Best linear equivariant estimation and prediction in location-scale families

In this paper, we derive representations of the best (affine) linear equivariant estimators (BLEEs) of the location and scale parameters in the location, scale, and location-scale set-ups. Furthermore, we show that the best linear equivariant predictor (BLEP) with respect to standardized mean squared error has a representation similar to the well known best linear unbiased predictor (BLUP). By analogy with the BLUP which depends on the best linear unbiased estimators (BLUEs) of the location and scale parameters, the BLEP depends on the corresponding BLEEs. Finally, we establish a relation between BLEEs based on the original sample X and the extended sample X0. As a byproduct, the result shows that the BLEEs remain unchanged when future observations are replaced by its BLEPs. The same property was established earlier for BLUEs and BLUPs by Doganaksoy and Balakrishnan (1997). © 2008, Indian Statistical Institute.