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

(is paper has investigated the estimation problem for the competing risks model where the data are adaptive progressively typeII 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.(e 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.


Introduction
e investigator may not be able to receive complete information on failure times for all experimental units via lifetesting 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. e 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 � (R 1 , . . . , R m ). If the m th failure happens before the threshold T (i.e., X m: m: n < T), the test ends at the time X m: 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. us, we should leave as numerous surviving units on the experiment as possible. Suppose J is the number of failures before T, that is, X J: m: n < T < X J+1: m: n , J � 0, 1, . . . m, where X 0: m: n ≡ 0 and X m+1: m: n ≡ ∞. When the experiment passed the predetermined time T, we put R J+1 � · · · � R m− 1 � 0 and R m � n − m − J i�1 R i . is formulation directs us to end the test as soon as possible if the (J + 1)th failure time is larger than T for J + 1 < m. e threshold of T plays a significant role in determining the values of R i 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. e 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. is 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. is distribution is chosen because it is a logical extension of the exponential distribution and hence plays a key role in acoustics, lifetesting 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.
e failure rate of this distribution is a linear function of time, which is a major characteristic. e 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. e 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. e 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. e 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. e remaining part of the paper proceeds as follows: In Section 2, 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. e 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.

Description and Notation of the Model
Without loss of generality, suppose that we have n identical items in a lifetime experiment. Moreover, assume that there are only two causes of failure, and then, X i � min X 1i , X 2i , 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 (X 1i , X 2i ) are independent identically distributed random variables. In addition, it is assumed that X ki , i � 1, . . . , n and k � 1, 2 are independent and follow the RD with the following probability density function: 2 Mathematical Problems in Engineering e corresponding cumulative distribution function is given by We can write the observations under adaptive progressively type-II censored competing risks data as follows: where R * m � n − m − J i�1 R i and c i ∈ (1, 2), which indicates the cause of failure of item i. Let then m 1 � m i�1 I(c i � 1) and m 1 � m i�1 I(c i � 2) refer to the number of failures owed to cause 1 and 2, respectively, and m � m 1 + m 2 . Based on the independence of X 1i and X 2i , we can obtain the relative risk due to cause 1 as follows: Similarly, the relative risk due to cause 2 can be obtained as follows: Given a progressive censoring scheme (R 1 , . . . , R J , 0, . . . , 0, R * m ), then the likelihood function of the observed data can be expressed as follows: where F k � 1 − F k (x), k � 1, 2, x i � x i: m: n for the sake of simplicity, and

Maximum Likelihood Estimation
Based on (1)-(3), the likelihood function, ignoring the normalized constant, can be written as follows: e natural logarithm of the likelihood function (ℓ � log L(λ 1 , λ 2 )) in (4) takes the form To obtain the MLEs of the λ 1 and λ 2 , we find the firstorder partial derivatives of the log-likelihood function in (5) with respect to λ 1 and λ 2 , and then, by equating each of them to zero, we get where Mathematical Problems in Engineering From (6) and (7), we can acquire the MLEs λ 1 and λ 2 of λ 1 and λ 2 , respectively, as follows: To obtain the ACIs for the unknown parameters λ 1 and λ 2 , we require to obtain the second derivatives of the loglikelihood function as follows: us, the asymptotic variance-covariance matrix is obtained by inverting the observed Fisher information matrix as follows:

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. e 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. e Bayes estimates are obtained based on the assumption that the random variables λ 1 and λ 2 have gamma prior distributions with known shape and scale parameters a k and b k , k � 1, 2, with pdf given by 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 λ 1 and λ 2 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.
erefore, independent priors are more widespread in statistics under the Bayesian approach for the sake of simplicity. e joint posterior density of λ 1 and λ 2 can be expressed using the likelihood function in (4) and the joint prior distribution in (12) as follows: where e marginal posterior densities of λ 1 and λ 2 are given, respectively, by From (15) and (16), it can easily be noted that the marginal posterior distributions of λ 1 and λ 2 are gamma distributions, that is, Gamma(a 1 + m 1 , b 1 + A(x)) and Gamma(a 2 + m 2 , b 2 + A(x)), respectively. Now, to obtain the Bayes estimators of λ 1 and λ 2 , 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.

Bayesian Estimators Using SE Loss Function.
First, we consider the SE loss function. Based on the SE loss function, the Bayesian estimators of λ 1 and λ 2 , denoted by λ SE1 and λ SE2 , are the mean of the posterior densities (15) and (16), which are given by It is observed that in the case of non-informative priors, i.e., a 1 � b 1 � a 2 � b 2 � 0, the Bayesian estimators of λ 1 and λ 2 corresponds to the MLEs given by (9).

Bayesian Estimators Using LINEX Loss Function.
e 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 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 λ 1 and λ 2 using the LINEX loss function, respectively, as follows:

Bayesian Estimators Using GE Loss Function.
e 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 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 λ 1 and λ 2 using the GE loss function are, respectively, given by Here, we assume three particular cases, corresponding to q � − 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 λ 1 and λ 2 under the WSE loss function as and when q � − 2, the Bayesian estimators using the Pr loss function of λ 1 and λ 2 , denoted by λ Pr1 and λ Pr2 , are as follows:

Confidence Intervals
In this section, five methods are proposed to construct the confidence intervals for the parameters λ 1 and λ 2 . e proposed methods are the ACIs, two kinds of bootstrap confidence intervals, the BCIs, and HPD intervals.

