# Estimation Based on Adaptive Progressively Censored under Competing Risks Model with Engineering Applications

Authors: Mazen Nassar; Refah Alotaibi; Sanku Dey
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper has investigated the estimation problem for the competing risks model where the data are adaptive progressively type-II censored and follow the Rayleigh distribution. Maximum likelihood and Bayesian methods are used to estimate the unknown parameters. To generate the interval estimates of the model parameters, approximate confidence intervals and two bootstrap confidence intervals are also considered based on the classical setup. Bayes estimates are obtained using independent gamma priors based on various loss functions including squared error, LINEX, general entropy, weighted SE, and precautionary loss functions. Moreover, Bayes credible intervals and the highest posterior density intervals of the model parameters are obtained. A numerical investigation has been carried out to assess the performance of the classical and Bayes estimates as well as the associated confidence and credible intervals. Finally, one simulated and two real data sets, one for breaking strengths of wire connections and the other for times to failure of small electrical appliances, have been analyzed for illustrative purposes. The results showed that the Bayes estimates using the LINEX loss function provide more reasonable estimates than the classical and Bayes estimates using squared error, general entropy, and precautionary loss functions.


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## Body

## 1. Introduction

The investigator may not be able to receive complete information on failure times for all experimental units via life-testing and reliability investigations, and thus, censoring is inevitable. In clinical practice, for example, patients may be removed from the analysis before the experiment is completed, or the analysis may have to be discontinued at a specific time threshold. Units in industrial investigations may fail by accident before a predetermined time point. In many cases, however, the discarding of units before failure is planned ahead of time to preserve the duration and expense of testing. In the past decades, in order to remove units during a lifetime experiment, various progressive censoring (PC) approaches have been presented in the statistical literature (see Balakrishnan [1] and Balakrishnan and Aggarwala [2]). In the course of time, several modifications on PC have been undertaken by many authors (see Kundu and Joarder [3]; Childs et al. [4]). To improve the efficiency of statistical investigation as well as for reducing overall test time, adaptive progressive type-II censoring (APCS-TII) was introduced by Ng et al. [5]. In APCS, the effective sample size m is predetermined in this scheme, while the testing duration is permitted to pass the threshold time T>0. The following is a description of an APCS-TII: suppose n identical units are placed in a life-testing experiment, and the effective sample size m<n is designated as the starting value of the experiment with a predetermined progressive censoring scheme R=R1,…,Rm. If the mth failure happens before the threshold T (i.e., Xm:m:n<T), the test ends at the time Xm:m:n. On the other hand, once the testing time exceeds the time T but the number of observed failures has not reached m, by modifying the number of items progressively withdrawn from the test upon failure, the researcher can terminate it as soon as possible. Thus, we should leave as numerous surviving units on the experiment as possible. Suppose J is the number of failures before T, that is, XJ:m:n<T<XJ+1:m:n, J=0,1,…m, where X0:m:n≡0 and Xm+1:m:n≡∞. When the experiment passed the predetermined time T, we put RJ+1=⋯=Rm−1=0 and Rm=n−m−∑i=1JRi. This formulation directs us to end the test as soon as possible if the J+1th failure time is larger than T for J+1<m. The threshold of T plays a significant role in determining the values of Ri and also as a compromise between a shorter testing time and a greater opportunity to get extreme failures. When T tends to infinity, the APCS-TII reduces to a conventional progressive type-II censoring scheme. Moreover, if T=0, this variant reduces to the usual type II censoring scheme. Several investigations based on the APCS-TII have been conducted; readers can refer to the works of Hemmati and Khorram [6], Amein [7], Nassar and Abo-Kasem [8], Ateya and Mohammed [9], and Nassar et al. [10], among many others.Researchers have recently shown a significant interest in studies on competing risk models, which include data from survival analysis, lifetime studies, reliability studies, and other areas. The independence of causes of failure appears to be restricted in a competing risks model; however, when the reason for failure is dependent, identifiability issues of the underlying model may occur. This must be highlighted that without knowledge of covariates, it is impossible to evaluate the assumption of the failure time distributions of competing causes using just observed data (see Kalbfleisch and Prentice [11], Crowder [12]). Several authors have studied competing risks model under different scenarios. In this regard, readers may refer to the works of Kundu et al. [13]. Competing risks model under APCS-TII data is studied by Ashour and Nassar [14] and Hemmati and Khorram [6]. Recently, Dutta and Kayal [15] studied competing risk model based on improved APCS-TII sample under independent exponential distributions.We consider that the lifetimes follow the Rayleigh distribution (RD) in this investigation. This distribution is chosen because it is a logical extension of the exponential distribution and hence plays a key role in acoustics, life-testing of electro-vacuum systems, reliability analysis, communication engineering, and so on. Rayleigh [16] wa is the first to utilize this distribution in the literature, using it in the context of acoustics. It can be employed to model numerous paths of densely scattered signals while reaching a receiver. It is also used in reliability engineering for the lifetime modelling of an object, electromagnetic wave propagation, and physiological sensing systems, see Hossain et al. [17] for more details. The failure rate of this distribution is a linear function of time, which is a major characteristic. The Rayleigh distribution’s dependability function degrades at a considerably higher rate than the exponential distribution’s reliability function. Many researchers have written on situations in which data reveal increasing hazard rates. Besides, it has some relationship with other distributions such as Weibull, generalized extreme value, and Chi-squared distributions, and thus, its utility in different fields is significant. The RD with one parameter has been considerably investigated by several researchers (see Dey and Das [18] and Liao and Gui [19]) concerning the estimation, prediction, and several other inferences.In this paper, our objective is to analyse competing risk data under APCS-TII where the lifetimes of the various risk factors are distributed independently and follow RD with a known cause of failure. Here, it is further assumed that Cox [20] latent failure time model is used for studying competing risks data. The competing causes of failure are considered to be distributed independently in the latent failure time modelling. According to the aforementioned censoring scheme, the maximum likelihood estimates (MLEs), approximate confidence intervals (ACIs), and two bootstrap confidence intervals of the unknown parameters are investigated in the context of the classical phenomenon. The Bayes estimates are obtained under different loss functions, namely, squared error (SE), LINEX, general entropy (GE), weighted SE (WSE), and precautionary (Pr) loss functions based on the assumption of independent gamma priors of the unknown parameters. Furthermore, for comparison purposes, Bayes credible intervals (BCIs) and highest posterior density (HPD) intervals are obtained via the MCMC sampling approach. Simulation studies are conducted to evaluate the performance of different point and interval estimates. Two real data sets are investigated for illustration purposes. To the best of our knowledge, no studies have been done on the aforementioned estimation methods for the Rayleigh distribution using an APCS-TII with competing risks data.The remaining part of the paper proceeds as follows: In Section2, we present the competing risk model and the corresponding likelihood function based on APCS-TII data. In Section 3, the MLEs and asymptotic variance-covariance matrix are obtained. The Bayes estimators based on various loss functions of the unknown parameters are studied in Section 4. We also provide the ACIs, two parametric bootstrap methods for obtaining CIs, the BCIs, and HPD intervals in Section 5. In Section 6, a simulation study is implemented to compare the efficiency of the different estimators and the different confidence intervals. In Section 7, we investigate simulated and real data sets to show the importance of the proposed methods and how these methods work in practice, and finally, we conclude the paper in Section 8.


