A class of cross-validatory model selection criteria

In this paper, we define a class of cross-validatory model selection criteria as an estimator of the predictive risk function based on a discrepancy between a candidate model and the true model. For a vector of unknown parameters, n estimators are required for the definition of the class, where n is the sample size. The ith estimator (i = 1,..., n) is obtained by minimizing a weighted discrepancy function in which the ith observation has a weight of 1 - lambda and others have weight of 1. Cross-validatory model selection criteria in the class are specified by the individual lambda. The sample discrepancy function and the ordinary cross-validation (CV) criterion are special cases of the class. One may choose lambda to minimize the biases. The optimal lambda makes the bias-corrected CV (CCV) criterion a second-order unbiased estimator for the risk function, while the ordinary CV criterion is a first-order unbiased estimator of the risk function.