akram kohansal

In this paper, we delve into Bayesian inference related to multi-component stress-strength parameters, focusing on non-identical component strengths within a two-parameter Rayleigh distribution under the progressive first failure censoring scheme. We explore various scenarios: the general case, and instances where the common location parameter is either unknown or known. For each scenario, point and interval estimates are derived using methods including the MCMC method, Lindley's approximation, exact Bayes estimates, and HPD credible intervals. The efficacy of these methods is evaluated using a Monte Carlo simulation, and their practical applications are demonstrated with a real data set.

This paper describes the point and interval estimation of the unknown parameters of modified Weibull distribution under the adaptive Type-II progressive censored samples‎. ‎First‎, ‎we obtain the maximum likelihood estimation of parameters‎. ‎Because maximum likelihood estimations should be solved in numerical methods and cannot be derived in a closed form‎, ‎the approximate maximum likelihood estimations of the parameters are achieved‎. ‎Also‎, ‎asymptotic confidence intervals are obtained by earning the asymptotic distribution of the parameters‎. ‎Moreover‎, ‎two bootstrap confidence intervals are derived‎. ‎Second‎, ‎the Bayesian estimation of parameters is approximated using the Markov chain Monte Carlo algorithm and Lindley's method‎. ‎Furthermore‎, ‎the highest posterior density credible intervals of the parameters are derived‎. ‎Finally‎, ‎the different proposed estimations have been compared by the simulation studies and one data set is analyzed to illustrative aims.

In order to produce more flexible models in the reliability theory field, the Bayesian inference of 𝑚-component reliability model with the non-identical-component strengths for modified Weibull distribution under the progressive censoring scheme is considered. One of the key benefits is the generality of this model, so it includes some cases studied previously, such as multi-component stress-strength model with one and two non-identical-component and stress-strength models. In addition, the study of progressive censored data discussed in this paper is critical in many practical situations. The problem is considered in three cases: when the two common parameters for strengths and stress variables are unknown, known, and general. In each case, the approximation methods, such as the MCMC and Lindley’s approximation, are used to consider the -component stress-strength parameter. The Monte Carlo simulation study compares the performance of different methods—finally, a demonstration of how the proposed model may be utilized to analyse real data sets.

‎In this paper, under adaptive hybrid progressive censoring samples, Bayes estimation of the multi-component reliability, with the non-identical-component strengths, in unit generalized Gompertz distribution is considered. This problem is solved in three cases. In the first case, strengths and stress variables are assumed to have unknown, uncommon parameters. In the second case,  it is assumed that strengths and stress variables have two common and one uncommon parameter, so all of these parameters are unknown. In the third case, it is assumed that strengths and stress variables have two known common parameters and one unknown uncommon parameter. In each of these cases, Bayes estimation of the multi-component reliability, with the non-identical-component strengths, is obtained with different methods. Finally, different estimations are compared using the Monte Carlo simulation, and the results are implemented on one real data set.

The statistical inference of the multi-component stress-strength parameter, $R_{s,k}$, is considered in the three-parameter Weibull distribution. The problem is studied in two cases. In the first case, assuming that the stress and strength variables have common shape and location parameters and non-common scale parameters and all these parameters are unknown, the maximum likelihood estimation and the Bayesian estimation of the parameter $‎R_{s,k}$ are investigated. In this case, as the Bayesian estimation does not have a closed form, it is approximated by two methods, Lindley and $mbox{MCMC}$. Also, asymptotic confidence intervals have been obtained. In the second case, assuming that the stress and strength variables have known common shape and location parameters and non-common and unknown scale parameters, the maximum likelihood estimation, the uniformly minimum variance unbiased estimators, the exact Bayesian estimation of the parameter $‎R_{s,k}$ and the asymptotic confidence interval is calculated. Finally, using Monte Carlo simulation, the performance of different estimators has been compared.

The estimation of unknown parameters of two-parameter Rayleigh distribution based on Type-II progressive censoring with binomial removals is studied. Maximum likelihood estimators of the parameters and their confidence intervals are derived. By applying Markov Chain Monte Carlo techniques, Bayes estimators, and corresponding highest posterior density confidence intervals of parameters are obtained. The expected time required to complete the life test under this censoring scheme is investigated. Monte Carlo simulations are performed to compare the performances of the different methods, and one data set is analyzed for illustrative purposes.

The Bayesian estimation of the stress-strength parameter in Lomax distribution under the progressive hybrid censored sample is considered in three cases. First, assuming the stress and strength are two random variables with a common scale and different shape parameters. The Bayesian estimations of these parameters are approximated by Lindley method and the Gibbs algorithm. Second, assuming the scale parameter is known, the exact Bayes estimation of the stress-strength parameter is obtained. Third, assuming all parameters are unknown, the Bayesian estimation of the stress-strength parameter is derived via the Gibbs algorithm. Also, the maximum likelihood estimations are calculated, and the usefulness of the Bayesian estimations is confirmed, in comparison with them. Finally, the different methods are evaluated utilizing the Monte Carlo simulation and one real data set is analyzed.

‎Bucket recursive trees are an interesting and natural generalization of recursive trees‎. ‎In this model the nodes are buckets that can hold up to b>= 1 labels‎. ‎The (modified) Zagreb index of a graph is defined as the sum of‎ ‎the squares of the outdegrees of all vertices in the graph‎. ‎We give the mean and variance of this index in random bucket recursive trees‎. ‎Also‎, ‎two limiting results on this index are given‎.

‎The aim of this paper is to introduce some results for the F-index of the tree structures without any information on the exact values of vertex degrees‎. Three martingales related to the first Zagreb index and F-index are given‎.

‎The estimation of R=P(X<Y) in the case that X and Y are two independent Kumaraswamy distributed random variables with different parameters for progressively Type-II censored samples is studied‎. ‎First assuming the same second shape parameters of two distributions‎, ‎the maximum likelihood estimation and different confidence intervals are considered‎. ‎Moreover‎, ‎in case the second shape parameters of two distributions are known‎, ‎MLE‎, ‎UMVUE‎, ‎Bayes estimation of R and confidence intervals are derived‎. ‎Finally‎, ‎the Maximum Likelihood and Bayes estimations of R in general case are obtained‎. ‎Performance comparisons of different methods are carried out utilizing Monte Carlo simulations‎. ‎Besides‎, ‎the proposed approach is employed for reliability analysis on a real strength-stress dataset to demonstrate its application‎.