Advances and Challenges in Inferences for Elliptically Contoured t Distributions

When a multivariate elliptical such as t response is taken from each of n individuals, the inference for the parameters of the t distribution including the location (or regression effects), scale and degrees of freedom (or shape) depends on the assumption whether n multi-dimensional responses are independent or uncorrelated but dependent. In the former case, that is, when responses are independent, the exact sampling theory based inference is extremely complicated, whereas in the later case the derivation of the exact sampling distributions for the standard statistics is manageable but the estimators based on certain standard statistics such as sample covariance matrix may be inconsistent for the respective parameters. In this paper we provide a detailed discussion on the advances and challenges in inferences using uncorrelated but dependent t samples. We then propose a clustered regression model where the multivariate t responses in the cluster are uncorrelated but such clustered responses are taken from a large number of independent individuals. The inference including the consistent estimation of the parameters of this proposed model is also presented.