Small Confidence Sets for the Mean of a Spherically Symmetric Distribution

Abstract

Summary Suppose that X has a k-variate spherically symmetric distribution with mean vector θ and identity covariance matrix. We present two spherical confidence sets for θ, both centred at a positive part Stein estimator TS+(X). In the first, we obtain the radius by approximating the upper α-point of the sampling distribution of ‖TS+(X)−θ‖2 by the first two non-zero terms of its Taylor series about the origin. We can analyse some of the properties of this confidence set and see that it performs well in terms of coverage probability, volume and conditional behaviour. In the second method, we find the radius by using a parametric bootstrap procedure. Here, even greater improvement in terms of volume over the usual confidence set is possible, at the expense of having a less explicit radius function. A real data example is provided, and extensions to the unknown covariance matrix and elliptically symmetric cases are discussed.