2, generate a sample of size 10. Then, the true parameters’ value is computed to be the sample mean ω=α1,α2,λ = 0.2,0.05,2.2.(3)
Generate type-II competing risks’ sample from NRMW with (α1+α2) shape and λ scale parameters to be t¯={0.0273, 0.0349, 0.0604, 0.0779, 0.0834, 0.0902, 0.1604, 0.167, 0.1674, 0.1682, 0.1892, 0.2384, 0.2431, 0.2786, 0.2786, 0.2958, 0.3827, 0.452, 0.5049, 0.5735, 0.5845, 0.5996, 0.6032, 0.6201, 0.6279, 0.6376, 0.6461, 0.6529, 0.6668, 0.6726}; m1 and m2 are generated from binomial distribution to be m1=25 and m2=5.(4)
The graphs of profile log-likelihood function and empirical posterior distribution under MCMC are given in Figures5–7, respectively.(5)
The results of ML and Bayes estimate are reported in Table3 for point and 95% interval estimate.Figure 5
Profile log-likelihood function ofλ.Figure 6
Simulation number generated by the MCMC method forα1, α2, and λ, respectively.
(a)(b)(c)Figure 7
Histogram generated by the MCMC method forα1, α2, and λ, respectively.
(a)(b)(c)Table 3
Point and 95% interval ML and Bayes estimate.
Pa.⋅ML⋅B−MCMC95% ACILength95% CILengthα1=0.200.12740.1467(0.0334, 0.2215)0.1881(0.0742, 0.2469)0.1727α2=0.050.02550.0320(0.0000, 0.0529)0.0529(0.0104, 0.0690)0.0586λ=2.201.28392.4580(1.4654, 3.8878)2.4224(1.5523, 3.4745)1.9222


