Jointly Rayleigh lifetime products in the presence of competing risks model

1 Introduction

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. [1], Basu [2], Johnson and Mehrotra [3], Mehrotra and Johnson [4], Bhattacharyya and Mehrotra [5], and Mehrotra and Bhattacharyya [6]. Also, this problem in modern time was studied by Balakrishnan and Rasouli [7], Rasouli and Balakrishnan [8], and Shafay et al. [9]. Recently, it was studied by Bekheet and Abd-Elmougod [10], Soliman et al. [11], Hanaa [12], and Algarni et al. [13]. Also, the balanced case of the joint censoring scheme was proposed by Mondal and Kundu [14].

Suppose that the first line of production is denoted by the line- Ω 1 (L- Ω 1 ) and the line- Ω 2 (L- Ω 2 ) and the joint censoring scheme by JSC. So, two independent samples of products from L- Ω 1 and L- Ω 2 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 m from the L- Ω 1 and n from L- Ω 2 . And, let m units have a life { X 1 , … , X m } (from L- Ω 1 ) distributed with cumulative distribution function (CDF) F 1 and probability density function (PDF) f 1 . Also, n units with life { Y 1 , … , Y n } (from L- Ω 2 ) are distributed with CDF F 2 and PDF f 2 . Suppose that S is the prior integer that represents the number of failure units needed for statistical inference. So, the ordered lifetime sample { T 1 , T 2 , … , T S } is taken from { X 1 , … , X S m } and { Y 1 , … , Y S n } , S = S m + S n . The random lifetimes with the corresponding type (mean from L- Ω 1 or L- Ω 2 ) denoted by ( T , φ ) = { ( T 1 , φ 1 ) , ( T 2 , φ 2 ) , … , ( T S , φ S ) } are called type-II JC sample. The values φ i = { 1 , 2 } denoted the line L- Ω 1 or L- Ω 2 , respectively. The joint likelihood function of type-II JCS sample ( T , φ ) is given by

where S j ( ⋅ ) = 1 − F j ( ⋅ ) , j = 1 , 2 is the reliability function and φ i = 1 or 2, from L- Ω 1 or L- Ω 2 , 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. [15] and Alsulami [16]. 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 [17] and Crowder [18]. Also, for more details about the competing risks model see the studies of Balakrishnan and Han [19], Modhesh and Abd-Elmougod [20], and Bakoban and Abd-Elmougod [21]. And recently, see the studies of Ganguly and Kundu [22], Abu-Zinadah and Neveen [23], Algarn et al. [24], Almalki et al. [25], and Soliman et al. [26]. 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 N is tested under life testing experiment. The pre-fixed number S needed for statistical inference is proposed. Considering only two independent causes of failure the type-II competing risks sample is defined by ( T , ρ ) = { ( T 1 , ρ 1 ) , ( T 2 , ρ 2 ) , … , ( T S , ρ S ) } , ρ 2 = { 1 , 2 } . So, the likelihood function of type-II competing risks sample ( T , ρ ) is given by

where 0 < t 1 < t 2 < … < t S < ∞ and

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 [29]. The Rayleigh random variable T with parameter β 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 [29], and increasing linear failure rate function. Several authors discussed the statistical inferences of the RD; Harter and Moore [30] derived an explicit form for the maximum likelihood (ML) estimator based on Type II censored data; and Dyer and Whisen [27] and Dyer and Whisen [28] 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 [31] and Kong and Fei [32]. Bayesian estimation and prediction problems are also important and have been investigated, among others, by Howlader and Hossain [33] and Fernandez [34]. In addition, AL-Hussaini and Ahmad [35] and AL-Hussaini and Ahmad [36] 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 [10].

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. [37], for Burr XII by Abushal et al. [38], and for Weibull by Alghamdi et al. [39]. 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- P (PBCI) and bootstrap- t (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 N be randomly selected from two lines L- Ω 1 and L- Ω 2 to satisfy m from L- Ω 1 and n from L- Ω 2 . Before the test, the integer S which satisfies the number of failures in statistical inferences procedures is selected. The unit type φ i = { 1 , 2 } (mean from L- Ω 1 or L- Ω 2 ) and the corresponding cause of failure ρ i = { 1 , 2 } (mean first or second cause) are recorded at each observed failure time. Say, the random sample ( T i , φ i , ρ i ) , i = 1 , 2 , … , S is observed. So, the competing risks type-II JC sample is presented by

where 1 < S ≤ N . This scheme has the following assumptions:

1. Number of unit failures from the L- Ω 1 is given by κ 1 = ∑ i = 1 S ( 2 − φ i ) .

2. Number of unit failures from the L- Ω 2 is given by κ 2 = ∑ i = 1 S ( φ i − 1 ) .

3. Number of unit failures from the L- Ω 1 under the cause j is given by ξ 1 j = ∑ i = 1 S ( 2 − φ i ) ∗ ω ( ρ i = j ) , j = 1 , 2 .

4. Number of unit failures from the L- Ω 2 under the cause j is given by ξ 2 j = ∑ i = 1 S ( φ i − 1 ) ∗ ω ( ρ i = j ) , j = 1 , 2 .