## 2. Description and Notation of the Model

Without loss of generality, suppose that we haven identical items in a lifetime experiment. Moreover, assume that there are only two causes of failure, and then, Xi=minX1i,X2i,i=1,…,n and k=1,2 refer to the latent failure time of the i−th unit under the k−th cause of failure. We also assume that X1i,X2i are independent identically distributed random variables. In addition, it is assumed that Xki,i=1,…,n and k=1,2 are independent and follow the RD with the following probability density function:(1)fkx=2λkxe−λkx2,λk>0,x>0.The corresponding cumulative distribution function is given by(2)Fkx=1−e−λkx2.We can write the observations under adaptive progressively type-II censored competing risks data as follows:(3)X1:m:n,c1,R1,…,XJ:m:n,cJ,RJ,XJ+1:m:n,cJ+1,0,…Xm−1:m:n,cm−1,0,Xm:m:n,cm,Rm∗,where Rm∗=n−m−∑i=1JRi and ci∈1,2, which indicates the cause of failure of item i. Let(4)Ici=1=1,ci=1,0,else,Ici=2=1,ci=2,0,else,then m1=∑i=1mIci=1 and m1=∑i=1mIci=2 refer to the number of failures owed to cause 1 and 2, respectively, and m=m1+m2. Based on the independence of X1i and X2i, we can obtain the relative risk due to cause 1 as follows:(5)PX1i≤X2i=∫0∞2λ1xe−λ1x2e−λ2x2dx=λ1λ1+λ2.Similarly, the relative risk due to cause 2 can be obtained as follows:(6)PX1i≤X2i=∫0∞2λ2xe−λ1x2e−λ2x2dx=λ1λ1+λ2.Given a progressive censoring schemeR1,…,RJ,0,…,0,Rm∗, then the likelihood function of the observed data can be expressed as follows:(7)L=C∏i=1mf1xiF¯2xiIci=1f2xxiF¯1xiIci=2∏i=1JF¯1xiF¯2xiRiF¯1xmF¯2xmRm∗,where F¯k=1−Fkx,k=1,2, xi=xi:m:n for the sake of simplicity, and(8)C=nn−R1−1n−R1−R2−2…n−J−R1−…−RJ×n−J−R1−⋯−RJ+1…n−m−R1−⋯−RJ+1.


## 3. Maximum Likelihood Estimation

Based on (1)–(3), the likelihood function, ignoring the normalized constant, can be written as follows:(9)L=2mλ1m1λ2m2exp−λ1+λ2∑i=1mxi2+∑i=1JRixi2+Rm∗xm2∏i=1mxi.The natural logarithm of the likelihood functionℓ=logLλ1,λ2 in (4) takes the form(10)ℓ∝m1logλ1+m2logλ2−λ1+λ2∑i=1mxi2+∑i=1JRixi2+Rm∗xm2.To obtain the MLEs of theλ1 and λ2, we find the first-order partial derivatives of the log-likelihood function in (5) with respect to λ1 and λ2, and then, by equating each of them to zero, we get(11)∂ℓ∂λ1=m1λ1−Ax=0,(12)∂ℓ∂λ2=m2λ2−Ax=0,where(13)Ax=∑i=1mxi2+∑i=1JRixi2+Rm∗xm2.From (6) and (7), we can acquire the MLEs λ^1 and λ^2 of λ1 and λ2, respectively, as follows:(14)λ^1=m1Ax,λ^2=m2Ax.To obtain the ACIs for the unknown parametersλ1 and λ2, we require to obtain the second derivatives of the log-likelihood function as follows:(15)∂ℓ2∂λ12=−m1λ12,(16)∂ℓ2∂λ22=−m2λ22,(17)∂ℓ2∂λ1∂λ2=∂ℓ2∂λ2∂λ1=0.Thus, the asymptotic variance-covariance matrix is obtained by inverting the observed Fisher information matrix as follows:(18)−∂ℓ2∂λ12−∂ℓ2∂λ1∂λ2−∂ℓ2∂λ1∂λ2−∂ℓ2∂λ22λ1=λ^1,λ2=λ^2−1=varλ^100varλ^2.


## 4. Bayesian Estimation

In this section, we use the Bayesian estimation approach to estimate the parameters of the RD under adaptive progressively type-II censored competing risks data. In a statistical study, the Bayesian technique has some benefits when compared with the usual maximum likelihood method. The capability of possessing prior details in analysis makes the Bayesian approach incredibly useful in reliability studies and many other areas where one of the important challenges is the limited availability of data. The Bayes estimates are obtained based on the assumption that the random variablesλ1 and λ2 have gamma prior distributions with known shape and scale parameters ak and bk, k=1,2, with pdf given by(19)gλk=bkakΓakλkak−1e−λkbk,λk>0,ak,bk>0.Here, we consider the independent gamma priors in Bayesian analysis for the RD due to its flexibility, see, for examples, about independent gamma priors Kundu [21] and Ashour and Nassar [22]. In addition, some possible motivations for employing the independent priors for parameters λ1 and λ2 are furnished as follows: First, independent priors are fairly straightforward and concise, which may not produce many complicated inferential and computational problems. Second, in numerous practical problems, although the dependent prior appears more appealing, the dependent property between parameters cannot be explained from a statistical viewpoint due to historical knowledge and expert background where such prior may be very rare. Therefore, independent priors are more widespread in statistics under the Bayesian approach for the sake of simplicity. The joint posterior density of λ1 and λ2 can be expressed using the likelihood function in (4) and the joint prior distribution in (12) as follows:(20)gλ1,λ2|x=1Dλ1m1+a1−1λ2m2+a2−1e−λ1b1+Ax+λ2b2+Ax,where(21)D=Γa1+m1Γa2+m2b1+Axa1+m1b2+Axa2+m2.The marginal posterior densities ofλ1 and λ2 are given, respectively, by(22)gλ1|x=b1+Axa1+m1Γa1+m1λ1a1+m1−1e−λ1b1+Ax,(23)gλ2|x=b2+Axa2+m2Γa2+m2λ2a2+m2−1e−λ1b2+Ax.From (15) and (16), it can easily be noted that the marginal posterior distributions of λ1 and λ2 are gamma distributions, that is, Gammaa1+m1,b1+Ax and Gammaa2+m2,b2+Ax, respectively. Now, to obtain the Bayes estimators of λ1 and λ2, we suggest using three distinct loss functions, the first one is the SE loss function, the second function is the LINEX loss function, and the third is the GE loss function.


### 4.1. Bayesian Estimators Using SE Loss Function

First, we consider the SE loss function. Based on the SE loss function, the Bayesian estimators ofλ1 and λ2, denoted by λ˜SE1 and λ˜SE2, are the mean of the posterior densities (15) and (16), which are given by(24)λSE1=a1+m1b1+Ax,(25)λSE2=a2+m2b2+Ax.It is observed that in the case of non-informative priors,i.e.,a1=b1=a2=b2=0, the Bayesian estimators of λ1 and λ2 corresponds to the MLEs given by (9).


### 4.2. Bayesian Estimators Using LINEX Loss Function

The LINEX loss function was introduced by Varian [23] as an asymmetric loss function and got its popularity due to Zellner [24]. Let θ be an unknown parameter to be estimated, and then, the Bayesian estimator of the unknown parameter θ using the LINEX loss function, denoted by θ˜LIN, is(26)θ˜LINEX=−1τlogEθe−τθ,where τ≠0, and for τ closed to zero, the LINEX loss function may be considered as a natural extension of the SE loss function. In our case and from (15), (16), and (19), we can obtain the Bayesian estimators of λ1 and λ2 using the LINEX loss function, respectively, as follows:(27)λLINEX1=−a1+m1τlogb1+Axτ+b1+Ax,(28)λLINEX2=−a2+m1τlogb2+Axτ+b2+Ax.


