In this article, we are applying the competing risks model of product from two different lines of production. So, the comparative life test is done under type-II censoring scheme with consideration of only two independent causes of failure. The statistical analysis procedures are developed considering joint sample of production and its life distributed with the Rayleigh lifetime distribution. The point estimation and the corresponding asymptotic confidence interval of the model parameters under maximum likelihood are constructed. Two bootstrap confidence intervals, bootstrap- , and bootstrap- , are discussed. Also, Bayesian approach to estimate point and credible interval is constructed. The estimation results are discussed through data set analyses. The validity of theoretical results is assessed and compared through Monte Carlo study. Finally, some of the points are reported as a brief comment.
The units of product produce from different lines of production under the same facility in a competing duration test under joint censoring scheme. Also, the comparative life testing is done on this product to measure the relative merits of two competing durations of life products. Practice, if we have two lines of production then, select two independent sets of product to test under joint censoring scheme. For the time and cost considerations, the experimenter terminates the experiment after recording some failure times. These data and the corresponding statistical inferences were studied previously by Rao et al. , Basu , Johnson and Mehrotra , Mehrotra and Johnson , Bhattacharyya and Mehrotra , and Mehrotra and Bhattacharyya . Also, this problem in modern time was studied by Balakrishnan and Rasouli , Rasouli and Balakrishnan , and Shafay et al. . Recently, it was studied by Bekheet and Abd-Elmougod , Soliman et al. , Hanaa , and Algarni et al. . Also, the balanced case of the joint censoring scheme was proposed by Mondal and Kundu .
Suppose that the first line of production is denoted by the line- (L- ) and the line- (L- ) and the joint censoring scheme by JSC. So, two independent samples of products from L- and L- are combined to test under life testing experiment. Hence, the relative merits are measured of two competing durations of life products, which is known by comparative life testing. Under type-II censoring scheme, the JSC is called type-II joint censoring scheme (type-II JCS), which is described as follows.
The total random sample of size N of product consists of from the L- and from L- . And, let units have a life (from L- ) distributed with cumulative distribution function (CDF) and probability density function (PDF) . Also, units with life (from L- ) are distributed with CDF and PDF . Suppose that is the prior integer that represents the number of failure units needed for statistical inference. So, the ordered lifetime sample is taken from and , The random lifetimes with the corresponding type (mean from L- or L- ) denoted by are called type-II JC sample. The values denoted the line L- or L- , respectively. The joint likelihood function of type-II JCS sample is given by
where , is the reliability function and or 2, from L- or L- , respectively.
In general, in life testing experiment for survival or medical trial the unit may fail due to different causes of failure, and for more details see the studies of AbdullahiBaba et al.  and Alsulami . The assessment of one cause of failure with respect to other causes is carried out by the competing risks model. Several authors have been interested in this model and its properties, and for the exponential model see the studies of Cox  and Crowder . Also, for more details about the competing risks model see the studies of Balakrishnan and Han , Modhesh and Abd-Elmougod , and Bakoban and Abd-Elmougod . And recently, see the studies of Ganguly and Kundu , Abu-Zinadah and Neveen , Algarn et al. , Almalki et al. , and Soliman et al. . In the competing risks model the observed data consist of unit failure time and the corresponding indicator denoting the cause of failure.
For the type-II competing risks sample, suppose the sample of size is tested under life testing experiment. The pre-fixed number needed for statistical inference is proposed. Considering only two independent causes of failure the type-II competing risks sample is defined by , . So, the likelihood function of type-II competing risks sample is given by
Rayleigh distribution (RD) is defined as a special case of the Weibull distribution, which has wide applications in engineering and communication, for instance, see the studies of Dyer and Whisenand [27,28]. The RD has different applications in life testing of electrovacuum devices, see the study of Polovko . The Rayleigh random variable with parameter has CDF, which is given by
RD has a decreasing reliability function with higher rate than in the case of exponential distribution, for more details see Polovko , and increasing linear failure rate function. Several authors discussed the statistical inferences of the RD; Harter and Moore  derived an explicit form for the maximum likelihood (ML) estimator based on Type II censored data; and Dyer and Whisen  and Dyer and Whisen  provided the best linear unbiased estimator based on complete sample, censored sample, and selected order statistics. Doubly censored samples were considered, among other authors, by Lalitha and Mishra  and Kong and Fei . Bayesian estimation and prediction problems are also important and have been investigated, among others, by Howlader and Hossain  and Fernandez . In addition, AL-Hussaini and Ahmad  and AL-Hussaini and Ahmad  studied Bayesian predictive densities and prediction bounds of generalized order statistics and future records. Also, the estimation under two Rayleigh lifetime population was done by Al-Matrafi and Abd-Elmougod .
