In the present paper, statistical inference problem is considered for the geometric process (GP) by assuming the distribution of the first arrival time is generalized Rayleigh with the parameters $\alpha$ and $\lambda$. We use the maximum likelihood method for obtaining the ratio parameter of the GP and distributional parameters of the generalized Rayleigh distribution. By a series of Monte-Carlo simulations evaluated through the different samples of sizes small, moderate and large, we also compare the estimation performances of the maximum likelihood estimators with the other estimators available in the literature such as modified moment, modified L-moment, and modified least squares. Furthermore, we present two real-life dataset analyzes to show the modeling behavior of GP with generalized Rayleigh distribution.

The geometric process (GP) plays an important role in the reliability theory and life span models. It has been used extensively as a stochastic model in many areas of application. Therefore, the parameter estimation problem is very crucial in a GP. In this study, the parameter estimation problem for GP is discussed under the assumption that [Formula: see text] has a Lindley distribution with parameter [Formula: see text]. The maximum likelihood (ML) estimators of [Formula: see text] and [Formula: see text] of the GP and their asymptotic distributions are derived. A test statistic is developed based on ML estimators for testing whether [Formula: see text] or not. The same problem is also studied by using Bayesian methods. Bayes estimators of the unknown model parameters are obtained under squared error loss function (SELF) using uniform and gamma priors on the ratio [Formula: see text] and [Formula: see text] parameters. It is not possible to obtain Bayes estimators in explicit forms. Therefore, Markov Chain Monte Carlo (MCMC), Lindley (LD), and Tierney–Kadane (T–K) methods are used to estimate the parameters [Formula: see text] and [Formula: see text] in GP. The efficiencies of the ML estimators are compared with Bayes estimators via an extensive Monte Carlo simulation study. It is seen that the Bayes estimators perform better than the ML estimators. Two real-life examples are also presented for application purposes. The first data set concerns the coal mining disaster. The second is the number of COVID-19 patients in Turkey.

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In this paper, the classical and Bayesian estimators of the unknown parameters and the reliability function of the inverse Weibull distribution are considered. The maximum likelihood estimators (MLEs) and modified maximum likelihood estimators (MMLEs) are used in the classical parameter estimation. Bayesian estimators of the parameters are obtained by using symmetric and asymmetric loss functions under informative and non-informative priors. Bayesian computations are derived by using Lindley approximation and Markov chain Monte Carlo (MCMC) methods. The asymptotic confidence intervals are constructed based on the maximum likelihood estimators. The Bayesian credible intervals of the parameters are obtained by using the MCMC method. Furthermore, the performances of these estimation methods are compared concerning their biases and mean square errors through a simulation study. It is seen that the Bayes estimators perform better than the classical estimators. Finally, two real-life examples are given for illustrative purposes.

The main objective of this paper is to determine the best estimators of the shape and scale parameters of the two parameter Weibull distribution. Therefore, both classical and Bayesian approximation methods are considered. For parameter estimation of classical approximation methods maximum likelihood estimators (MLEs), modi…ed maximum likelihood estimators-I (MMLEs-I), modi…ed maximum likelihood estimators-II (MMLEs-II), least square estimators (LSEs), weighted least square estimators (WLSEs), percentile estimators (PEs), moment estimators (MEs), L-moment estimators (LMEs) and TL-moment estimators (TLMEs) are used. Since the Bayesian estimators don't have the explicit form. There are Bayes estimators are obtained by using Lindley's and Tierney Kadane's approximation methods in this study. In Bayesian approximation, the choice of loss function and prior distribution is very important. Hence, Bayes estimators are given based on both the non-informative and informative prior distribution. Moreover, these estimators have been calculated under di¤erent symmetric and asymmetric loss functions. The performance of classical and Bayesian estimators are compared with respect to their biases and MSEs through a simulation study. Finally, a real data set taken from Turkish State Meteorological Service is analysed for better understanding of methods presented in this paper.