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‎In this paper‎, ‎a new five-parameter so-called Beta-Gompertz Geometric (BGG) distribution is introduced that can have a decreasing‎, ‎increasing‎, ‎and bathtub-shaped failure rate function depending on its parameters‎. ‎Some mathematical properties of the this distribution‎, ‎such as the density and hazard rate functions‎, ‎moments‎, ‎moment generating function‎, ‎R and Shannon entropy‎, ‎Bonferroni and Lorenz curves and the mean deavations are provided‎. ‎We discuss maximum likelihood estimation of the BGG parameters from one observed sample‎. ‎At the end‎, ‎in order to show the BGG distribution flexibility‎, ‎an application using a real data set is presented‎.

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‎In this study‎, ‎E-Bayesian and hierarchical Bayesian of parameter of Rayleigh distribution under progressive type-II censoring sampales and the efficiency of the proposed methods has been compared with each and Bayesian estimator using Monte Carlo simulation‎.

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Bootstrap is a computer-based resampling and statistical inference method that can provide an estimate for the uncertainty of distribution parameters and quantiles in a frequency analysis. In this article, in addition to calculating the bootstrap confidence interval for the quantiles with percentile bootstrap(BP), accelerated bias-corrected bootstrap(BCA) and t-bootstrap methods, calculating of the confidence interval is proposed using the bootstrap highest density method(HDI) for the probability distributions used in the hydrology data and we obtain the average length of the confidence interval as a criterion for evaluating the methods. To calculate the average length of the confidence interval with different methods, first, the best distribution among the widely used distributions is fitted to the original data, and the parameters of the fitted distribution are estimated by the maximum likelihood method, and from that we obtain the quantiles. Then we continue by repeating the simulated bootstrap samples until the probability of covering the real quantile reaches the nominal confidence level of 0.95. The simulation results show that the bootstrap highest density method gives the lowest average confidence interval length among all methods.
In previous studies, the probability of coverage with the same number of samples as the original sample and the number of bootstrap repetitions (for example, 1000) have been obtained, and finally the coverage probability closest to the nominal value of 0.95 was chosen as the optimal state, while in this article, for Avoiding a large number of bootstrap iterations, considering the number of different samples n = 20, 40, 60, ..., 160 that are simulated from the mother distribution, we continue the number of iterations of the bootstrap samples only until reaching the coverage probability of 0.95. Usually, two-parameter distributions require fewer samples than three-parameter distributions. In terms of the number of necessary repetitions (B) to reach the 95% confidence level, the t-bootstrap method usually required fewer repetitions, although the implement time of this method was longer in R software. The best distribution for the data used in this research is the two-parameter distribution of Frechet, which was selected as the best distribution in 80% of the stations, which can be used for similar studies that deal with the maximum annual precipitation values. According to the fitted distributions for the data of different stations, the two-parameter distributions always fit the data better than the three-parameter distributions. In general, the widest confidence intervals were obtained with BCA and tbootstrap methods, and the shortest confidence intervals were obtained with HDI and BP methods. Also, in terms of the distribution used and the length of the confidence interval, the two-parameter Gamma distribution has provided shorter confidence intervals and the three-parameter GEV and LP3 distributions have provided wider confidence intervals.