Selection of error probability laws by generalized modified profile likelihood

Although error probability law selection of models of location–scale forms is of importance in some sense, the commonly used model selection procedures, such as AIC and BIC, do not apply to it. By treating error probability law as a “parameter” of interest, location and scale as nuisance parameters, this paper proposes that generalized modified profile likelihood (GMPL), considered as a quasi-likelihood function of error probability law, be used to select the error probability laws. The GMPL method achieves minimax rate optimality and proves to be consistent. Simulations show its good performance for finite and even small samples. Note that it is straightforward to generalize the GMPL of location–scale models to various models of location–scale forms particularly including the various linear regression models and their variations, to select their error probability laws. The author believes that GMPL and its variations would be quite promising for various model selection problems.