Under consideration that the failure time of the unit is distributed with Rayleigh lifetime distribution with PDF given by

where k and j denote the type and cause of the failure, respectively. Also, the observed failure times t i , i = 1 , 2 , … , S are described as t i = min { t i k 1 , t i k 2 } and t i k j is the i th failure time from the line k and cause j . Hence, the minimum value t i has distribution G ( t ) given by G ( ⋅ ) = F t k 1 ( ⋅ ) + F t k 2 ( ⋅ ) − F t k 1 ( ⋅ ) ∗ F t k 2 ( ⋅ ) . So, the latent failure time is distributed with RD with scale parameters β k 1 + β k 2 . Also, the numbers ξ 1 j and ξ 2 j have the binomial distributions described by

and

From the previous model assumptions the joint likelihood function of ( T , φ , ρ ) is given by

where S ( ⋅ ) = 1 − F ( ⋅ ) , h ( ⋅ ) = f ( ⋅ ) S ( ⋅ ) , β ̲ = ( β 11 , β 12 , β 21 , β 22 ) , 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.

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 [41] and Efron and Tibshirani [42]. 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- p and bootstrap- t methods. For more details, see the studies of Efron [43] and Hall [44]. The confidence interval estimation can be formulated with the following algorithms:

1. For given original competing risks type-II JC sample ( t , φ , ρ ) = { ( t 1 , φ 1 , ρ 1 ) , ( t 2 , φ 2 , ρ 2 ) , … , ( t S , φ S , ρ S ) } , the estimators of parameter vector β ˆ ̲ = { β ˆ 11 , β ˆ 12 , β ˆ 21 , β ˆ 22 } are obtained.

2. Considering that the censoring scheme N , n , m , and S and estimate β ˆ ̲ generate a sample of size n from RD with scale parameters β ˆ 11 + β ˆ 12 and a sample of size m from RD with scale parameters β ˆ 21 + β ˆ 22 . The S -bootstrap competing risks type-II JC sample is obtained from the generated joint sample as a small S failure denoted by ( T ∗ , φ ∗ ) = { ( T 1 ∗ , φ 1 ∗ ) , ( T 2 ∗ , φ 2 ∗ ) , … , ( T S ∗ , φ S ∗ ) } , 1 < S ≤ N .

3. For the bootstrap random sample ( T ∗ , φ ∗ ) = { ( T 1 ∗ , φ 1 ∗ ) , ( T 2 ∗ , φ 2 ∗ ) , … , ( T S ∗ , φ S ∗ ) } , 1 < S ≤ N , the two numbers κ 1 ∗ and κ 2 ∗ (number of failures taken from L- Ω 1 and L- Ω 2 , respectively) are reported.

4. The four numbers ξ 1 j ∗ and ξ 2 j ∗ , j = 1 , 2 are randomly generated from binomial distribution with size S − κ 3 − k ∗ and probability β ˆ k j β ˆ k j + β ˆ k j , k , j = 1 , 2 .

5. The last steps are used to obtain the bootstrap estimate sample β ˆ ̲ ̲ ∗ = { β ˆ 11 ∗ , β ˆ 12 ∗ , β ˆ 21 ∗ , β ˆ 22 ∗ } .

6. Repeat steps 2–5 M times.

7. The values ( β ˆ 11 [ i ] ∗ , β ˆ 12 [ i ] ∗ , β ˆ 21 [ i ] ∗ , β ˆ 22 [ i ] ∗ ) , i = 1 , 2 , … , M , are arranged in ascending order to obtain β ˜ ̲ ∗ = { β ˆ 11 ( i ) ∗ , β ˆ 12 ( i ) ∗ , β ˆ 21 ( i ) ∗ , β ˆ 22 ( i ) ∗ } .

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. [45]. Therefore, we consider the competing risks type-II JC sample obtained from life testing experiment under random joint sample selected from two lines L- Ω 1 and L- Ω 2 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 β ̲ = { β 11 , β 12 , β 21 , β 22 } and ( a i , b i ), i = 1 , 2 , … , 4 are prior parameters that present prior information of the model parameters. The joint prior distribution is given by h ∗ ( β ̲ ) = ∏ i = 1 4 h i ∗ ( β i ) . 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

6 Data analysis

In this section, we analyzed a real data set presented by Hoel [46], 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 300 r at an age of 5–6 weeks. Let L- Ω 1 be considered as the first group which lived in a conventional laboratory environment and L- Ω 2 be the second group lived in a germ-free environment. The complete data of two lines are presented in ref. [46]. 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. [47]. 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 β k 1 + β k 2 , k = 1 , 2 ( k mean from L- Ω 1 or L- Ω 2 ). 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- Ω 1 and (K-S) equal to 0.1805 under L- Ω 2 . From Figures 1 and 2, the values of (K-S) and the tested P -value, we accept that the null hypothesis from transformed data set is drawn from Rayleigh lifetime distribution with scale parameter β k 1 + β k 2 .

Figure 1

The empirical and fitted survival.

Figure 2

The empirical and fitted survival.

Therefore, the observed joint type-II competing risks data taken from two lines of production L- Ω 1 and L- Ω 2 with censoring scheme N = 181 , m = 99 , n = 82 , and S = 80 are reported in Table 1. The data show that ( κ 1 , κ 2 ) = ( 50 , 30 ) and ( ξ 11 , ξ 11 , ξ 11 , ξ 11 ) = { 26 , 24 , 25 , 5 } (Table 2). The point estimators under ML, bootstrap, and Bayes estimators with non-informative prior information (mean a i = b i = 0.001 , i = 1 , 2 , 3 , 4 ) are reported in Table 3. The 95% approximate ML, two bootstrap and credibly intervals, respectively, are reported in Table 4.