Inference Of Gamma Distribution And Its Environment Factor

Gamma distribution is an important model in statistics.It has been widely used to fit data in reliability engineering and other fields.Many scholars have studied inferential procedures of Gamma distribution.However,there are still some problems.For example,when the shape parameter is small,the actual coverage of the confidence interval of the existing method has a large deviation from the nominal coverage.This paper mainly discusses the inferential procedures of gamma distribution and its environment factor.Firstly,based on the Cornish-Fisher expansion and the pivotal cumulative distribution function theorem,the approximate confidence interval of the shape parameter is derived.Secondly,we discuss the generalized confidence interval of scale parameters and other reliability features(such as mean).And the prediction interval of the next measurement and at least p of m measurements at each of r locations is studied based on the proposed generalized confidence interval process.Monte Carlo simulation is used to evaluate the performance of the proposed methods.The simulation results show that the proposed methods have very satisfactory results.Finally,we study the point estimation and interval estimation of environment factor of Gamma distribution.In this part,we first study the point estimation,including the maximum likelihood estimation and the unbiased estimation.The simulation results show that the performance of the unbiased estimation is better.In addition,we study the confidence interval of environment factor,including the generalized confidence interval and Bootstrap-t confidence interval.The simulation results show that the generalized confidence interval and the Bootstrap-t confidence interval both have good performance.But when the shape parameter and sample size are small,the generalized confidence interval is superior to the Bootstrap-t confidence interval.The calculation results of practical example are given.The results show that the example results are consistent with the simulation results.In this paper,the inferential procedures proposed effectively solves the problem that the confidence interval's actual coverage of the parameters and other quantities of Gamma distribution does not match the nominal coverage when the gamma shape parameter is small.There is some innovation in the procedures of constructing confidence interval of the Gamma shape parameter.