The type-II JCS in this article, is implemented on two independent samples taken from two Rayleigh lifetime populations to obtain type-II JC sample. We develop classical and Bayesian inference of type-II JC sample when the lifetime of the units has Rayleigh lifetime distribution. Therefore, studying a population which have different lines of production under the same facility, particularly, in competing duration when Rayleigh failure time is happening to two independent causes of failure is the aim of this article. This problem was handheld recently, for exponential by Almarashi et al. , for Burr XII by Abushal et al. , and for Weibull by Alghamdi et al. . So, we built under these assumptions the model and the corresponding likelihood function. And, the statistical inferences of type-II competing risks samples are presented with classical and Bayesian approach. The point estimators of model parameters are constructed with the corresponding asymptotic confidence/credible intervals (ACIs/CIs). Also, two confidence interval estimation with bootstrap- (PBCI) and bootstrap- (PTCI) are discussed. The validity of results of point estimators is measured with the mean squared error (MSE). But interval estimators are measured with coverage probability (CPs) and interval lengths (ILs)
The outline of this article is presented as follows. The model formulation and the corresponding likelihood function are presented in Section 2. Estimation by the ML method for point and the corresponding ACIs are developed in Section 3. Bootstrap confidence intervals of the model parameters with two different methods are constructed in Section 4. Bayesian approach of model parameters are discussed in Section 5. The joint data set is analyzed for illustrative purposes in Section 6. Finally, the numerical simulation study is used to assess the developed methods in Section 7.
2 Model building
Let a joint sample of size be randomly selected from two lines L- and L- to satisfy from L- and from L- . Before the test, the integer which satisfies the number of failures in statistical inferences procedures is selected. The unit type (mean from L- or L- ) and the corresponding cause of failure (mean first or second cause) are recorded at each observed failure time. Say, the random sample , , is observed. So, the competing risks type-II JC sample is presented by
where . This scheme has the following assumptions:
Number of unit failures from the L- is given by .
Number of unit failures from the L- is given by .
Number of unit failures from the L- under the cause is given by , .
Number of unit failures from the L- under the cause is given by , .
Under consideration that the failure time of the unit is distributed with Rayleigh lifetime distribution with PDF given by
where and denote the type and cause of the failure, respectively. Also, the observed failure times are described as and is the th failure time from the line and cause . Hence, the minimum value has distribution given by . So, the latent failure time is distributed with RD with scale parameters . Also, the numbers and have the binomial distributions described by
From the previous model assumptions the joint likelihood function of is given by
where , , , and
3 Maximum likelihood estimation (MLE)
The point estimation and the corresponding ACIs are computed in this section under given competing risks type-II JC sample. Each sample point has the failure time with the corresponding type and failure cause.
3.1 Point estimation
The joint likelihood function (8) under distribution (6) is reduced to
The natural log-likelihood function can be presented by
Hence, the likelihood equations are obtained after taking the partial derivatives of (10) with respect to model parameters as follows:
So, the estimators under L- and L- , respectively, are given by
Remark: Eqs (12) and (13) have shown that the estimators depend on the non-zero values of numbers and . But, if we set or and or , it means the MLEs of or do not exist. Also, the exact probability distributions of the estimators and are defined as mixture of discrete and continuous distributions, which are difficult to be derived (see, e.g., Kundu and Joarder ).
3.2 Interval estimation
The minus expectation of second derivatives of the log-likelihood function given by (10) presents the Fisher information matrix of the model parameters. We denote the Fisher information matrix by , , which is given by
The second derivative of log-likelihood function of the model parameters is presented by
Then, the Fisher information matrix when is given by
Hence, the variance–covariance matrix of the MLE of model parameters can be obtained from the inverse Fisher information matrix when the diagonal elements have non-zero values. Under the property that ML estimators have a normal distribution with mean and variance obtained from from (18). Therefore, the approximate ( )% CIs of the parameters , and under normal property of MLEs , and are given by
where is the percentile value computed from standard normal distribution.
