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

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. +e 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 MCMCmethods are constructed. +e real and simulated datasets are analyzed for illustrative purposes. Finally, the estimators are compared with the Monte Carlo simulation study.


Introduction
In the past few years, different modifications of Weibull distribution were constructed with a bathtub-shaped hazard rate function. ese modifications were discussed with different authors, see [1][2][3][4]. Some modification models of Weibull distribution are discussed with with two or three parameters' vectors [4][5][6]. Also, some of the modifications of Weibull distribution were discussed with four parameters (see [7]) and the modifications with five parameters (see [8]). en, the large sets of distribution are generalized with generalized linear exponential distribution as the bathtubshaped hazard rate presented by Sarhan et al. [9]. e unimodal, decreasing, increasing, or bathtub-shaped hazard rate beta-Weibull distribution was proposed by Lee et al. [10]. e 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 where α 1 and α 2 are called shape parameters, β 1 and β 2 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 β 1 � β 2 � (1/2), with CDF given by where α 1 and α 2 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 α 1 � α 2 � α as follows: e density, survival, and hazard failure rate functions are given, respectively, by where α and λ are nonnegative parameters. e 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.
If the random variable T is distributed with NRMW, then the following two properties are satisfied.
(1) e rth moment of the random variable of NRMW is given by where Γ(·) is the gamma function. (2) e moment generating function is given by e lifetime data of product units are presented in complete or censored data dependent on some consideration of the test time or cost. e 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. e 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][15][16][17][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.
e 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. en, 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. e theoretical results are discussed through simulation Monte Carlo study and real data analysis. e paper is organized as follows. Section 2 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.

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.

Remarks 1
(1) e latent failure time is distributed with NRMW distribution with α 1 + α 2 and λ shape and accelerate parameters, respectively (2) e value m 1 is distributed with binomial distribution with sample size m 1 and probability of success (α 1 /α 1 + α 2 ) but m 1 is binomial distributed with sample size m 2 and probability of success (α 2 /α 1 + α 2 )

Maximum Likelihood Estimations
e 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.

Point MLE.
e natural logarithms of likelihood function (10) is reduced to After taking the partial derivative of the log-likelihood function with respect to parameters' vector α 1 and α 2 , we have the likelihood equations which are defined by the following theorem. e conditional ML estimators of parameters α 1 and α 2 , given λ > 0 and two numbers m 1 and m 2 > 0, can be written by where Remark 2. If m j � 0, there are no failures due to cause j. en, the observed data t � {(t 1 , η 1 ), (t 2 , η 2 ), . . ., (t m , η m )} does not provide information about parameter α j .
Also, the partial derivative of the log-likelihood function with respect to parameter λ m i�1 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.
e ML estimators of parameter λ presented by the nonlinear equation can be given by where 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 en, from theorems (1) and (2), the ML estimate of the parameters' vector ω � (α 1 ,

Interval Estimation.
e second derivative of the log likelihood function with respect to ω � (α 1 , α 2 , λ) is given by 4 Complexity where Hence, from equations (19) to (21), the Fisher information matrix (IM) is defined as minus expectation of equations (19)- (21) to be given by e approximate value of the Fisher information matrix is defined by Under limiting properties of the MLEs is using it to present the asymptotic distribution of ω � (α 1 , α 2 , λ). en, the asymptotically normally distributed mean ω � (α 1 , α 2 , λ) and variance covariance matrix IM − 1 0(ω) describe the asymptotic distribution of ω � (α 1 , α 2 , λ) erefore, the interval estimate with confidence level 100(1 − 2c)% is given by where the vector (ϵ 11 , ϵ 22 , ϵ 33 ) presents the diagonal of inverse of IM 0(ω) − 1 and the value Z c presents the percentile of the standard normal distribution with right-tail probability c.

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. en, we aim to formulate these two data in the form of probability distribution, known by posterior distribution, as follows.
Suppose that the parameters' vector ω � (α 1 , α 2 , λ) is distributed with independent gamma priors. erefore, the joint prior distribution can be presented by , a i and b i > 0 and i � 1, 2, and 3.
Generally, the posterior distribution can be formulated by Also, generally the ratio of two integrals (26), especially, in the high-dimensional case, cannot obtain analytical. en, 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 Ω(ω) under squared error loss (SEL) function is given by 4.1. MCMC Approach. e posterior distribution presented by (26) with prior distribution (24) and likelihood function (10) is reduced to where C is the normalized constant. From the posterior distribution the full-conditional PDFs of the vector ω, ω � (α 1 , α 2 , λ) is given by e 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]) 6 Complexity Also, for the recent review of MH algorithms under Gibbs, see [25] and [26].
Step 6: the point Bayes estimators for any function Ω(ω) are defined by and integer M is the number needed to achieve the stationary distribution (burn-in). e corresponding posterior variance of Ω(ω) is given by (34) Step 7: for interval estimators with confidence level 100(1 − 2c)% of Ω(ω) is given by

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

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. e 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. e 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][25][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) For given t 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 loglikelihood 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. e noninformative prior is used for prior information as a i � b i � 0.0001, i � 1, 2, 3, and 4. e 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." e empirical posterior distribution under the MCMC approach is described with Figures 3 and 4. e MCMC convergence to the posterior distribution is described by Figures 3 and 4.

Example 2: Simulated Dataset.
e algorithms used to generate and analyze the type-II competing risks sample from NRMW distribution is described as follows: (1) Suppose a sample size n � 50 and effected sample size m � 30.  Table 3 for point and 95% interval estimate.

Simulation Studies
e 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 size n affect sample size m Complexity Table 1: e failure data of 61 male mice under radiation dose or 300 r at age 5-6 weeks.  ymic  159  189  191  198  200  207  220  235  245  250  256  lymphoma  261  265  266  280  343  356  383  403  414  428  432   Other causes   40  42  51  62  163  179  206  222  228  252  249  282  324  333  341  366  385  407  420  431  441  461  462  482  517  517  524  564  567  586  619  620  621  622  647  651  686  761    }. e simulation study is carried out with respect to 1000 simulated datasets. e prior parameters are selected to satisfy the property that E(ω i )≃(a i /b i ). 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). e results of the simulation study are reported in Tables 4-7.
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.

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]. e 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. is 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) e proposed model under the type-II censoring scheme for competing risks' model serves well for all choice of censoring schemes and parameters' choices (2) e Bayes estimation under noninformative prior is more closed to maximum likelihood estimation (3) e informative priors serve better than noninformative prior and maximum likelihood estimations (4) By increasing the value of the affect sample size m, the values of the MSE and MIL reduce. (5) e numerical results are more suitable when the proportion (m/n) is increasing.

Data Availability
No data were used to support the findings of this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest regarding the publication of the paper.