Minimax prediction in the linear model with a relative squared error

Arnold and Stahlecker (Stat Pap 44:107–115, 2003) considered the prediction of future values of the dependent variable in the linear regression model with a relative squared error and deterministic disturbances. They found an explicit form for a minimax linear affine solution d* of that problem. In the paper we generalize this result proving that the decision rule d* is also minimax when the class D{\mathcal{D}} of possible predictors of the dependent variable is unrestricted. Then we show that d* remains minimax in D{\mathcal{D}} when the disturbances are random with the mean vector zero and the known positive definite covariance matrix.