4 Bootstrap confidence intervals
Bootstrap technique is a commonly resembling method used in parameter estimation as well as estimation of the bias and variance of an estimator or calibrate hypothesis tests. Parametric and nonparametric bootstrap techniques are defined in the studies by Davison and Hinkley  and Efron and Tibshirani . In this section, we adopted the problem of statistical inference in bootstrap techniques in the form of parameter estimation. To formulate confidence intervals under parametric bootstrap technique, we are adopted the percentile bootstrap- and bootstrap- methods. For more details, see the studies of Efron  and Hall . The confidence interval estimation can be formulated with the following algorithms:
For given original competing risks type-II JC sample , the estimators of parameter vector are obtained.
Considering that the censoring scheme , , , and and estimate generate a sample of size from RD with scale parameters and a sample of size from RD with scale parameters . The -bootstrap competing risks type-II JC sample is obtained from the generated joint sample as a small failure denoted by .
For the bootstrap random sample , , the two numbers and (number of failures taken from L- and L- , respectively) are reported.
The four numbers and , are randomly generated from binomial distribution with size and probability , .
The last steps are used to obtain the bootstrap estimate sample .
Repeat steps 2–5 times.
The values , , , are arranged in ascending order to obtain .
4.1 Percentile bootstrap confidence interval (PBCI)
Define the empirical distribution as a CDF of , where mean and others. So, the point bootstrap estimators are defined by
Also, the PBCIs are given by
4.2 Bootstrap-t confidence interval (PTCI)
Also, from the sample , we built the order statistics values , where
There the PTCIs are given by
where the value is given by
and is the CDF of .
5 Bayesian estimation
Considering the available prior information about the model parameters, Bayesian approach can be used not only in parameter estimation but also in the problem of estimating a common functions of the parameters such as readability, hazard rate, and lifetime performance index functions; see ref. . Therefore, we consider the competing risks type-II JC sample obtained from life testing experiment under random joint sample selected from two lines L- and L- of production . The point and corresponding CI estimates are constructed. The common prior densities of the model parameters are considered to be independent gamma distributions. The independent gamma distributions are chosen for each of the model parameters as follows:
where and ( ), are prior parameters that present prior information of the model parameters. The joint prior distribution is given by . In gereneral, the posterior distribution of model parameter is given by
Hence, from (9) and (26) the corresponding posterior distribution is given by
The function given by (27) shows that the joint posterior distribution has independent gamma distributions with shape parameters given by
and scale parameters given by
5.1 Point estimation
The results of the point estimators of model parameters are given by
Under squared error (SE) loss function, the point estimators (PEs) of model parameters are given by(30)
Under LINEX loss function, the PEs of the model parameters are given by(31)
5.2 Interval estimation
In this subsection, we built interval estimation of model parameters under consideration that the posterior distribution is joint-independent gamma distribution. Therefore, the probability credible interval (PCI) of model parameters is given by
where is the marginal posterior distribution (marginal gamma density). So, Bayes credible intervals are constructed based on the following lemma.
Lemma: If has gamma distribution as parameters and , then the pivotal quantity is distributed with distribution with 2 degrees of freedom (see Kundu and Joarder ).
Therefore, the parameter has gamma distribution with shape parameter and scale parameter . So, the pivotal quantity is distributed with distribution with 2 degrees of freedom. And hence, the PCI is reduced to
6 Data analysis
In this section, we analyzed a real data set presented by Hoel , which has presented the failure times and the corresponding cause of failure for two groups of strain male mice under laboratory experiment, which is received a radiation dose of at an age of 5–6 weeks. Let L- be considered as the first group which lived in a conventional laboratory environment and L- be the second group lived in a germ-free environment. The complete data of two lines are presented in ref. . Let the data be classified into two causes of failure: thymic lymphoma with reticulum cell sarcoma is the first cause of death (failure) and the second cause is presented by other causes of death (failure), and more details are presented in ref. . These data are divided by 100 and after that, the validity of modeling these data by Rayleigh lifetime distribution is tested by plotting the fitted survival function with empirical survival function. The data are tested to follow Rayleigh lifetime distribution with scale parameters , ( mean from L- or L- ). Figures 1 and 2 show the fitted survival function with empirical survival function. The Kolmogorov-Smirnov distances of fit data between the empirical distribution functions and the fitted distribution functions satisfy (K-S) equal to 0.1513 under L- and (K-S) equal to 0.1805 under L- . From Figures 1 and 2, the values of (K-S) and the tested -value, we accept that the null hypothesis from transformed data set is drawn from Rayleigh lifetime distribution with scale parameter .
