Maximum Likelihood Estimation Of The Parameters Of The Multivariate Kotz-type Distribution Under The Simple Partly Order Restrictions

Kotz type distribution is an important kind of elliptically symmetric distributions, It wasintroduced by Samuel Kotz as a generalization of the multivariate normal distribution. Since1990there has been a surge of activity relating to this distribution, it has been studied bymany scholars. Kotz type distribution constructs many models to which the usual normalityassumption is not applicable. In addition, Kotz type distribution also has an extensiveapplication in the fields such as economic mathematics, repeated measurement, and so on.In statistical analysis, statistical deduce under order restriction becomes an importantconstituent part because of it’s profound application background. Multivariate isotonicregression theory plays a key role in the field of statistical inference under order restriction formultivariate parameters.This paper firstly using combined the theory of elliptical distributions and the theory of thematrix Kotz distribution, the maximum likelihood estimations of the means vector of themultivariate Kotz-type distribution Kp(μ,Σ), under the certain conditions are given, accordingto the means vector known or unknown, the maximum likelihood estimations of thecovariance matrixes of the distribution are studied in two case respectively. Using the theoryand the method of the multivariate isotonic regression, according to the covariance matrixesof multivariate Kotz-type distribution Kxp(θ,Σ), is different, the problem of the algorithm ofthe maximum likelihood estimation of a set of means vector of the distribution under thesimple order restriction are further studied in three case respectively, consider the propertiesof the distribution, when the covariance matrixes are known or unknown and equal, themultivariate isotonic regression can be obtained easily by applying the methods of computingunivariate isotonic regression and use the PAVA. When the covariance matrixes are unknownand unequal, the multivariate isotonic regression can be obtained by applying the methods ofcomputing univariate isotonic regression and giving the initial value of parameters and get theMLE by the use of iterative algorithm repeatedly. Finally, the maximum likelihoodestimations of parameters under the order restrictions are obtained.