On the residual dependence index of elliptical distributions

The residual dependence index of bivariate Gaussian distributions is determined by the correlation coefficient. This tail index is of certain statistical importance when extremes and related rare events of bivariate samples with asymptotic independent components are being modeled. In this paper we calculate the partial residual dependence indices of a multivariate elliptical random vector assuming that the associated random radius is in the Gumbel max-domain of attraction. Furthermore, we discuss the estimation of these indices when the associated random radius possesses a Weibull-tail distribution.Comment: 11 pages, case \theta=1 now include

Asymptotics for Kotz Type III Elliptical Distributions

In this paper we derive the tail asymptotics of a Kotz Type III elliptical random vector. As an application of our asymptotic expansion we derive an approximation for the conditional excess distribution. Furthermore, we discuss the asymptotic dependence of Kotz Type III triangular arrays and provide some details on the estimation of conditional excess distribution and survivor function.Comment: 10 page

Optimal Linear Shrinkage Estimator for Large Dimensional Precision Matrix

In this work we construct an optimal shrinkage estimator for the precision matrix in high dimensions. We consider the general asymptotics when the number of variables $p\rightarrow\infty$ and the sample size $n\rightarrow\infty$ so that $p/n\rightarrow c\in (0, +\infty)$. The precision matrix is estimated directly, without inverting the corresponding estimator for the covariance matrix. The recent results from the random matrix theory allow us to find the asymptotic deterministic equivalents of the optimal shrinkage intensities and estimate them consistently. The resulting distribution-free estimator has almost surely the minimum Frobenius loss. Additionally, we prove that the Frobenius norms of the inverse and of the pseudo-inverse sample covariance matrices tend almost surely to deterministic quantities and estimate them consistently. At the end, a simulation is provided where the suggested estimator is compared with the estimators for the precision matrix proposed in the literature. The optimal shrinkage estimator shows significant improvement and robustness even for non-normally distributed data.Comment: 26 pages, 5 figures. This version includes the case c>1 with the generalized inverse of the sample covariance matrix. The abstract was updated accordingl

Estimation of the precision matrix of multivariate Kotz type model

In this paper, the problem of estimating the precision matrix of a multivariate Kotz type model is considered. First, using the quadratic loss function, we prove that the unbiased estimator , where denotes the sample sum of product matrix, is dominated by a better constant multiple of , denoted by . Secondly, a new class of shrinkage estimators of is proposed. Moreover, the risk functions of , and the proposed estimators are explicitly derived. It is shown that the proposed estimator dominates , under the quadratic loss function. A simulation study is carried out which confirms these results. Improved estimator of is also obtained.primary, 62H12 secondary, 62C15 Multivariate Kotz type model Estimation of the precision matrix Quadratic loss Decision theoretic estimation