Generalised Elliptically Contoured Distributions And Their Properties

Some generalized elliptical matrix distributions are defined and their properties about non-singular matrix transformation are studied.The tail dependence properties are studied exten-sively for many multivariate generalized t distributions family.In this thesis it is taken up a revision and characterization of the class of absolutely continuous elliptical distributions based on the left-spherical distribution.In chapter 2 the property of the left-spherical distribution under affine transformation is considered.Then random matrix F and t are defined under the same stochastic representations in their definition under the left-spherical matrix distribution.The probability density functions are derived by using the relation between the distribution and the moments of any Borel function.And we get that the new random matrixes have the same probability density functions as defined by the spherical distribution.And prove that the probability functions have no relation with the density functions generating them.These invariant properties not only extent the matrix F and t distribution to an enlarging scope,but also ensure that the matrix Beta,Inverse Beta,Dirichlet,Inverse Dirichlet,F and t distributions are all invariant about any nonsingular transformation.In chapter 3 some multivariate generalized t distributions defined by the scale mixture of the multivariate spherical distribution and the inverse generalized gamma distribution.The upper orthant tail dependence and the extremal tail dependence index for these multivariate generalized t distributions are deduced by calculating the corresponding probability.The explicit representations of the tail dependence parameters are represented under the correlation structure based on their copula property,which are more simple than those obtained in former research about the tail dependence indices of the multivariate t distributions.And the local monotonicity of these indexes about the tail index and the linear correlation coefficient are discussed.For the multivariate generalized t distributions defined by the inverse gamma distribution and the mul-tivariate normal,we get a sufficient condition for the monotone property of the tail dependence indices.and get that the upper extremal dependence index is increasing about the correlation coefficient,but the monotonicity of upper orthant tail dependence index is complex,the corre-sponding results is demonstrated in the paper.Some simulations axe performed by Monte Carlo method to verify the obtained results,which are found to be satisfied.Furthermore,For the multivariate generalized t distributions generated from the the multivariate power exponential distribution and the inverse generalized gamma distribution,the corresponding tail dependence parameters are presented in simple formulas.The monotonicity of them are investigated.Some simulation are performed to illustrate the results.In chapter 4 a new family of multivariate generalized t distributions is defined via the scale mixture of the regularly varying distribution and the multivariate spherical distribution.This distribution family may has a fatter or a thinner tail than the general elliptical distribution.And it is shown that all the multivariate generalized t distributions defined in chapter 3 are included in this family.So,the new family may be regarded as a useful extension of the multivariate generalized t distributions,we call it as" regularly varying scale mixture multivariate generalized t distributions".We use tail dependence functions to study tail dependence for this multivariate generalized t distribution.First,tail dependence functions are deduced through the intensity measure.Then,the relation between the tail dependence function and the intensity measure is established:they are bi-uniquely determined.Finally,we obtain the expressions of the tail dependence parameters based on the expectation of the marginal components of the spherical distribution random vector in the stochastic representation.These expressions are coincided with those obtained via the conditional probability.Some simulation examples are demonstrated to verify the results we established in this paper.