Estimation Of The Parameters Of The Generalized Gamma Distribution

As a representative statistical model, generalized Gamma distribution has the characte-ristic of diversity. Some important and widely used distributions, such as Weibull distribu-tion, Rayleigh distribution and Gamma distribution, are the special cases of the distribution under some conditions. Besides, generalized Gamma distribution is flexible to describe the data obeying Rayleigh distribution and the real data with a severe tail. Therefore, this dis-tribution focus more and more attention. However, besides the diversity of the form and the flexibility of its description the parameters bring, they also bring the problem of estimation to us. Though, there have been some methods to estimate the parameters of generalized Gamma distribution, such as maximum likelihood method, method of moment, method of logarithmic cumulants and scale-independent shape estimation. However, we maybe face some problems when using these methods. Therefore, it’s necessary to study the methods of estimating these parameters of generalized Gamma distribution. In this thesis, we’ll give two methods to estimate the parameters of the distribution.The first one is based on the thought of percentile. According to it we establish two kinds of models:percentile point estimation and maximum cumulative likelihood method. Percentile point estimation is discussed from two aspects:the percentile of the sample and the percentile of probability. By searching the most suitable quantile number of sample and probability, we can obtain the best estimated values of parameters. Then compared the K-S distances gotten from Matlab simulation under the two methods, we find that the method of probability percentile performs better under different numbers of samples, while the method of sample percentile is invalid when the sample size is small. Another model based on per-centile is established under the relationship between the likelihood function and the cumula-tive function. Just as maximum likelihood method, we can estimate the parameters by max-imum the cumulative function. The maximum value of cumulative function is estimated by sample percentile, based on which we can estimate these parameters. Simulation results show that this method perform well even under the condition of a small sample size.Another one is based on the Bayesian statistics. Using this method, we should take prior information of parameters into consideration, as well as general information and sample information. So at first, we discuss the prior information of three parameters. Based on Jef-freys prior and subjective experience prior, two kinds of parameters’ prior distributions are get. Then we can obtain posterior distributions according to the prior. Using M-H algorithm and adaptive rejection sampling method respectively to sample the samples of parameters. The average values of the samples we get are what we want. Simulation results from Mat-lab show this method performs well when the sample size is large, In addition, we don’t need to solve equations to estimate parameters when using this method.