## 6. Simulation Studies

The quality of the proposed model and the corresponding developed results in this paper is assessed through the Monte Carlo simulation study as follows. In our study, we measure the effect of change of each sample sizen affect sample size m and model parameters ω=α1,α2,λ. So, different combinations of the censoring scheme are reported in the simulation table. For the parameters’ values, two sets are used ω = 0.01,0.02,1.5,1.0,1.5,2.0. The simulation study is carried out with respect to 1000 simulated datasets. The prior parameters are selected to satisfy the property that Eωi≃ai/bi. For the point estimate, we compute mean (ME) and the square root of mean squared error (MSE). For the interval estimate, we compute mean interval length (MIL) and probability coverage (PC). We run chan (MCMC iteration) 11000 times discarding the first 1000 values as brun-in (number of iterations needed to reach the stationary distribution). The results of the simulation study are reported in Tables 4–7.Table 4
MEs and MSEs fromω  =  0.01,0.02,1.5.
n,mTollsMLEBayesP0BayesP1αβλαβλαβλ(30, 10)MEs0.01350.02541.84200.01280.02451.81510.01190.02251.6241MSEs0.00750.01250.37500.00730.01220.34010.00540.01090.2841(30, 20)MEs0.01200.02451.81800.01170.02411.80700.01150.02141.6110MSEs0.00600.01130.35010.00590.01090.31820.00400.00950.2480(30, 30)MEs0.01080.02351.70100.01090.02371.70210.01110.02121.6030MSEs0.00450.01010.33220.00420.01000.32500.00320.00680.2171(50, 15)MEs0.01230.02471.82410.01180.02391.81100.01130.02121.6132MSEs0.00620.01150.35410.00580.01110.32100.00410.00920.2491(50, 30)MEs0.01090.02341.70400.01110.02351.70300.01090.02141.6051MSEs0.00410.00990.32910.00390.09800.3260.00280.00590.2131(50, 50)MEs0.00980.02111.68710.01010.02171.65510.00970.02211.5550MSEs0.00280.00700.30210.00250.06800.28900.00150.00480.1900(70, 35)MEs0.01200.02181.69800.01130.02251.66020.01020.02281.5672MSEs0.00370.00820.31300.00340.07510.29410.00240.00510.1971(70, 50)MEs0.01010.02121.6890.01020.02181.65100.00960.02191.552MSEs0.00240.00680.30400.00230.06410.28700.00130.00420.1910(70, 70)MEs0.00990.02081.60310.00990.02101.5550.00980.02191.5400MSEs0.00140.00590.27500.00110.05510.27100.00090.00350.1641Table 5
MILs and CPs fromω=0.01,0.02,1.5.
n,mTollsMLEBayesP0BayesP1αβλαβλαβλ(30, 10)MILs0.02990.04753.41520.02870.04643.41490.02150.04223.4109CPs(0.89)(0.88)(0.89)(0.90)(0.89)(0.90)(0.90)(0.89)(0.90)(30, 20)MILs0.02680.04513.41270.02540.04313.41220.01850.04013.4090CPs(0.90)(0.90)(0.92)(0.91)(0.96)(0.92)(0.93)(0.91)(0.92)(30, 30)MILs0.02370.04193.41000.02210.04023.41000.01570.03703.4065CPs(0.93)(0.91)(0.94)(0.93)(0.95)(0.93)(0.92)(0.95)(0.94)(50, 15)MILs0.02720.04573.41310.02550.04343.41270.01900.04083.4092CPs(0.90)(0.89)(0.90)(0.91)(0.90)(0.91)(0.91)(0.91)(0.90)(50, 30)MILs0.02410.04153.41010.02190.04083.41030.01550.03723.4061CPs(0.92)(0.92)(0.93)(0.91)(0.91)(0.93)(0.90)(0.94)(0.93)(50, 50)MILs0.02250.04033.40850.02070.04003.40880.01420.03613.4048CPs(0.93)(0.90)(0.92)(0.93)(0.93)(0.95)(0.93)(0.92)(0.96)(70, 35)MILs0.02310.04093.40920.02140.04073.40970.01490.03723.4054CPs(0.90)(0.91)(0.92)(0.92)(0.93)(0.90)(0.93)(0.94)(0.92)(70, 50)MILs0.02250.04013.40820.02040.04023.40830.01440.03653.4045CPs(0.92)(0.92)(0.93)(0.94)(0.93)(0.94)(0.97)(0.93)(0.95)(70, 70)MILs0.02090.03823.40410.01790.03763.40450.01120.03313.4015CPs(0.94)(0.90)(0.95)(0.93)(0.93)(0.92)(0.93)(0.92)(0.95)Table 6
MEs and MSEs fromω  =  0.1,0.05,1.0.
n,mTollsMLEBayesP0BayesP1αβλαβλαβλ(30, 10)MEs0.08420.04321.35420.08460.04351.35390.09210.04891.2874MSEs0.03450.01420.21140.03410.01400.21110.02750.00950.1874(30, 20)MEs0.08820.04811.34980.08900.04951.34820.09810.04821.2811MSEs0.03180.01230.20890.03150.01190.20900.02540.00740.1836(30, 30)MEs0.09210.04911.34500.09110.05211.3440.09920.04821.2711MSEs0.02870.01000.20420.02770.00980.20320.02110.00690.1801(50, 15)MEs0.08740.04711.35110.08800.04821.34900.09940.04871.2815MSEs0.03250.01340.20960.03240.01310.20970.02620.00810.1841(50, 30)MEs0.09240.04931.34550.09090.05191.35100.09900.04871.2707MSEs0.02840.01030.20460.02730.00950.20370.02150.00710.1804(50, 50)MEs0.12010.05111.32220.12170.05091.30010.09970.05031.2110MSEs0.02490.00740.20010.02420.00640.19870.01820.00500.1641(70, 35)MEs0.09600.04991.34410.09390.05111.34510.09980.04971.2670MSEs0.02710.00980.20330.02650.00870.20260.02090.00630.1795(70, 50)MEs0.11500.05081.32000.12070.04801.29000.09940.05091.2121MSEs0.02440.00710.20040.02390.00560.19810.01760.00470.1640(70, 70)MEs0.10700.05031.14100.10020.05101.15910.10700.05031.1100MSEs0.02010.00250.17510.01850.00230.16900.01210.00110.1574Table 7
MILs and CPs fromω=0.1,0.05,1.0.
n,mTollsMLEBayesP0BayesP1αβλαβλαβλ(30, 10)MILs0.23420.12341.72410.23290.12231.72190.23290.12231.7219CPs(0.88)(0.90)(0.88)(0.91)(0.89)(0.90)(0.92)(0.90)(0.91)(30, 20)MILs0.23230.12151.72190.23110.12041.72020.23110.12011.7200CPs(0.89)(0.91)(0.90)(0.91)(0.91)(0.92)(0.92)(0.91)(0.93)(30, 30)MILs0.22800.11911.71710.22850.11721.71690.22930.11771.7168CPs(0.92)(0.91)(0.92)(0.93)(0.94)(0.93)(0.91)(0.93)(0.91)(50, 15)MILs0.23270.12181.72230.23170.12111.72080.23190.12071.7206CPs(0.91)(0.90)(0.92)(0.93)(0.91)(0.94)(0.93)(0.92)(0.92)(50, 30)MILs0.22770.11901.71680.22820.11751.71710.22900.11741.7163CPs(0.92)(0.92)(0.91)(0.94)(0.91)(0.93)(0.92)(0.94)(0.92)(50, 50)MILs0.22240.11631.71320.22270.11611.71250.22390.11261.7125CPs(0.93)(0.93)(0.92)(0.92)(0.93)(0.94)(0.97)(0.92)(0.94)(70, 35)MILs0.22680.11811.71490.22750.11701.71630.22810.11671.7157CPs(0.91)(0.92)(0.92)(0.92)(0.93)(0.92)(0.93)(0.92)(0.93)(70, 50)MILs0.22270.11611.71290.22250.11581.71260.22410.11231.7122CPs(0.93)(0.94)(0.95)(0.94)(0.94)(0.94)(0.95)(0.94)(0.95)(70, 70)MILs0.22010.11431.71140.21990.11421.71090.22250.11111.7104CPs(0.92)(0.91)(0.95)(0.94)(0.92)(0.95)(0.92)(0.92)(0.93)Models with the bathtub-shaped or increasing failure rate function have modeled a different lifetime data. In this section, we adopted the simulation study for one reducing form of the modified Weibull distribution.


## 7. Conclusion

Different types of censoring schemes are available, see [27], and the type-II censoring scheme presents the simple type of censoring, saving the number of failures needed for statistical inference. Also, simulation studies are constructed to check the validity of the developed theoretical result, see [28, 29]. The problem of “time-to-failure” under different causes of failure is common in reliability studying. In natural, causes of failure may be dependent but in our modeling, we proposed that the causes of failure are independent. This model considers that the unit failure time is distributed with NRMW distribution with type-II censoring scheme. In this section, the simulation results have reported some points described as follows.(1)
The proposed model under the type-II censoring scheme for competing risks’ model serves well for all choice of censoring schemes and parameters’ choices(2)
The Bayes estimation under noninformative prior is more closed to maximum likelihood estimation(3)
The informative priors serve better than noninformative prior and maximum likelihood estimations(4)
By increasing the value of the affect sample sizem, the values of the MSE and MIL reduce.(5)
The numerical results are more suitable when the proportionm/n is increasing.



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