Loss estimation for spherically symmetric distributions
Consider the general statistical decision problem of estimating a parameter \(\theta \in \mathbb{R}^p\) by some decision procedure \(\delta\) under a loss function \(L (\theta, \delta) \). Classical decision theory advocates that one should use a decision rule \(\delta^*\) if it has suitable properties with respect to the frequentist risk \(R (\theta, \delta^*)\). Given the observation \(x\), in \(\mathbb{R}^p\), it is important to calculate the loss incurred by using \(\delta^* (x)\). However, \(L(\theta, \delta^* (x))\) is not available since it depends on the parameter \(\theta\). Hence it is of interest to estimate the loss of \(\delta^* (x)\). Therefore we wish to construct a data dependent measure of the loss incurred for the particular data at hand. We consider estimating the loss incurred when using the minimax estimator of a location parameter of a spherically symmetric distribution. We first develop an unbiased estimator of the loss, which is the spherically symmetric analog of \textit{I. Johnstone}'s [S. S. Gupta and J. O. Berger (eds.), Statistical Decision Theory Relat. Topics IV, Vol. 1, 361-379 (1988; Zbl 0647.62018)] unbiaded estimator \(p\) in the \(p\)-dimensional normal distribution, with mean \(\theta\) and identity covariance matrix \((N_p (\theta,I))\). Next we construct a shrinkage-type estimator which dominates this unbiased estimator, in terms of squared-error loss. The shrinkage-type estimator will be constructed by solving a certain second order differential inequality. These results complement the James-Stein estimator of a location parameter of a spherically symmetric distribution given by \textit{A. C. Brandwein} and \textit{W. E. Strawderman} [Ann. Stat. 19, No. 3, 1639-1650 (1991; Zbl 0741.62058)].
scientific article; zbMATH DE number 769841