### 4.3. Bayesian Estimators Using GE Loss Function

The GE loss function was introduced by Calabria and Pulcini [25]. Accordingly, the Bayesian estimator of the unknown parameter θ based on the GE loss function can be defined as(29)θ˜GE=Eθθ−q−1/q,where θ˜GE is the Bayesian estimator of θ under GE loss function. It is to be noted that for q=−1, the GE loss function reduces to SE loss function. When q=1, it reduces to the WSE loss function. For q=−2, it reduces to the Pr loss function (Norstrom [26]) which is an asymmetric loss function. Based on (15), (16), and (22), the Bayes estimators of λ1 and λ2 using the GE loss function are, respectively, given by(30)λ˜GE1=1b1+AxΓa1+m1−qΓa1+m1−1/q,(31)λ˜GE2=1b2+AxΓa2+m1−qΓa2+m1−1/q.Here, we assume three particular cases, corresponding toq=−1, q=1, and q=−2. When q=−1, (23) and (24) correspond to the Bayesian estimators under the SE loss function given by (17) and (18). When q=1, we can get the Bayesian estimators of λ1 and λ2 under the WSE loss function as(32)λWSE1=a1+m1−1b1+Ax,(33)λ˜WSE2=a2+m2−1b2+Ax,and when q=−2, the Bayesian estimators using the Pr loss function of λ1 and λ2, denoted by λ˜Pr1 and λ˜Pr2, are as follows:(34)λ˜Pr1=a1+m12+a1+m1b1+Ax,(35)λ˜Pr2=a2+m22+a1+m1b2+Ax.


## 4.1. Bayesian Estimators Using SE Loss Function

First, we consider the SE loss function. Based on the SE loss function, the Bayesian estimators ofλ1 and λ2, denoted by λ˜SE1 and λ˜SE2, are the mean of the posterior densities (15) and (16), which are given by(24)λSE1=a1+m1b1+Ax,(25)λSE2=a2+m2b2+Ax.It is observed that in the case of non-informative priors,i.e.,a1=b1=a2=b2=0, the Bayesian estimators of λ1 and λ2 corresponds to the MLEs given by (9).


## 4.2. Bayesian Estimators Using LINEX Loss Function

The LINEX loss function was introduced by Varian [23] as an asymmetric loss function and got its popularity due to Zellner [24]. Let θ be an unknown parameter to be estimated, and then, the Bayesian estimator of the unknown parameter θ using the LINEX loss function, denoted by θ˜LIN, is(26)θ˜LINEX=−1τlogEθe−τθ,where τ≠0, and for τ closed to zero, the LINEX loss function may be considered as a natural extension of the SE loss function. In our case and from (15), (16), and (19), we can obtain the Bayesian estimators of λ1 and λ2 using the LINEX loss function, respectively, as follows:(27)λLINEX1=−a1+m1τlogb1+Axτ+b1+Ax,(28)λLINEX2=−a2+m1τlogb2+Axτ+b2+Ax.


## 4.3. Bayesian Estimators Using GE Loss Function

The GE loss function was introduced by Calabria and Pulcini [25]. Accordingly, the Bayesian estimator of the unknown parameter θ based on the GE loss function can be defined as(29)θ˜GE=Eθθ−q−1/q,where θ˜GE is the Bayesian estimator of θ under GE loss function. It is to be noted that for q=−1, the GE loss function reduces to SE loss function. When q=1, it reduces to the WSE loss function. For q=−2, it reduces to the Pr loss function (Norstrom [26]) which is an asymmetric loss function. Based on (15), (16), and (22), the Bayes estimators of λ1 and λ2 using the GE loss function are, respectively, given by(30)λ˜GE1=1b1+AxΓa1+m1−qΓa1+m1−1/q,(31)λ˜GE2=1b2+AxΓa2+m1−qΓa2+m1−1/q.Here, we assume three particular cases, corresponding toq=−1, q=1, and q=−2. When q=−1, (23) and (24) correspond to the Bayesian estimators under the SE loss function given by (17) and (18). When q=1, we can get the Bayesian estimators of λ1 and λ2 under the WSE loss function as(32)λWSE1=a1+m1−1b1+Ax,(33)λ˜WSE2=a2+m2−1b2+Ax,and when q=−2, the Bayesian estimators using the Pr loss function of λ1 and λ2, denoted by λ˜Pr1 and λ˜Pr2, are as follows:(34)λ˜Pr1=a1+m12+a1+m1b1+Ax,(35)λ˜Pr2=a2+m22+a1+m1b2+Ax.


## 5. Confidence Intervals

In this section, five methods are proposed to construct the confidence intervals for the parametersλ1 and λ2. The proposed methods are the ACIs, two kinds of bootstrap confidence intervals, the BCIs, and HPD intervals.


### 5.1. ACIs

Utilizing the asymptotic normality of the MLEs, the1001−α% ACIs of λ1 and λ2, are, respectively, given by(36)λ^1±zα/2λ^1m1,λ^2±zα/2λ^2m2,where zα/2 is the upper α/2 percentile of a standard normal distribution, and λ^k/mk,k=1,2 is the standard error of λ^k, which can be computed by carrying the square root of the main diagonal in (11).


### 5.2. Bootstrap Confidence Intervals

For a small sample size, the ACIs may not perform well. Hence, we offer to employ two parametric bootstrap confidence intervals forλk,k=1,2, namely, the percentile bootstrap (p-boot) and the studentized-t bootstrap (t-boot) confidence intervals. The p-boot and t-boot confidence intervals are obtained through the following steps:


#### 5.2.1. p-boot

(1)
From the original data, the MLEs ofλk,k=1,2are obtained.(2)
λ^k is used to get a bootstrap adaptive progressively type-II censored competing risks sample, and the bootstrap estimates of λk, say, λ^kb are computed.(3)
Step 2 is repeatedB times to calculate λ^kb1,…,λ^kbB.(4)
Allλ^kb’s in ascending order as λ^kb1,…,λ^kbB are arranged.(5)
The two-sided1001−α% p-boot confidence interval for λk is given by(37)λ^kbBα/2,λ^kbB1−α/2,k=1,2.


#### 5.2.2. t-boot

(1)
As in (1) and (2) in p-boot,(2)
thet-statistic is computed, where Tk=mkλ^kb−λ^k/λ^kb,k=1,2.(3)
Step 2-3 is repeatedB times to get Tk1,…,TkB.(4)
Tk1,…,TkB is arranged in ascending order as Tk1,…,TkB.(5)
The two-sided1001−α% t-boot confidence intervals for λk are(38)λ^k+TkBα/2λ^kmk,λ^k+TkB1−α/2λ^kmk,k=1,2.


### 5.3. Bayes Credible Intervals

The credible intervals with equal tails ofλ1 and λ2 can be obtained using the posterior distributions of λ1 and λ2. From (15) and (16), if mk,k=1,2, are integers, we can see that Z1=2λ1b1+Ax and Z1=2λ2b2+Ax follow Chi-squared distribution with 2a1+m1 and 2a2+m2 degrees of freedom, respectively. Therefore, 1001−α% BCIs for λ1 and λ2 are(39)χ1−α/2,2a1+m122b1+Ax,χα/2,2a1+m122b1+Ax,(40)χ1−α/2,2a2+m222b2+Ax,χα/2,2a2+m222b2+Ax.Ifmk,k=1,2 are not integer values, then gamma distribution can be employed to create the desired credible intervals.


### 5.4. Highest Posterior Density Intervals

The1001−α% HPD intervals HLλ1,HUλ1 and HLλ2,HUλ2 for λ1 and λ2 are given by solving the next two nonlinear equations, respectively,(41)∫HLλ1HUλ1gλ1|xdλ1=1−α,gHLλ1|x=gHLλ1|x,(42)∫HLλ1HUλ1gλ2|xdλ2=1−α,gHLλ2|x=gHLλ2|x.On simplifications, we can write (32) and (33) as(43)Γa1+m1,HUλ1b1+Ax−Γa1+m1,HUλ1b1+AxΓa1+m1=1−α,HUλ1HLλ1a1+m1−1=exp−b1+AxHLλ1−HUλ1,(44)Γa2+m2,HUλ1b1+Ax−Γa2+m2,HUλ1b2+AxΓa1+m1=1−α,HUλ2HLλ2a2+m2−1=exp−b2+AxHLλ2−HUλ2,where Γc,d=∫0dxc−1e−xdx is the incomplete gamma function. Expressions in (34) and (35) can be solved numerically to get the required HPD intervals of λ1 and λ2.


