{'fraction_non_alphanumeric': 0.04323122529644269, 'fraction_numerical': 0.020421607378129116, 'mean_word_length': 4.3150984682713345, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Analysis of the improved adaptive type-II progressive censoring based on competing risk data

In this paper, a competing risk model is analyzed based on the improved adaptive type-II progressive censored sample (IAT-II PCS). Only two competing causes of failures from independent exponential distributions are considered. Maximum likelihood estimates (MLEs) for the unknown model parameters are obtained. By using the asymptotic normality properties of the MLEs, the approximate confidence intervals are constructed. The existence and uniqueness of the MLEs are studied. The Bayes estimates are obtained under the symmetric and asymmetric loss functions with non-informative and independent gamma prior distributions. Further, the highest posterior density (HPD) credible intervals are obtained by using the MCMC technique. Coverage probabilities for each confidence interval are computed. A Monte Carlo simulation study is carried out to compare the performance of the proposed estimates. Finally, a real data set is considered for illustrative purposes.


Introduction
In the study of survival analysis, life-time experimental units may fail because of more than one causes. In such studies, the experimenter can observe the time of failure of these experimental units along with the corresponding causes of failure. These causes compete with each other for the failure of the experimental units. Such studies are known as the competing risks problem in statistical literature. Competing risks data have a wide variety of applications in the fields of engineering, biological and medical studies. In recent years, inferences of competing risks data have accepted a lot of attention by several authors in survival analysis. For example, one can refer to Kundu and Basu (2000), Sarhan et al. (2010), Wang (2018), Ren and Gui (2021) and Mahto et al. (2021). Due to improvement in the reliability of lifetime products, time and cost constraints, it is impossible for an experimenter to collect complete sample information from a lifetime experiment. In such situations of survival studies, censoring became a common phenomenon in past years. The conventional censoring schemes, namely, type-I and type-II censoring schemes have been widely used in reliability studies. Epstein (1954) introduced the mixture of these two censoring schemes, which is known as the type-I hybrid censoring schemes. Childs et al. (2003) introduced the type-II hybrid censoring scheme. The major drawback of these conventional censoring schemes is that it does not allow to remove experimental units before the terminal points of the experiments. To overcome such drawback, Cohen (1963) introduced progressive censoring scheme, where one can remove survival units at various stages of experiments. Extensive work on these censoring schemes has been done in past years. See, for instance, Balakrishnan et al. (2003), Balakrishnan and Kundu (2013), and Rastogi and Tripathi (2012).
Due to improvement in manufacturing technologies, now-a-days products are more reliable, sometimes the experiment under progressive type-II censoring schemes takes a longer time to get a pre-fixed number of failure time units. To overcome such drawback, Kundu and Joarder (2006) introduced type-II progressive hybrid censoring scheme. This scheme has been studied by many researchers. See, for example, Lin et al. (2009), Lin et al. (2012), Hemmati and Khorram (2013), and Tomer and Panwar (2015). Under this censoring scheme, experimental time is prefixed, because of that the number of effective sample size becomes random. Even sometimes the number of effective sample size becomes zero, due to that the efficiency of statistical inference decreases. To overcome such inefficiency in experimental results, Ng et al. (2009) proposed the adaptive type-II progressive censoring scheme (AT-II PCS). To know more about that censoring scheme, one can refer Ye et al. (2014), Sobhi and Soliman (2016), and El-Sagheer et al. (2018). In AT-II PCS, the experiment will terminate after getting a prefixed number of failure-time data. If the experimental units are taken as highly reliable units, then it will take very long time to complete that experiment. To overcome this drawback, Figure 1: Schematic representation of the improved adaptive type-II progressive censored scheme.
The main aim of this paper is to analyze the statistical inference of the unknown parameters based on IAT-II PCS data in presence of competing risk data. Here it is considered that, the lifetime distributions due to competing causes of failure follow a latent failure time model introduced by Cox (1959). Further it is assumed that, competing risks data occurs due to two types of causes of failures only. It is considered that, the competing causes follow independent exponential distributions with mean τ 1 and τ 2 , respectively. The MLEs and asymptotic confidence intervals of the unknown parameters are obtained in the context of classical phenomenon. Further, the Bayes estimates are obtained based on non-informative and informative priors. For comparison purpose, the HPD credible intervals are also constructed by using MCMC sampling technique.
The rest of the paper is organized as follows. In Section 2, the competing risk model and the corresponding likelihood function based on IAT-II PCS data are described. In Section 3, the MLEs and their existence and uniqueness are discussed. The asymptotic confidence intervals are constructed. The Bayes estimates under different loss functions and the HPD credible intervals are discussed in Section 4. In Section 5, a Monte Carlo simulation study is performed to compare the proposed methods. In Section 6, a real life data set is analyzed for illustrative purposes. Finally, conclusion is made in Section 7.
Suppose n identical units are put on an IAT-II PHC life-testing experiment. Lifetimes of these experimental units are described as independent and identical random variables X 1 , · · · , X n . Only two causes of failures are considered here, and the latent failure times are given as, where X ji denotes the latent failure time of the ith experimental units when jth cause of failure is taken part of, where j = 1 and 2. Further, it is assumed that the latent failure times X 1i and X 2i are independent.
Let us consider, δ i = 1, if the failure occurs due to cause 1 and δ i = 0, if the failure occurs due to cause 2. Therefore, D 1 = D i=1 I(δ i = 1) and D 2 = D i=1 I(δ i = 0) denotes the total number of failures due to cause 1 and cause 2 respectively, with D 1 + D 2 = D. The cumulative distribution function (CDF), probability density function (PDF), survival function, and hazard rate of the latent failure times X ji , for i = 1, 2, · · · , n and j = 1 and 2, are considered as F j (·), f j (·), S j (·), and h j (·) respectively. The likelihood function based on the observations can be written as By using f k (x) = h k (x)F k (x), further, the likelihood function can be written as (2.1) where x i represents the i-th failure time x i:m:n . Further, for case-I, D = m and R * = 0 , for case-II, D = k 1 and R * = n−m− k 1 i=1 R i , and for case-III, D = k 1 and R * = n−k 2 − k 1 i=1 R i . If the failure times follow one parameter exponential distribution then the CDF and hazard rate of latent failure times can be written as where τ j > 0 for j = 1 and 2.

