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 α₀ A^{- 1}, where A denotes the sample sum of product matrix, is dominated by a better constant multiple of A^{- 1}, denoted by α₀^{{star operator}} A^{- 1}. Secondly, a new class of shrinkage estimators of Σ^{- 1} is proposed. Moreover, the risk functions of α₀ A^{- 1}, α₀^{{star operator}} A^{- 1} and the proposed estimators are explicitly derived. It is shown that the proposed estimator dominates α₀^{{star operator}} A^{- 1}, under the quadratic loss function. A simulation study is carried out which confirms these results. Improved estimator of tr (Σ^{- 1}) is also obtained.