## 5.1. ACIs

Utilizing the asymptotic normality of the MLEs, the1001−α% ACIs of λ1 and λ2, are, respectively, given by(36)λ^1±zα/2λ^1m1,λ^2±zα/2λ^2m2,where zα/2 is the upper α/2 percentile of a standard normal distribution, and λ^k/mk,k=1,2 is the standard error of λ^k, which can be computed by carrying the square root of the main diagonal in (11).


## 5.2. Bootstrap Confidence Intervals

For a small sample size, the ACIs may not perform well. Hence, we offer to employ two parametric bootstrap confidence intervals forλk,k=1,2, namely, the percentile bootstrap (p-boot) and the studentized-t bootstrap (t-boot) confidence intervals. The p-boot and t-boot confidence intervals are obtained through the following steps:


### 5.2.1. p-boot

(1)
From the original data, the MLEs ofλk,k=1,2are obtained.(2)
λ^k is used to get a bootstrap adaptive progressively type-II censored competing risks sample, and the bootstrap estimates of λk, say, λ^kb are computed.(3)
Step 2 is repeatedB times to calculate λ^kb1,…,λ^kbB.(4)
Allλ^kb’s in ascending order as λ^kb1,…,λ^kbB are arranged.(5)
The two-sided1001−α% p-boot confidence interval for λk is given by(37)λ^kbBα/2,λ^kbB1−α/2,k=1,2.


### 5.2.2. t-boot

(1)
As in (1) and (2) in p-boot,(2)
thet-statistic is computed, where Tk=mkλ^kb−λ^k/λ^kb,k=1,2.(3)
Step 2-3 is repeatedB times to get Tk1,…,TkB.(4)
Tk1,…,TkB is arranged in ascending order as Tk1,…,TkB.(5)
The two-sided1001−α% t-boot confidence intervals for λk are(38)λ^k+TkBα/2λ^kmk,λ^k+TkB1−α/2λ^kmk,k=1,2.


## 5.2.1. p-boot

(1)
From the original data, the MLEs ofλk,k=1,2are obtained.(2)
λ^k is used to get a bootstrap adaptive progressively type-II censored competing risks sample, and the bootstrap estimates of λk, say, λ^kb are computed.(3)
Step 2 is repeatedB times to calculate λ^kb1,…,λ^kbB.(4)
Allλ^kb’s in ascending order as λ^kb1,…,λ^kbB are arranged.(5)
The two-sided1001−α% p-boot confidence interval for λk is given by(37)λ^kbBα/2,λ^kbB1−α/2,k=1,2.


## 5.2.2. t-boot

(1)
As in (1) and (2) in p-boot,(2)
thet-statistic is computed, where Tk=mkλ^kb−λ^k/λ^kb,k=1,2.(3)
Step 2-3 is repeatedB times to get Tk1,…,TkB.(4)
Tk1,…,TkB is arranged in ascending order as Tk1,…,TkB.(5)
The two-sided1001−α% t-boot confidence intervals for λk are(38)λ^k+TkBα/2λ^kmk,λ^k+TkB1−α/2λ^kmk,k=1,2.


## 5.3. Bayes Credible Intervals

The credible intervals with equal tails ofλ1 and λ2 can be obtained using the posterior distributions of λ1 and λ2. From (15) and (16), if mk,k=1,2, are integers, we can see that Z1=2λ1b1+Ax and Z1=2λ2b2+Ax follow Chi-squared distribution with 2a1+m1 and 2a2+m2 degrees of freedom, respectively. Therefore, 1001−α% BCIs for λ1 and λ2 are(39)χ1−α/2,2a1+m122b1+Ax,χα/2,2a1+m122b1+Ax,(40)χ1−α/2,2a2+m222b2+Ax,χα/2,2a2+m222b2+Ax.Ifmk,k=1,2 are not integer values, then gamma distribution can be employed to create the desired credible intervals.


## 5.4. Highest Posterior Density Intervals

The1001−α% HPD intervals HLλ1,HUλ1 and HLλ2,HUλ2 for λ1 and λ2 are given by solving the next two nonlinear equations, respectively,(41)∫HLλ1HUλ1gλ1|xdλ1=1−α,gHLλ1|x=gHLλ1|x,(42)∫HLλ1HUλ1gλ2|xdλ2=1−α,gHLλ2|x=gHLλ2|x.On simplifications, we can write (32) and (33) as(43)Γa1+m1,HUλ1b1+Ax−Γa1+m1,HUλ1b1+AxΓa1+m1=1−α,HUλ1HLλ1a1+m1−1=exp−b1+AxHLλ1−HUλ1,(44)Γa2+m2,HUλ1b1+Ax−Γa2+m2,HUλ1b2+AxΓa1+m1=1−α,HUλ2HLλ2a2+m2−1=exp−b2+AxHLλ2−HUλ2,where Γc,d=∫0dxc−1e−xdx is the incomplete gamma function. Expressions in (34) and (35) can be solved numerically to get the required HPD intervals of λ1 and λ2.


