| Pareto distribution was first explicitly introduced by Italian economist-Vilfredo Pareto(1979).As a kind of distribution with shape and scale parameters,it can be used to describe various socio-economic,physical and biological phenomena,such as stock price,disaster prediction,equipment components failure,enterprise life,etc.,and also has applications in astronomy,military fields.Therefore,it attracts much attention in modern research.In the study of Pareto distribution,interval estimation and confidence region estimation of parameters provide a range of parameters reflecting the degree of uncertainty in the estimation,which is more intuitive,more accurate,more practical and more valuable than the parameter point estimation.In this dissertation,we study the parameter estimation of truncated Pareto distribution,interval estimation of reliability parameters of Pareto distribution,minimum volume confidence region of parameters for one Pareto sample and two Pareto samples.The main content of research is as follows:(1)The parameter estimation problem of the truncated Pareto distribution is studied.The parameter estimation for the truncated Pareto distribution is much more complicated than that for the Pareto distribution.Traditional estimation methods such as maximum likelihood estimation have been widely used,but the bias is large.To solve this problem,Using Jackknife method to obtain the parameter estimation.It is proved that the estimation has smaller bias and simpler form.The accuracy of several estimation methods is compared through Monte Carlo simulation.(2)The interval estimation of reliability R = P(X > Y)is researched.Reliability R is a measure of system performance and most literature often uses simulation methods(Bootstrap,Generalized variable method,etc.)to obtain interval estimation of R,which requires a lot of calculation.This dissertation establishes the confidence interval of R based on the method of variance estimates recovery(MOVER),whichhas the closed form espression of the confidence interval,and greatly reduces the computational complexity.The simulation results show that the MOVER confidence interval based on Fieller's theorem performs better.(3)The joint confidence region of Pareto distribution parameter based on a double censored sample is concerned.Under some constraints,the minimum volume confidence region of parameters is established,which can be used for complete sample,left(right)censoring sample and double sample.The computation of the optimal confidence region in literature is difficult in left censoring case.However,the minimum volume confidence region established in this dissertation is convenient to calculate,easy to obtain critical value and has smallest area.The size of the confidence region in the case of left,right and double censoring is compared through Monte Carlo simulation.(4)The joint confidence region of Pareto distribution parameter based on a progressive censoring sample is concerned.This dissertation starts from the progressive censoring sample and sufficient statistics,and establishes a minimum volume confidence region with non-equal tail confidence region without predetermined confidence coefficient,which can be used for complete sample,right censoring sample and progressive censoring sample.The area of two common confidence regions is optimized.The performance of confidence region under different censoring mechanisms is conducted through Monte Carlo simulation and by real examples.(5)The confidence region based on two independent Pareto samples is concerned.This dissertation establishes the minimum volume confidence region of parameters for two samples and considers the minimum volume confidence region of shape parameters,scale parameters and high dimensional vectors formed by these parameters.The critical value of the minimum volume confidence region of each case has a specific formula,which is easy to be solved.The minimum volume confidence region is compared by numerical simulation and example,which can verifythe effectiveness of the method. |
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