The James–Stein procedure for a conditionally Gaussian regression

The paper considers the problem of estimating a $p$-dimensional ($p\ge2$) mean vector of a multivariate conditionally normal distribution under quadratic loss. The problem of this 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 procedure of the form $\theta^*(Y)=(1-c/\|Y\|)Y$, where $Y$ is an observation and $c>0$ is a special constant. This estimate allows one to derive an explicit upper bound for the quadratic risk and has a significantly smaller risk than the usual maximum likelihood estimator for the dimensions $p\ge2$. This procedure is applied to the problem of parametric estimation in a continuous time conditionally Gaussian regression model and to that of estimating the mean vector of a multivariate normal distribution when the covariance matrix is unknown and depends on some nuisance parameters.