Lindley Type Estimators When the Norm is Restricted to an Interval

Consider the problem of estimating a $p{\times}1$ mean vector $\theta(p\geq4)$ under the quadratic loss, based on a sample $X_1$, $X_2$, $\cdots$, $X_n$. We find a Lindley type decision rule which shrinks the usual one toward the mean of observations when the underlying distribution is that of a variance mixture of normals and when the norm $\parallel\;{\theta}-\bar{{\theta}}1\;{\parallel}$ is restricted to a known interval, where $bar{{\theta}}=\frac{1}{p}\;\sum\limits_{i=1}^{p}{\theta}_i$ and 1 is the column vector of ones. In this case, we characterize a minimal complete class within the class of Lindley type decision rules. We also characterize the subclass of Lindley type decision rules that dominate the sample mean.