Kubokawa, Tatsuya and Muni S. Srivastava

The problem of estimating the covariance matrix of normal and non-normal distributions is addressed when both the sample size and the dimension of covariance matrix tend to innity. In this paper, we consider a class of ridge-type estimators which are linear combinations of the unbiased estimator and the identity matrix multiplied by a scalor statistic, and we derive a leading term of their risk functions relative to a quadratic loss function. Within this class, we obtain the optimal ridge-type estimator by minimizing the leading term in the risk approximation. It is interesting to note that the optimal weight is based on a statistic for testing sphericity of the covariance matrix.