The Statistical Analysis Of Pareto Distribution

Many investigators pay more and more attention to Pareto Distribution for its good qualities. In this thesis some qualities and applications of Pareto distribution will be studied on the base of much other relative research. Furthermore, some available methods are given, which are proved by examples or simulations.In chapter 1, we give a detailed description of Pareto Distribution on its research foundation, present research works and its main qualities. We make a special effort to introduce how to estimate the unknown parameters (6, a) based on order samples. Arnold B.C. (1983) has given their least squares estimate (LSE), maximum likelihood estimate (MLE), uniformly minimum variance unbiased estimate (UMVUE), and so on, based on ordinary samples. In this thesis, we deduce two methods of estimation based on censored samples especially type II, one is uniformly minimum variance unbiased estimate (UMVUE), another is best linear unbiased estimate (BLUE) based on Two-parameter Exponential Distribution. The coefficient table of BLUE and its variance, covariance (see attached table 1) are given. And then we compare the degree of accuracy between the two methods of estimation. The UMVUE has high degree of accuracy and it is simple and convenient to get; the BLUE also has high degree of accuracy, but we need to translate the original data to its logarithm form and look up the coefficients from relative table to get it. We still think the BLUE has its applied value in life testing research because of its linear form.In chapter 2, we study several testing methods of Pareto Distribution such as probability chart method, correlation coefficient R2 method and x2 method based on Two-parameter Exponential Distribution. In order to use at the testing ground, we design a special Pareto Probability Paper. According to the thought of W testing statistics for Normal Distribution, we deduce a R2 testing statistics for Pareto Distribution. But we can hardly get the accuracy probability distribution of this statistics, so we compute the R2 testing quantile by simulations (see attached table2). Furthermore, we study the testing power of R2 testing towards several Non-pareto Distributions and it is proved to be available. And then we compare the two testing methods, R2 and x2, with their capacity of discernment towards several Non-pareto Distributions. At last we give an example to these methods.In chapter 2, we try to lead Pareto Distribution into the research of accelerated life testing and completely discuss the statistical inference in constant-stress accelerated life testing on Pareto Distribution case under type I and type II censoring.First, we introduce the arrangements of constant-stress accelerated life testing and some basic hypothesis. We deduce a statistics based on Bartlett testing statistics to test the hypothesis that all shape parameters are equal. And we give the statistics a refusal area and study its effect by simulations.Under type II censoring, we make a special effort to estimate the common shape parameter a and the coefficients a, b in accelerated equation. Using the thought of MLE, we can deduce the UMVUE of the common a. There are two kinds of estimate of the coefficients a, b in accelerated equation, LSE and BLUE. We simulate order samples at an ideal case and compare the accuracy of these two estimates. We can say they are all available from the simulation.Under type I censoring, we make a special effort to find good estimate to the unknown parameters. According to some results of Two-parameter Exponential Distribution, we deduce approximate unbiased estimate (AUE) of the parameters on Pareto Distribution case under type I censoring. We compute the most proper iterative times N for each n, r, and give the coefficients of AUE and variance (see attached table 3). Through the simulation it seems that AUE has good quality and it is a very good estimate under type I censoring.