Adaptive Estimation in Elliptical Distributions with Extensions to High Dimensions

The goal of this paper is to propose efficient and adaptive regularized estimators for the nonparametric component, mean and covariance matrix in both high and fixed dimensional situations. Although, semiparametric estimation of elliptical distribution has also been discussed in [8], we wish to expand the model in two ways. First, study adaptive estimation methods with a novel scheme of estimating the nonparametric component and second, we perform regularized estimation of Euclidean parameters of the elliptical distribution such that high dimensional inference of the Euclidean parameters under certain additional structural assumption can be carried out. Some methods have already been developed. But we extend the work in [5] [6] [10] [18] [19]. The estimate of elliptic densities can also be used to approximately estimate certain sub-class of log-concave densities by using results from convex geometry. The problem of estimation of mixture of elliptical distributions is also important in clustering, as the level sets produce disjoint elliptical components, which can be viewed as model of clusters of specific shape high dimensional space. The regularized estimation of mixture of elliptical distributions will also lead to an algorithm for finding elliptical clusters in high dimensional space under highly relaxed tail conditions.