Therefore, the observed joint type-II competing risks data taken from two lines of production L- and L- with censoring scheme , , , and are reported in Table 1. The data show that and (Table 2). The point estimators under ML, bootstrap, and Bayes estimators with non-informative prior information (mean ) are reported in Table 3. The 95% approximate ML, two bootstrap and credibly intervals, respectively, are reported in Table 4.
|Group 1||Cause 1||59||189||191||198||200||207||220||235||245||250||256||261||265|
|Group 2||Cause 1||158||192||193||194||195||202||212||215||229||230||237||240||244|
|(0.0094, 0.0212)||(0.0102, 0.0325)||(0.0085, 0.0215)||(0.0107, 0.0206)|
|(0.0085, 0.0198)||(0.0102, 0.0215)||(0.0075, 0.0191)||(0.0098, 0.0192)|
|(0.0097, 0.0223)||(0.0120, 0.0291)||(0.0082, 0.0235)||(0.0112, 0.0217)|
|(0.0004, 0.0060)||(0.0011, 0.0095)||(0.0011, 0.0067)||(0.0011, 0.0052)|
7 Simulation study
The performances of developed results presented in this article are assessed and compared through Monte Carlo simulations study. Therefore, we test the estimators by changing sample size , effected sample size , and different choices of the initial parameter values. For the point estimate, we reported the mean estimate (ME) and MSE. But, for interval estimate, we reported the mean interval length (MIL) and coverage percentage (CP). So, we used two sets of parameter values and . Different censoring schemes are used in the simulation tables. About the prior information in Bayesian approach, we adopted non-informative priors (mean ) and informative priors (mean take non-zero values). For the first choice of the parameter values the prior information is taken to be . Also, for the second choice of the parameter values the prior information is taken to be . Our studying dependent on simulate samples with iteration number equal to 1,000. Monte Carlo simulation study has algorithm described as follows:
For given prior information, the initial parameter values can be selected randomly by generating value from prior distribution. Also, some authors selected the mean of some generated values. In this article, we selected the initial parameter value to satisfy almost the mean of prior distribution. Therefore, for given parameter vector , the informative prior information satisfy E .
Generate two samples of size and from RD with scale parameters , , respectively. Hence, from the joint sample of size select a small value of size
From Step 2 compute the two integer values and .
The binomial distributions are used to generate discrete random variables , , sample size and under probability of success and , respectively.
Steps 2–4 are repeated times, then competing risks type-II JC samples with the corresponding times of , , and are available.
The MLE, bootstrap, and Bayes point and interval estimates are computed for each sample.
|(20, 20, 15)||1.312||0.234||1.412||0.324||1.291||0.218||1.200||0.189|
|(20, 20, 25)||1.295||0.202||1.397||0.301||1.275||0.190||1.189||0.157|
|(30, 30, 25)||1.288||0.195||1.391||0.303||1.271||0.187||1.184||0.152|
|(30, 30, 40)||1.252||0.151||1.361||0.258||1.225||0.139||1.141||0.114|
|(50, 50, 50)||1.233||0.122||1.343||0.229||1.204||0.109||1.120||0.085|
|(50, 50, 70)||1.195||0.101||1.301||0.210||1.185||0.091||1.089||0.066|
|(50, 50, 100)||1.145||0.070||1.254||0.177||1.137||0.073||1.065||0.042|
|(75, 75, 100)||1.138||0.068||1.249||0.179||1.132||0.070||1.061||0.041|
|(75, 75, 130)||1.122||0.053||1.231||0.162||1.118||0.051||1.035||0.031|
|(20, 20, 15)||3.124||0.88||3.171||0.89||3.111||0.90||3.103||0.89||3.071||0.91|
|(20, 20, 25)||3.085||0.89||3.124||0.90||3.063||0.91||3.063||0.93||3.024||0.92|
|(30, 30, 25)||3.082||0.89||3.133||0.89||3.054||0.93||3.051||0.92||3.017||0.91|
|(30, 30, 40)||3.037||0.93||3.100||0.91||3.003||0.93||3.012||0.97||2.980||0.92|
|(50, 50, 50)||3.019||0.90||3.087||0.92||2.987||0.94||2.985||0.92||2.957||0.94|