## 6. Simulation Study

In this section, we report the simulation outcomes to evaluate the performance of the different estimators and the different confidence intervals discussed in the previous sections. The simulation study is conducted by choosing the parameter (Par) valuesλ1=1 and λ2=2, and various values of n,m,T and the following censoring schemes (Sch)(i)
Sch 1:R1=⋯=Rm−1=0andRm=n−m.(ii)
Sch 2:R1=⋯=Rm−1=1andRm=n−2m+1.(iii)
Sch 3:R1=⋯=Rm=n−m/m.For different cases, we obtain the MLEs and Bayesian estimates under SE, LINEX (withτ=0.5), GE (with q=0.5), WSE, and Pr loss functions based on 1000 simulations. The Bayesian estimates are obtained by assuming gamma prior distributions for λ1 and λ2 with parameters 1,1 and 2,1, respectively. These values are specified in such a way that prior averages are exact to the actual averages. Table 1 reports the mean values of estimates and mean square errors (MSEs). The confidence lengths and the associated coverage probabilities for the different confidence intervals are displayed in Table 2. The simulation study is carried out by using the MATHCAD program Ver.15. From Tables 1 and 2, we can conclude the following:(1)
The Bayes estimates using the LINEX loss function serve more satisfactory results than other estimates in terms of the lowest MSEs followed by Bayes estimates based on the WSE loss function.(2)
It is also to be noted that the MSEs decrease asm increases in all the cases, which indicate that all estimators are consistent.(3)
Whenn and m remain constant and T increases, no specific pattern in the MSEs is observed because the number of failures is predetermined and no more information is collected when T increases.(4)
It is observed that the HPD intervals have the shortest confidence lengths followed by the BCI among all the interval estimation methods.(5)
The BT intervals have the largest confidence lengths when compared with other methods.(6)
Asm grows, the confidence lengths fall in all the interval estimation methods.(7)
All the interval estimation methods are capable to hold the nominal coverage probabilities, except the t-boot intervals which have coverage probabilities always below the nominal level.(8)
Sch 1 outperforms the other two schemes in terms of minimum MSEs and minimum confidence lengths.Table 1
The average values of estimates and the corresponding MSEs (in parentheses) forλ1=1 and λ2=2 under various censoring schemes.
n,mSchParMLEsBayes estimatesSELINEXGEWSEPrT=0.2(50,5)1λ11.352(0.749)1.221(0.322)1.138(0.247)1.013(0.245)0.939(0.242)1.354(0.421)λ22.695(1.683)2.441(0.683)2.277(0.467)2.231(0.490)2.159(0.446)2.578(0.860)2λ11.445(0.917)1.279(0.368)1.188(0.275)1.057(0.266)0.979(0.257)1.421(0.489)λ22.991(2.432)2.628(0.947)2.441(0.631)2.403(0.659)2.327(0.586)2.774(1.191)3λ11.669(1.382)1.410(0.502)1.303(0.363)1.172(0.329)1.088(0.299)1.562(0.676)λ23.273(3.582)2.777(1.243)2.566(0.816)2.537(0.859)2.455(0.757)2.934(1.559)(50,10)1λ11.070(0.478)1.020(0.240)0.960(0.203)0.846(0.230)0.785(0.243)1.130(0.282)λ22.120(1.061)2.029(0.499)1.910(0.411)1.854(0.457)1.795(0.458)2.143(0.563)2λ11.081(0.481)1.032(0.251)0.970(0.214)0.853(0.243)0.790(0.278)1.146(0.293)λ22.232(1.087)2.117(0.513)1.990(0.425)1.936(0.484)1.875(0.487)2.235(0.595)3λ11.351(0.819)1.212(0.338)1.131(0.258)1.008(0.256)0.936(0.290)1.343(0.438)λ22.640(1.879)2.384(0.749)2.223(0.526)2.177(0.561)2.107(0.519)2.518(0.921)(60,15)1λ11.061(0.276)1.038(0.173)0.993(0.152)0.907(0.166)0.862(0.173)1.122(0.198)λ22.177(0.652)2.118(0.397)2.026(0.330)1.987(0.352)1.943(0.345)2.204(0.447)2λ11.145(0.334)1.101(0.201)1.051(0.169)0.965(0.173)0.918(0.173)1.189(0.240)λ22.292(0.778)2.205(0.457)2.104(0.364)2.067(0.382)2.021(0.366)2.295(0.528)3λ11.353(0.581)1.256(0.317)1.191(0.254)1.101(0.237)1.047(0.222)1.355(0.397)λ22.683(1.484)2.495(0.783)2.366(0.583)2.339(0.605)2.286(0.556)2.597(0.927)(100,25)1λ10.993(0.137)0.988(0.107)0.962(0.100)0.909(0.108)0.882(0.112)1.040(0.113)λ21.986(0.307)1.977(0.237)1.925(0.221)1.898(0.234)1.871(0.235)2.030(0.247)2λ11.054(0.142)1.043(0.109)1.014(0.109)0.960(0.112)0.931(0.123)1.098(0.121)λ22.115(0.362)2.090(0.271)2.032(0.239)2.006(0.248)1.978(0.244)2.145(0.294)3λ11.312(0.368)1.260(0.259)1.218(0.221)1.161(0.205)1.127(0.191)1.324(0.305)λ22.543(0.834)2.453(0.580)2.372(0.472)2.354(0.478)2.320(0.448)2.518(0.659)T=0.4(50,5)1λ11.379(0.823)1.240(0.345)1.155(0.243)1.029(0.240)0.955(0.244)1.374(0.419)λ22.721(1.738)2.459(0.606)2.292(0.481)2.247(0.506)2.174(0.439)2.597(0.888)2λ11.467(0.957)1.292(0.377)1.199(0.280)1.069(0.268)0.991(0.256)1.433(0.503)λ22.996(2.681)2.620(0.991)2.432(0.663)2.395(0.699)2.319(0.624)2.766(1.239)3λ11.627(1.272)1.382(0.473)1.277(0.343)1.145(0.313)1.062(0.286)1.532(0.639)λ23.264(3.577)2.770(1.237)2.561(0.812)2.532(0.855)2.451(0.753)2.926(1.551)(50,10)1λ11.109(0.531)1.049(0.249)0.986(0.207)0.871(0.232)0.808(0.243)1.163(0.298)λ22.195(1.115)2.085(0.528)1.960(0.424)1.905(0.467)1.844(0.463)2.202(0.606)2λ11.096(0.566)1.042(0.258)0.979(0.215)0.862(0.245)0.799(0.259)1.156(0.302)λ22.245(1.262)2.117(0.569)1.988(0.449)1.935(0.493)1.874(0.483)2.235(0.658)3λ11.315(0.722)1.189(0.312)1.109(0.240)0.986(0.248)0.914(0.268)1.318(0.405)λ22.627(1.805)2.378(0.724)2.219(0.508)2.173(0.541)2.104(0.499)2.511(0.893)(60,15)1λ11.063(0.263)1.039(0.168)0.994(0.147)0.910(0.160)0.866(0.166)1.122(0.192)λ22.138(0.648)2.086(0.403)1.995(0.340)1.956(0.364)1.912(0.359)2.171(0.447)2λ11.130(0.311)1.091(0.189)1.041(0.160)0.954(0.166)0.907(0.168)1.179(0.225)λ22.320(0.865)2.226(0.497)2.124(0.394)2.089(0.415)2.042(0.397)2.317(0.573)3λ11.370(0.625)1.268(0.337)1.203(0.270)1.113(0.254)1.059(0.237)1.368(0.419)λ22.684(1.630)2.491(0.824)2.362(0.616)2.335(0.645)2.282(0.596)2.593(0.970)(100,25)1λ11.010(0.147)1.003(0.114)0.976(0.106)0.924(0.112)0.897(0.115)1.055(0.122)λ22.001(0.305)1.990(0.239)1.937(0.184)1.910(0.233)1.883(0.234)2.043(0.250)2λ11.055(0.157)1.042(0.118)1.013(0.107)0.958(0.115)0.929(0.118)1.098(0.131)λ22.165(0.372)2.134(0.274)2.074(0.236)2.050(0.243)2.021(0.236)2.190(0.303)3λ11.275(0.315)1.230(0.223)1.189(0.190)1.132(0.176)1.099(0.165)1.293(0.264)λ22.522(0.858)2.434(0.593)2.355(0.485)2.336(0.494)2.304(0.465)2.499(0.670)Table 2
The 95% confidence lengths and the coverage probabilities (in parentheses) forλ1=1 and λ2=2 under various censoring schemes.
n,mSchParACIBayesBootstrapBCIHPDBPBTT=0.2(50,5)1λ12.822(95.000)2.218(97.500)2.093(97.100)2.822(95.400)9.633(86.900)λ24.066(97.200)3.205(98.300)2.047(96.300)4.066(96.000)12.17(93.700)2λ13.052(96.900)2.344(96.900)2.203(97.600)3.052(97.400)9.952(90.800)λ24.486(97.700)3.436(96.000)2.133(96.500)4.486(97.600)12.44(95.000)3λ13.471(97.100)2.552(96.300)2.384(97.300)3.471(94.000)11.02(93.200)λ24.954(96.900)3.656(95.500)2.307(95.300)4.954(95.700)14.71(97.700)(50,10)1λ12.229(94.600)1.850(94.200)1.746(94.500)2.229(88.300)10.24(87.300)λ23.202(93.600)2.667(92.500)1.711(93.100)3.202(90.500)14.09(84.900)2λ12.280(92.200)1.890(95.800)1.782(93.500)2.280(84.300)10.51(81.700)λ23.348(92.900)2.768(94.600)1.743(92.200)3.348(73.700)14.39(88.700)3λ12.806(93.200)2.193(96.100)2.071(95.400)2.806(93.000)10.67(88.400)λ23.997(94.200)3.139(96.200)2.006(94.200)3.997(93.000)14.49(91.600)(60,15)1λ11.839(90.300)1.625(96.200)1.559(94.000)1.839(89.000)6.412(86.900)λ22.675(92.200)2.364(95.500)1.551(93.700)2.675(90.800)8.792(88.200)2λ11.974(92.600)1.717(95.200)1.648(94.300)1.974(91.900)6.732(83.100)λ22.829(93.200)2.469(94.900)1.637(94.100)2.829(86.300)8.866(80.900)3λ12.325(94.800)1.953(94.600)1.871(96.000)2.325(96.100)7.081(90.100)λ23.317(93.600)2.797(92.600)1.845(95.700)3.317(96.000)9.523(93.200)(100,25)1λ11.334(90.000)1.246(93.600)1.182(90.200)1.334(92.900)3.711(90.000)λ21.902(90.300)1.782(92.500)1.214(91.200)1.902(93.700)5.097(92.800)2λ11.419(94.100)1.316(95.800)1.243(93.900)1.419(85.700)4.207(78.700)λ22.025(92.000)1.883(93.500)1.280(94.500)2.025(84.800)5.772(82.700)3λ11.744(93.100)1.574(90.100)1.456(91.200)1.744(96.700)4.670(92.500)λ22.448(91.300)2.220(87.600)1.534(93.000)2.448(96.200)6.493(93.800)T=0.4(50,5)1λ12.866(95.400)2.206(96.800)2.111(97.600)2.866(96.700)9.721(86.500)λ24.010(97.000)3.232(97.600)2.039(96.300)4.015(96.200)11.43(92.700)2λ13.086(96.300)2.358(97.500)2.215(98.200)3.086(96.800)9.868(89.600)λ24.502(96.100)3.432(95.900)2.166(97.100)4.502(97.500)13.12(94.900)3λ13.412(97.000)2.516(96.700)2.359(98.200)3.412(95.200)9.906(91.500)λ24.924(96.800)3.638(94.500)2.268(95.800)4.924(96.500)13.16(97.300)(50,10)1λ12.307(93.400)1.902(94.800)1.790(91.900)2.307(91.600)10.31(89.100)λ23.314(93.700)2.740(93.100)1.763(91.200)3.314(92.800)12.52(86.500)2λ12.317(92.500)1.903(95.700)1.793(94.100)2.307(91.200)9.609(89.500)λ23.372(93.700)2.773(93.300)1.767(93.100)3.372(90.800)13.77(87.600)3λ12.753(92.200)2.163(96.400)2.042(95.100)2.753(92.800)9.592(85.900)λ23.965(95.300)3.123(96.700)1.992(94.100)3.965(92.600)12.19(92.000)(60,15)1λ11.830(90.000)1.619(94.900)1.552(93.100)1.830(92.700)6.022(92.200)λ22.636(92.000)2.334(93.800)1.548(92.700)2.636(94.800)8.291(91.900)2λ11.961(92.300)1.710(95.800)1.639(93.500)1.961(90.600)6.581(90.600)λ22.851(93.800)2.485(94.600)1.622(93.200)2.851(91.400)8.919(92.800)3λ12.341(94.300)1.963(93.400)1.882(94.600)2.341(96.200)7.084(91.300)λ23.323(93.300)2.798(92.400)1.843(94.500)3.323(96.500)10.34(93.900)(100,25)1λ11.353(92.000)1.261(94.300)1.197(91.900)1.353(94.900)3.828(92.500)λ21.920(90.800)1.796(93.300)1.228(92.700)1.920(93.500)5.369(90.700)2λ11.431(94.000)1.325(95.800)1.262(94.400)1.431(94.600)3.946(92.900)λ22.066(94.500)1.917(95.200)1.289(95.100)2.066(95.500)5.530(91.900)3λ11.707(94.400)1.545(92.800)1.448(92.900)1.707(96.900)4.710(91.200)λ22.420(90.100)2.198(92.500)1.502(94.300)2.420(94.100)6.662(92.500)


