This article gives necessary and sufficient conditions for the equivalence of the simple linear unbiased estimator (SLUE) and the best linear unbiased estimator (BLUE) of the parameters when there arekindependent sources of Type II censored data from a location scale distribution in which the location parameter is an unknown linear function of certain given independent variables and the scale parameter is an unknown constant. It also shows that the BLUE can be obtained in a two-stage procedure that has the same model-checking features and computational simplicity of the SLUE.