ACIs.
Utilizing the asymptotic normality of the MLEs, the 100(1 − α)% ACIs of λ 1 and λ 2 , are, respectively, given by where z α/2 is the upper α/2 percentile of a standard normal distribution, and λ k / �� � m k √ , 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).

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

p-boot
(1) From the original data, the MLEs of λ k , k � 1, 2are obtained. terval for λ k is given by Mathematical Problems in Engineering

Bayes Credible Intervals.
e credible intervals with equal tails of λ 1 and λ 2 can be obtained using the posterior distributions of λ 1 and λ 2 . From (15) and (16), if m k , k � 1, 2, are integers, we can see that and [2(a 2 + m 2 )] degrees of freedom, respectively. erefore, 100(1 − α)% BCIs for λ 1 and λ 2 are , , If m k , k � 1, 2 are not integer values, then gamma distribution can be employed to create the desired credible intervals.

Highest Posterior Density Intervals.
e 100(1 − α)% HPD intervals (H On simplifications, we can write (32) and (33) as Γ a 2 + m 2 , H where is the incomplete gamma function. Expressions in (34) and (35) can be solved numerically to get the required HPD intervals of λ 1 and λ 2 .

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. e simulation study is conducted by choosing the parameter (Par) values λ 1 � 1 and λ 2 � 2, and various values of n, m, T and the following censoring schemes (Sch) For different cases, we obtain the MLEs and Bayesian estimates under SE, LINEX (with τ � 0.5), GE (with q � 0.5), WSE, and Pr loss functions based on 1000 simulations. e Bayesian estimates are obtained by assuming gamma prior distributions for λ 1 and λ 2 with parameters (1, 1) and (2, 1), respectively. ese 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 (1) e 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 as m increases in all the cases, which indicate that all estimators are consistent.

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.

Simulated Data Set.
In this subsection, we analyse one simulated data set. We generate adaptive progressively type-II censored competing risks data for λ 1 � 1 and λ 2 � 2 and by choosing n � 60, m � 20, T � 0.2 and the censoring scheme R 1 � · · · � R m− 1 � 1 and R m � n − 2m + 1, with the understanding that when the testing time passed the threshold T � 0.2, we replace R J+1 , . . . , R m− 1 by zero, and in this case, we have R m � n − m − J i�1 R i . e simulated observation is displayed in Table 3.
From Table 3, 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, m 1 � 6 and m 2 � 14. For these data, we obtain the MLEs and Bayes estimates under the different loss functions discussed before.
e Bayes estimates are obtained by considering informative priors, and the hyperparameters are taken to be (1, 1) for λ 1 and (2, 1) for λ 2 .
e different estimates and the corresponding MSEs are displayed in Table 4. e 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 λ 1 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 λ 2 . Furthermore, from Table 5, we conclude that the HPD intervals have the shortest confidence lengths for both parameters.

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. e first data set is given by King [27] and consists of the breaking strengths (mg) of 20 wire connections. ere 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. e letters W and B appear in Table 6 refer to bond lift (cause 1) and wire break (cause 2), respectively. ere 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. e 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. e results in Table 7 indicate that the RD can be assumed as an adequate model to fit the data set. Now, we use the original data given in Table 6 to generate an adaptive type-II progressively censored sample by choosing T � 1500, m � 10, and the censoring scheme R 1 � · · · � R 10 � 1. e generated observations are displayed in Table 8.
In this case, we have m 1 � 4, m 2 � 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. is result indicates that cause 2 is more severe than cause 1. e MLEs and Bayesian estimates of the unknown parameters are computed and shown in Table 9. e Bayesian estimates of λ 1 and λ 2 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. e 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.
(2) Example Two. e second data set is taken from Lawless [29] and also analyzed by Sarhan [30]. e 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. erefore, the data consist of two causes of failure: cause 1 (mode nine) and cause 2 (all other failure modes). e complete data set is displayed in Table 11. From Table 10, we have 17 failures due to cause 1 and 16 failures due to cause 2. e 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. e outcomes in Table 12 imply that the RD is a proper model for these data.
From the actual data shown in Table 11, an adaptive type-II progressively censored sample is generated by selecting T � 2000, m � 15 and the censoring scheme R 1 � · · · � R 14 � 1 and R 15 � 4. e generated sample is depicted in Table 13.
From Table 13, we have m 1 � 6, m 9 � 6, and J � 10, and the estimated relative risks due to causes 1 and 2 are 0.4 and 0.6, respectively. e MLEs and Bayes estimates of the unknown parameters are calculated and displayed in Table 14. e Bayes estimates of λ 1 and λ 2 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. e various intervals estimates are also obtained and displayed in Table 15. Also, we can see that the HPD intervals have the lowest confidence lengths.

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. e 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.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.