## 7. Numerical Illustration

In this section, we analyse simulated and real data sets to show the importance of the proposed methods and how these methods work in practice.


### 7.1. Simulated Data Set

In this subsection, we analyse one simulated data set. We generate adaptive progressively type-II censored competing risks data forλ1=1 and λ2=2 and by choosing n=60, m=20, T=0.2 and the censoring scheme R1=⋯=Rm−1=1andRm=n−2m+1, with the understanding that when the testing time passed the threshold T=0.2, we replace RJ+1,…,Rm−1 by zero, and in this case, we have Rm=n−m−∑i=1JRi. The simulated observation is displayed in Table 3.Table 3
Simulated adaptive type-II progressively censored competing risks data.
(0.016, 2)(0.032, 1)(0.086, 2)(0.114, 2)(0.185, 1)(0.188, 2)(0.218, 2)(0.224, 2)(0.243, 2)(0.244, 1)(0.261, 1)(0.267, 1)(0.273, 1)(0.277, 2)(0.279, 2)(0.291, 2)(0.292, 2)(0.313, 2)(0.315, 2)(0.361, 2)From Table3, it is to be noted that the first component refers to the lifetime, while the second one denotes the cause of failure. From the simulated data, we have 14 failures due to cause 1 and 6 failures due to cause 2; that is, m1=6 and m2=14. For these data, we obtain the MLEs and Bayes estimates under the different loss functions discussed before. The Bayes estimates are obtained by considering informative priors, and the hyperparameters are taken to be 1,1 for λ1 and 2,1 for λ2. The different estimates and the corresponding MSEs are displayed in Table 4. The different 95% confidence intervals are tabulated in Table 5. Based on the outcomes in Table 4, we observe that the Bayesian estimate using the LINEX loss function of λ1 has the lowest MSE among all other estimates, while the Bayes estimates based on the WSE loss function perform more satisfactory than other estimates in terms of minimum MSE for λ2. Furthermore, from Table 5, we conclude that the HPD intervals have the shortest confidence lengths for both parameters.Table 4
Estimated values and the corresponding MSEs (in parentheses) based on different methods for the simulated data.
ParMLEsBayes estimatesSELINEXGEWSEPrλ11.054(0.003)1.046(0.002)0.934(0.004)1.008(0.000 07)0.896(0.011)1.118(0.014)λ22.459(0.210)2.390(0.152)2.278(0.078)2.305(0.093 08)2.241(0.058)2.464(0.215)Table 5
The 95% confidence intervals based on different methods for the simulated data.
ParACIBayesBootstrapBCIHPDBPBTλ1(0.211, 1.897)(0.42, 1.951)(0.351, 1.832)(0.493 4, 2.313)(0.226, 3.215)λ2(1.171, 3.747)(1.366, 3.70)(1.285, 3.582)(1.648, 3.780 9)(1.489, 3.371)