Maximum likelihood estimation
In this section, MLEs of the unknown model parameters τ 1 and τ 2 are obtained. Based on Eqs. (2.1) and (2.2), the likelihood function of τ 1 and τ 2 can be written as Now, the log-likelihood function can be written as After differentiating (3.2) with respect to τ 1 and τ 2 and equating to zero, the likelihood equations can be obtained as and where l = log L(τ 1 , τ 2 ). From the above equations the MLEs τ 1 and τ 2 for the unknown parameters can be obtained, which are given as

Existence and uniqueness of MLEs
The existence and uniqueness of the MLEs for the unknown model parameters can be obtained through the following theorem.
Theorem 1 For D i > 0, where i = 1, 2, the MLE for each τ i exists and unique.
Proof. From (3.5), the MLEs of unknown model parameters can be written as For the existence of the MLE of each model parameter τ i , at least one failure should be occurred and observed due to ith cause. Therefore, for the existence of the MLE of τ i , D i > 0, where i = 1, 2. Further, to show the uniqueness of the MLEs we have to prove that the Hessian matrix H(τ 1 , τ 2 ) of l(τ 1 , τ 2 ) is negative definite. Here, the Hessian matrix can be written as H(τ 1 , τ 2 ) = ∂ 2 l ∂τ i ∂τ j , where i, j = 1, 2.
(3.6) From (3.3) and (3.4), we get Since the diagonal elements of the Hessian matrix, H(τ 1 , τ 2 ) are negative and rest of the terms are zero, then the Hessian matrix is negative definite. Hence the MLEs τ 1 and τ 2 exist and unique.

Asymptotic confidence intervals
Using asymptotic normality properties of the MLEs of τ 1 and τ 2 , the 100(1 − γ)% asymptotic confidence intervals are constructed. Under some mild regularity conditions, it is noticed that the asymptotic distribution of the MLE ( τ 1 , τ 2 ) ′ can be obtained as is the inverse of the observed Fisher information matrix of τ 1 and τ 2 and l ij = ∂ 2 l ∂ i τ 1 ∂ j τ 2 , and i, j = 0, 1, 2. Therefore, the 100(1 − γ)% asymptotic confidence intervals for τ 1 and τ 2 are given as where z γ 2 is the upper γ 2 th percentile point of a standard normal distribution.

Bayesian estimation
In this section, estimates of the unknown parameters τ 1 and τ 2 are obtained from the Bayesian point of view based on IAT-II PCS competing risk data. In doing so, three different types of loss functions are considered. In case of the symmetric loss function, the squared error loss function (SELF) is considered. Otherwise in case of asymmetric loss functions LINEX loss function (LLF), proposed by Varian (1975) and generalized entropy loss function (GELF) are used. Letθ be an estimator of θ. Then, SELF, LLF and GELF are respectively given by Under the loss functions given by (4.1), (4.2) and (4.3), the Bayes estimates of θ can be respectively written as In the study of Bayesian estimation, choosing priors for the unknown model parameters is an important as well as challenging problem. Here we consider the independent gamma priors following the unknown parameters as τ 1 ∼ Gamma(a, b) and where a, b, c, d > 0 are the hyper parameters which are chosen to reflect the prior knowledge. Then the joint prior density function of τ 1 and τ 2 can be written as Using the likelihood function (3.1) and the joint prior density function (4.7), the posterior density function can be written as From Eq. (4.8), the marginal posterior densities of τ 1 and τ 2 can be respectively obtained as and (4.10) Lemma 1 The posterior marginal PDF π(τ k |x) is log-concave for k = 1, 2.

Simulation study
In this section, a Monte Carlo simulation study has been employed to compare the performance of the proposed estimates based on 10000 IAT-II PCS. Different sample sizes (n) and different effective sample sizes (m), different time thresholds (T 1 , T 2 ) and different censoring schemes have been considered. These censoring schemes are  Tables 1, 3, 5, and 7, some points are clearly observed about point estimates. These results are given as follows : • Based on average values and MSEs the Bayes estimates under LLF and GELF give better results than other estimates for τ 1 and τ 2 , respectively.
• The Bayes estimates under informative priors give better results than non-informative priors.
• When sample size (n) and effective sample size (m) increase, then MSEs decrease.
• When the values of time thresholds T 1 and T 2 increase, then MSEs decrease.
In case of interval estimates, from Tables 2, 4, 6, and 8 the following points are observed.
• Based on average lengths and coverage probabilities HPD credible intervals perform better than asymptotic confidence intervals.
• When sample size (n) and effective sample size (m) increase, then length of intervals decrease.
• When the values of time thresholds T 1 and T 2 increase, then length of intervals decrease.