On A Multivariate Gamma Distribution And Parameter Estimation

For distribution theory of multivariate statistical analysis, multivariate normaldistribution theory has a relatively complete system. However, Pearson, America’smost famous statistician researched the random variables in the field of physics,biology and economy, found many special continuous form of the random variables,these variables do not obey the normal distribution, but to gamma distribution.Therefore, to study the distribution of the multivariate gamma distribution became anew research field in the multivariate statistical analysis. Multivariate gammadistribution is more complicated then multivariate normal distribution, but also hassome special properties.In multivariate gamma distribution, we first to study some special properties ofthe three parameters of the gamma distribution, and generalized it to the multivariategamma distribution through the total composition method. The parameter estimationis an important field of study to research the distribution, and the maximum likelihoodmethod is more commonly used method in parameter estimation. However, whenestimate the parameters of the multivariate gamma distribution in using the maximumlikelihood method, will encounter difficult problems, such as calculation, so here weuse the idea of EM algorithm, and at the same time through some change of thealgorithm, obtained a relatively simple conclusion. We used the MATLAB softwarewhich commonly used in mathematics in the conclusion, analysis the data, and get agood result.At present, the domestic and foreign literature has some special density functionis given in the multivariate gamma distribution, for estimation the parameters of thegamma distribution has a lot of research, but is less for estimate the parameters ofmultivariate gamma distribution. This article study the multivariate gammadistribution from the following several parts.The first chapter mainly introduces the current research achievements in thefield of multivariate gamma distribution, the problem which is put forward, as well asthis, and introduced some relative knowledge which is needed. The second chapter, given the gamma distribution model which includes threeparameters, and its related properties, research the relationship between the distancesof the two gamma distribution.The third chapter, from the form of gamma distribution, we construct a specialmultivariate gamma distribution. By the density function of multivariate gammadistribution, we estimate the parameters of multivariate gamma using the deformationof the EM algorithm, finally get the conclusion through the simulation.The fourth chapter, specialize some variables of model, gets a specialmultivariate gamma distribution, and studies the properties of conditional density andconditional expectation.