Optimal critical values of a preliminary test for linear restrictions in a regression model with multivariate Student-\(t\) disturbances (Q1965934)

For the estimation of a vector parameter \(\beta\) in the linear regression model \(y=X\beta +e\) the ordinary least squares estimator (OLSE) can always be used, and the restricted least squares estimator (RLSE) can be used if linear restrictions (\(H_0\)): \(R\beta=r\) hold. The preliminary test estimator (PTE) uses usual F-statistics to test \(H_0\) and then uses RLSE if \(H_0\) holds and OLSE otherwise. PTE is considered in this paper for the case of multivariate Student-t distribution of \(e\). It is shown that for the risk \(\rho={\mathbf E}\|X\tilde\beta - {\mathbf E}y\|^2\) under some conditions PTE can dominate both OLSE and RLSE (this is impossible for Gaussian \(e\)). The optimal choice of critical values in the preliminary test is discussed.