This study uses the adaptive Type-II progressively censored competing risks model to estimate the unknown parameters and the survival function of the Gompertz distribution. Where the lifetime for each failure is considered independent, and each follows a unique Gompertz distribution with different shape parameters. First, the Newton-Raphson method is used to derive the maximum likelihood estimators (MLEs), and the existence and uniqueness of the estimators are also demonstrated. We used the stochastic expectation maximization (SEM) method to construct MLEs for unknown parameters, which simplified and facilitated computation. Based on the asymptotic normality of the MLEs and SEM methods, we create the corresponding confidence intervals for unknown parameters, and the delta approach is utilized to obtain the interval estimation of the reliability function. Additionally, using two bootstrap techniques, the approximative interval estimators for all unknowns are created. Furthermore, we computed the Bayes estimates of unknown parameters as well as the survival function using the Markov chain Monte Carlo (MCMC) method in the presence of square error and LINEX loss functions. Finally, we look into two real data sets and create a simulation study to evaluate the efficacy of the established approaches.

In this paper, the bivariate extension of the so called Gompertz-G family was introduced and studied in detail. Marshall and Olkin shock model was used to build the proposed bivariate family. The new family was constructed from three independent Gompertz-H families using a minimisation process. Some of its statistical properties such as joint probability density function, coeffi cient of median correlation, moments, product moment, covariance, conditional probability density function, joint reliability function, stress-strength reliability and joint reversed (hazard) rate function were derived. After introducing the general class, three special models of the new family were discussed. Maximum likelihood method was used to estimate the family parameters. A simulation study was carried out to examine the bias and mean square error of the maximum likelihood estimators. Finally, the importance of the proposed bivariate family was illustrated by means of real dataset, and it was found that the proposed model provides better fi t than other well-known models in the statistical literature such as bivariate Gompertz, bivariate generalized Gompertz, bivariate Gumbel Gompertz, bivariate Burr X Gompertz and bivariate exponentiated Weibull-Gomperz

In this article we consider statistical inferences about the unknown parameters of the exponentiated Nadarajah-Haghighi (ENH) distribution based on progressively type-II censoring using classical and Bayesian procedures. For classical procedures, maximum likelihood (ML) and least square estimators of the unknown parameters are derived. The Bayes estimators are obtained based on both the symmetric (squared error) and asymmetric (LINEX, general entropy) loss functions. Furthermore, Markov chain Monte Carlo (MCMC) technique is used to compute the Bayes estimators and the associated credible intervals. Moreover, asymptotic confidence intervals are constructed using the normality property of the ML estimates. After that, the asymptotic confidence intervals using ML estimates and two parametric bootstrap confidence intervals are provided to compare with Bayes credible intervals. Finally, simulation study and a bladder cancer application are presented to illustrate the proposed methods of estimation.

In this research, the statistical inference of unknown lifetime parameters is proposed in the presence of independent competing risks using a progressive Type-II censored dataset. The lifetime distribution associated with a failure mode is assumed to follow the new Pareto distribution, with consideration given to two distinct competing failure reasons. Maximum likelihood estimators (MLEs) for the unknown model parameters, as well as reliability and hazard functions, are derived, noting that they are not expressible in closed form. The Newton–Raphson, expectation maximization (EM), and stochastic expectation maximization (SEM) methods are employed to generate maximum likelihood (ML) estimations. Approximate confidence intervals for the unknown parameters, reliability, and hazard rate functions are constructed using the normal approximation of the MLEs and the normal approximation of the log-transformed MLEs. Additionally, the missing information principle is utilized to derive the closed form of the Fisher information matrix, which, in turn, is used with the delta approach to calculate confidence intervals for reliability and hazards. Bayes estimators are derived under both symmetric and asymmetric loss functions, with informative and non-informative priors considered, including independent gamma distributions for informative priors. The Monte Carlo Markov Chain sampling approach is employed to obtain the highest posterior density credible intervals and Bayesian point estimates for unknown parameters and reliability characteristics. A Monte Carlo simulation is conducted to assess the effectiveness of the proposed techniques, with the performances of the Bayes and maximum likelihood estimations examined using average values and mean squared errors as benchmarks. Interval estimations are compared in terms of average lengths and coverage probabilities. Real datasets are considered and examined for each topic to provide illustrative examples.