|(50, 50, 70)||2.970||0.95||3.048||0.94||2.951||0.92||2.954||0.93||2.921||0.95|
|(50, 50, 100)||2.917||0.90||3.001||0.93||2.902||0.95||2.901||0.95||2.871||0.95|
|(75, 75, 100)||2.921||0.91||3.007||0.92||2.899||0.94||2.904||0.90||2.869||0.92|
|(75, 75, 130)||2.891||0.92||2.971||0.93||2.865||0.93||2.882||0.92||2.838||0.95|
|(20, 20, 15)||2.451||0.425||2.492||0.479||2.442||0.419||2.403||0.364|
|(20, 20, 25)||2.424||0.394||2.461||0.451||2.415||0.389||2.365||0.334|
|(30, 30, 25)||2.429||0.398||2.467||0.454||2.410||0.384||2.364||0.330|
|(30, 30, 40)||2.401||0.371||2.441||0.427||2.382||0.349||2.351||0.309|
|(50, 50, 50)||2.385||0.355||2.427||0.415||2.368||0.332||2.334||0.291|
|(50, 50, 70)||2.361||0.329||2.403||0.392||2.344||0.308||2.311||0.274|
|(50, 50, 100)||2.339||0.329||2.282||0.369||2.323||0.291||2.289||0.257|
|(75, 75, 100)||2.331||0.322||2.284||0.363||2.320||0.293||2.285||0.253|
|(75, 75, 130)||2.311||0.308||2.255||0.351||2.300||0.274||2.261||0.241|
|(20, 20, 15)||4.725||0.88||4.754||0.89||4.711||0.89||4.714||0.90||4.650||0.90|
|(20, 20, 25)||4.689||0.90||4.721||0.91||4.684||0.90||4.687||0.92||4.618||0.92|
|(30, 30, 25)||4.684||0.89||4.724||0.92||4.688||0.91||4.681||0.91||4.614||0.94|
|(30, 30, 40)||4.649||0.91||4.691||0.91||4.651||0.92||4.649||0.91||4.581||0.92|
|(50, 50, 50)||4.638||0.92||4.678||0.94||4.638||0.93||4.632||0.93||4.565||0.93|
|(50, 50, 70)||4.611||0.91||4.651||0.92||4.609||0.91||4.603||0.93||4.539||0.95|
|(50, 50, 100)||4.582||0.95||4.622||0.91||4.582||0.92||4.572||0.93||4.511||0.92|
|(75, 75, 100)||4.548||0.93||4.591||0.92||4.553||0.93||4.540||0.93||4.479||0.96|
|(75, 75, 130)||4.521||0.93||4.558||0.93||4.515||0.91||4.517||0.93||4.447||0.93|
8 Conclusions and remarks
Estimating problem under joint type-II competing risks sample is considered in this article. The unknown parameters under consideration of product life distributed with Rayleigh lifetime population are estimated under competing risks model with two different failure modes. Estimation problem is discussed with, three different estimation methods, ML, bootstrap, and Bayesian methods . Two different loss functions are considered in Bayesian context. The ACIs, two bootstrap, and Bayes credible intervals are also discussed. Bayesian estimation under SE loss function or LINEX loss functions for the parameters of two competing RDs was discussed for given competing risks type-II JC sample. Bayes results were compared with the ML and bootstrap estimates through a Monte Carlo simulation study. The computational results show that the Bayesian estimation under two proposed loss functions is more precise than the MLE. The results in both ML and Bayes with non-informative prior are closed for the information are obtained only from likelihood function. It is evident from the numerical results that while the MLEs are close to the ones based on the Bayes method under non-informative prior , the Bayes estimators under informative prior and bootstrap-t perform well when compared to the MLEs and bootstrap-p. Statistic defined by (22) show that bootstrap-t confidence intervals combined with the mean do not necessarily deal with skewness as well as other methods, which makes it better than other methods. It is observed that when the effective sample increases, the MSEs of the estimates and the ALs of the confidence intervals decreased. Moreover, the MSEs of the point estimates as well as the ILs and CPs of CIs, approximate boot-p and boot-t as well as CIs improve when is large. This observation is expected due to the fact that when increases, more information on the data is obtained.
Funding information: This work was funded by the University of Jeddah, Jeddah, Saudi Arabia, under grant No. (UJ-20-138-DR). The authors, therefore, acknowledge with thanks the University of Jeddah technical and financial support.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
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