Some results on parameter estimation in linear models with prior information (Q1309129)

Consider the linear model \(Y= X\beta+e\) subject to stochastic prior information about \(\beta\) by means of \(U= H\beta+ \varepsilon\). Two corollaries and three theorems are given for various estimations of \(\beta\) under the distribution assumption \[ {e \choose \varepsilon} \sim N \Biggl(0, \biggl( {{I\sigma^ 2} \atop 0} {0\atop W} \biggl)\Biggl), \] namely under different matrix loss functions \((d-\beta) (d- \beta)'\). One basic result is that a complete sufficient statistic does not exist. Under the loss function \(\sigma^{-2} (d- \beta)' (d-\beta)\), \(b:= (X' X)^{-1} X' Y\) is a minimax estimator of \(\beta\).