### 7.2. Real Data Analysis

In this Section 7.2, we reanalyse two real data sets to show the applicability of the point and interval estimates discussed in this paper.(1) Example One. The first data set is given by King [27] and consists of the breaking strengths (mg) of 20 wire connections. There are two causes of failures, the first cause is a bond lift, which is a failure of the bond and the second cause is a wire break, which is a failure of the wire, for more details about the data, see Nelson [28]. Table 6 illustrates the complete data set and the associated cause of failure.Table 6
Breaking strengths of wire connections.
Strength (mg)Cause of failureStrength (mg)Cause of failure550B1250B750W1350W950B1450B950W1450B1150W1450W1150B1550B1150B1550W1150W1550W1150W1850W1250B2050BThe letters W and B appear in Table6 refer to bond lift (cause 1) and wire break (cause 2), respectively. There were 10 failures due to each cause of failure. Before progressing further, we first compare the fit of the RD distribution with some other one-parameter models including Weibull (We), Lindley (L), and inverse Lindley (IL) distributions. The MLEs of the different models, the Kolmogorov-Smirnov (KS) distance, and the corresponding P-value are displayed in Table 7 for cause 1 and cause 2, respectively. The results in Table 7 indicate that the RD can be assumed as an adequate model to fit the data set.Table 7
The MLEs, KS, andP-value of the competitive models for breaking strengths data.
DataModelEstimateKSP-valueCause 1RD5.687×10−70.328 60.230 4We0.112 960.879 00.000 0IL1.212×1030.480 40.019 8L1.549×10−30.333 20.216 9Cause 2RD5.623×10−70.324 60.242 7We0.113 440.870 70.000 0IL1.147×1030.428 40.050 9L1.561×10−30.337 10.205 9Now, we use the original data given in Table6 to generate an adaptive type-II progressively censored sample by choosing T=1500, m=10, and the censoring scheme R1=⋯=R10=1. The generated observations are displayed in Table 8.Table 8
Generated adaptive type-II progressively censored competing risks data from breaking strengths data.
Cause of failureStrength (mg)B55095011501550W75011501150135015501850In this case, we havem1=4,m2=6, and J=7, and the estimated relative risk due to bond lift cause and wire break cause is 0.4 and 0.6, respectively. This result indicates that cause 2 is more severe than cause 1. The MLEs and Bayesian estimates of the unknown parameters are computed and shown in Table 9. The Bayesian estimates of λ1 and λ2 are obtained based on noninformative priors because no prior knowledge is known about the unknown parameters. It is observed that the MLEs and Bayesian estimates under the SE loss are coinciding under the noninformative priors, so we do not display estimates under the SE loss function in Table 9. Also, we can compute the MLEs of the meantime to failure for cause 1 as π/4λ^1=2568.912 and the meantime to failure for cause 2 as π/4λ^2=2097.508. The various intervals estimates are also obtained and reported in Table 10. Comparing the different confidence intervals, we observed that the HPD intervals have the minimum confidence lengths.Table 9
Estimated values of adaptive type-II progressively censored competing risks data (breaking strengths).
ParMLEsBayes estimatesLINEXτ=0.001GEq=0.5WSEPrλ11.190122×10−71.190124×10−79.69799×10−88.92591×10−81.3306×10−7λ21.785183×10−71.785185×10−71.56380×10−71.48765×10−71.92822×10−7Table 10
The 95% confidence intervals of adaptive type-II progressively censored competing risks data (breaking strengths).
Methodλ1λ2ACI2.38×10−9,2.36×10−73.57×10−8,3.22×10−7BCI3.24×10−8,2.61×10−76.55×10−8,3.47×10−7HPD5.50×10−8,1.29×10−71.63×10−7,1.83×10−7BP2.93×10−8,2.51×10−77.88×10−8,3.16×10−7BT0,2.59×10−71.07×10−8,2.68×10−7(2) Example Two. The second data set is taken from Lawless [29] and also analyzed by Sarhan [30]. The data consist of times to failure of 33 small electrical appliances subjected to an automatic life test. Failures were categorised into 18 various modes, and only modes six and nine occur more than twice. We concentrate on failure mode 9. Therefore, the data consist of two causes of failure: cause 1 (mode nine) and cause 2 (all other failure modes). The complete data set is displayed in Table 11.Table 11
Times to failure of small electrical appliances.
Times to failureCause of failureTimes to failureCause of failure1122471135225511492256811702269413292270223812276127082283129582303411062230592116713112115922321411925134781199013504122231432912327269761240017846124512From Table10, we have 17 failures due to cause 1 and 16 failures due to cause 2. The MLEs of the different models as well as the KS distance and the corresponding P-value are presented in Table 12 for cause 1 and cause 2, respectively. The outcomes in Table 12 imply that the RD is a proper model for these data.Table 12
The MLEs, KS, andP-value of the competitive models for times to failure data.
DataModelEstimateKSP-valueCause 1RD9.569×10−80.236 10.334 6We0.101 580.871 10.000 0IL2.577×1030.395 50.013 4L6.715×10−40.308 30.095 5Cause 2RD3.259×10−70.328 80.062 9We0.129 460.744 40.000 0IL102.4320.491 50.000 9L1.492×10−30.262 90.218 9From the actual data shown in Table11, an adaptive type-II progressively censored sample is generated by selecting T=2000, m=15 and the censoring scheme R1=⋯=R14=1 and R15=4. The generated sample is depicted in Table 13.Table 13
Generated adaptive type-II progressively censored competing risks data from times to failure data.
Cause of failureTimes to failure1116719251990222324002471211491703817081062159223272451From Table13, we have m1=6,m9=6, and J=10, and the estimated relative risks due to causes 1 and 2 are 0.4 and 0.6, respectively. The MLEs and Bayes estimates of the unknown parameters are calculated and displayed in Table 14. The Bayes estimates of λ1 and λ2 are obtained using noninformative priors. Under the noninformative priors, the MLEs and Bayes estimates under the SE loss are coinciding, so we do not show the estimates using the SE loss function in Table 14. Moreover, the MLEs of the meantime to failure for cause 1 and cause 2 are 3686.32 and 3009.87, respectively. The various intervals estimates are also obtained and displayed in Table 15. Also, we can see that the HPD intervals have the lowest confidence lengths.Table 14
Estimated values of adaptive type-II progressively censored competing risks data (times to failure).
ParMLEsBayes estimatesLINEXτ=0.001GEq=0.5WSEPrλ15.77967×10−85.78004×10−85.06292×10−84.81639×10−86.24276×10−8λ28.66950×10−88.67006×10−87.95069×10−87.70623×10−89.13846×10−8Table 15
The 95% confidence intervals of adaptive type-II progressively censored competing risks data (times to failure).
Methodλ1λ2ACI1.15×10−8,1.04×10−73.01×10−8,1.43×10−7BCI2.12×10−8,1.12×10−73.96×10−8,1.52×10−7HPD3.21×10−8,1.01×10−74.12×10−8,1.24×10−7BP1.21×10−8,1.35×10−73.20×10−8,1.59×10−7BT1.01×10−8,1.82×10−72.24×10−8,1.12×10−7


## 7.1. Simulated Data Set

In this subsection, we analyse one simulated data set. We generate adaptive progressively type-II censored competing risks data forλ1=1 and λ2=2 and by choosing n=60, m=20, T=0.2 and the censoring scheme R1=⋯=Rm−1=1andRm=n−2m+1, with the understanding that when the testing time passed the threshold T=0.2, we replace RJ+1,…,Rm−1 by zero, and in this case, we have Rm=n−m−∑i=1JRi. The simulated observation is displayed in Table 3.Table 3
Simulated adaptive type-II progressively censored competing risks data.
(0.016, 2)(0.032, 1)(0.086, 2)(0.114, 2)(0.185, 1)(0.188, 2)(0.218, 2)(0.224, 2)(0.243, 2)(0.244, 1)(0.261, 1)(0.267, 1)(0.273, 1)(0.277, 2)(0.279, 2)(0.291, 2)(0.292, 2)(0.313, 2)(0.315, 2)(0.361, 2)From Table3, it is to be noted that the first component refers to the lifetime, while the second one denotes the cause of failure. From the simulated data, we have 14 failures due to cause 1 and 6 failures due to cause 2; that is, m1=6 and m2=14. For these data, we obtain the MLEs and Bayes estimates under the different loss functions discussed before. The Bayes estimates are obtained by considering informative priors, and the hyperparameters are taken to be 1,1 for λ1 and 2,1 for λ2. The different estimates and the corresponding MSEs are displayed in Table 4. The different 95% confidence intervals are tabulated in Table 5. Based on the outcomes in Table 4, we observe that the Bayesian estimate using the LINEX loss function of λ1 has the lowest MSE among all other estimates, while the Bayes estimates based on the WSE loss function perform more satisfactory than other estimates in terms of minimum MSE for λ2. Furthermore, from Table 5, we conclude that the HPD intervals have the shortest confidence lengths for both parameters.Table 4
Estimated values and the corresponding MSEs (in parentheses) based on different methods for the simulated data.
ParMLEsBayes estimatesSELINEXGEWSEPrλ11.054(0.003)1.046(0.002)0.934(0.004)1.008(0.000 07)0.896(0.011)1.118(0.014)λ22.459(0.210)2.390(0.152)2.278(0.078)2.305(0.093 08)2.241(0.058)2.464(0.215)Table 5
The 95% confidence intervals based on different methods for the simulated data.
ParACIBayesBootstrapBCIHPDBPBTλ1(0.211, 1.897)(0.42, 1.951)(0.351, 1.832)(0.493 4, 2.313)(0.226, 3.215)λ2(1.171, 3.747)(1.366, 3.70)(1.285, 3.582)(1.648, 3.780 9)(1.489, 3.371)


