The James – Stein procedure for a conditionally gaussian regression

The paper considers the problem of estimating a p-dimensional (p ≥ 2) meanvector of a multivariate conditionally normal distribution under quadratic loss. The problem ofthis type arises when estimating the parameters in a continuous time regression model with a non-Gaussian Ornstein-Uhlenbeck process. We propose a modification of the James-Stein procedureof the form θ*(Y) = (1 - c/||Y||) Y, where Y is an observation and c > 0 is a special constant. Thisestimate allows one to derive an explicit upper bound for the quadratic risk and has a significantlysmaller risk than the usual maximum likelihood estimator for the dimensions p≥2. This procedureis applied to the problem of parametric estimation in a continuous time conditionally Gaussian regressionmodel and to that of estimating the mean vector of a multivariate normal distributionwhen the covariance matrix is unknown and depends on some nuisance parameters.

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