## 7.2. Real Data Analysis

In this Section 7.2, we reanalyse two real data sets to show the applicability of the point and interval estimates discussed in this paper.(1) Example One. The first data set is given by King [27] and consists of the breaking strengths (mg) of 20 wire connections. There are two causes of failures, the first cause is a bond lift, which is a failure of the bond and the second cause is a wire break, which is a failure of the wire, for more details about the data, see Nelson [28]. Table 6 illustrates the complete data set and the associated cause of failure.Table 6
Breaking strengths of wire connections.
Strength (mg)Cause of failureStrength (mg)Cause of failure550B1250B750W1350W950B1450B950W1450B1150W1450W1150B1550B1150B1550W1150W1550W1150W1850W1250B2050BThe letters W and B appear in Table6 refer to bond lift (cause 1) and wire break (cause 2), respectively. There were 10 failures due to each cause of failure. Before progressing further, we first compare the fit of the RD distribution with some other one-parameter models including Weibull (We), Lindley (L), and inverse Lindley (IL) distributions. The MLEs of the different models, the Kolmogorov-Smirnov (KS) distance, and the corresponding P-value are displayed in Table 7 for cause 1 and cause 2, respectively. The results in Table 7 indicate that the RD can be assumed as an adequate model to fit the data set.Table 7
The MLEs, KS, andP-value of the competitive models for breaking strengths data.
DataModelEstimateKSP-valueCause 1RD5.687×10−70.328 60.230 4We0.112 960.879 00.000 0IL1.212×1030.480 40.019 8L1.549×10−30.333 20.216 9Cause 2RD5.623×10−70.324 60.242 7We0.113 440.870 70.000 0IL1.147×1030.428 40.050 9L1.561×10−30.337 10.205 9Now, we use the original data given in Table6 to generate an adaptive type-II progressively censored sample by choosing T=1500, m=10, and the censoring scheme R1=⋯=R10=1. The generated observations are displayed in Table 8.Table 8
Generated adaptive type-II progressively censored competing risks data from breaking strengths data.
Cause of failureStrength (mg)B55095011501550W75011501150135015501850In this case, we havem1=4,m2=6, and J=7, and the estimated relative risk due to bond lift cause and wire break cause is 0.4 and 0.6, respectively. This result indicates that cause 2 is more severe than cause 1. The MLEs and Bayesian estimates of the unknown parameters are computed and shown in Table 9. The Bayesian estimates of λ1 and λ2 are obtained based on noninformative priors because no prior knowledge is known about the unknown parameters. It is observed that the MLEs and Bayesian estimates under the SE loss are coinciding under the noninformative priors, so we do not display estimates under the SE loss function in Table 9. Also, we can compute the MLEs of the meantime to failure for cause 1 as π/4λ^1=2568.912 and the meantime to failure for cause 2 as π/4λ^2=2097.508. The various intervals estimates are also obtained and reported in Table 10. Comparing the different confidence intervals, we observed that the HPD intervals have the minimum confidence lengths.Table 9
Estimated values of adaptive type-II progressively censored competing risks data (breaking strengths).
ParMLEsBayes estimatesLINEXτ=0.001GEq=0.5WSEPrλ11.190122×10−71.190124×10−79.69799×10−88.92591×10−81.3306×10−7λ21.785183×10−71.785185×10−71.56380×10−71.48765×10−71.92822×10−7Table 10
The 95% confidence intervals of adaptive type-II progressively censored competing risks data (breaking strengths).
Methodλ1λ2ACI2.38×10−9,2.36×10−73.57×10−8,3.22×10−7BCI3.24×10−8,2.61×10−76.55×10−8,3.47×10−7HPD5.50×10−8,1.29×10−71.63×10−7,1.83×10−7BP2.93×10−8,2.51×10−77.88×10−8,3.16×10−7BT0,2.59×10−71.07×10−8,2.68×10−7(2) Example Two. The second data set is taken from Lawless [29] and also analyzed by Sarhan [30]. The data consist of times to failure of 33 small electrical appliances subjected to an automatic life test. Failures were categorised into 18 various modes, and only modes six and nine occur more than twice. We concentrate on failure mode 9. Therefore, the data consist of two causes of failure: cause 1 (mode nine) and cause 2 (all other failure modes). The complete data set is displayed in Table 11.Table 11
Times to failure of small electrical appliances.
Times to failureCause of failureTimes to failureCause of failure1122471135225511492256811702269413292270223812276127082283129582303411062230592116713112115922321411925134781199013504122231432912327269761240017846124512From Table10, we have 17 failures due to cause 1 and 16 failures due to cause 2. The MLEs of the different models as well as the KS distance and the corresponding P-value are presented in Table 12 for cause 1 and cause 2, respectively. The outcomes in Table 12 imply that the RD is a proper model for these data.Table 12
The MLEs, KS, andP-value of the competitive models for times to failure data.
DataModelEstimateKSP-valueCause 1RD9.569×10−80.236 10.334 6We0.101 580.871 10.000 0IL2.577×1030.395 50.013 4L6.715×10−40.308 30.095 5Cause 2RD3.259×10−70.328 80.062 9We0.129 460.744 40.000 0IL102.4320.491 50.000 9L1.492×10−30.262 90.218 9From the actual data shown in Table11, an adaptive type-II progressively censored sample is generated by selecting T=2000, m=15 and the censoring scheme R1=⋯=R14=1 and R15=4. The generated sample is depicted in Table 13.Table 13
Generated adaptive type-II progressively censored competing risks data from times to failure data.
Cause of failureTimes to failure1116719251990222324002471211491703817081062159223272451From Table13, we have m1=6,m9=6, and J=10, and the estimated relative risks due to causes 1 and 2 are 0.4 and 0.6, respectively. The MLEs and Bayes estimates of the unknown parameters are calculated and displayed in Table 14. The Bayes estimates of λ1 and λ2 are obtained using noninformative priors. Under the noninformative priors, the MLEs and Bayes estimates under the SE loss are coinciding, so we do not show the estimates using the SE loss function in Table 14. Moreover, the MLEs of the meantime to failure for cause 1 and cause 2 are 3686.32 and 3009.87, respectively. The various intervals estimates are also obtained and displayed in Table 15. Also, we can see that the HPD intervals have the lowest confidence lengths.Table 14
Estimated values of adaptive type-II progressively censored competing risks data (times to failure).
ParMLEsBayes estimatesLINEXτ=0.001GEq=0.5WSEPrλ15.77967×10−85.78004×10−85.06292×10−84.81639×10−86.24276×10−8λ28.66950×10−88.67006×10−87.95069×10−87.70623×10−89.13846×10−8Table 15
The 95% confidence intervals of adaptive type-II progressively censored competing risks data (times to failure).
Methodλ1λ2ACI1.15×10−8,1.04×10−73.01×10−8,1.43×10−7BCI2.12×10−8,1.12×10−73.96×10−8,1.52×10−7HPD3.21×10−8,1.01×10−74.12×10−8,1.24×10−7BP1.21×10−8,1.35×10−73.20×10−8,1.59×10−7BT1.01×10−8,1.82×10−72.24×10−8,1.12×10−7


## 8. Conclusion Remarks

In this investigation, we worked with a competing risks model based on the Rayleigh distribution and used adaptive progressively Type-II censoring with a known cause of failure. Although we concentrated on only two causes of failure in this study, this research may be extended to include more causes of failure. We first obtain the maximum likelihood estimators of the unknown parameters. Bayesian estimators are also derived by considering both symmetric and asymmetric loss functions. Different procedures are adopted for building confidence intervals for the unknown parameters. A simulation study is implemented to assess the behaviour of classical and Bayesian estimates. The simulation results show that the Bayes estimators under LINEX loss function perform better than other estimators in terms of minimum mean square errors and the highest posterior density intervals give the shortest confidence length. Finally, a numerical illustration is offered to demonstrate the different methods involved in the article.



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