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The asymptotic optimality (a.o.) of various hyper-parameter estimators withdifferent optimality criteria has been studied in the literature forregularized least squares regression problems. The estimators include e.g., themaximum (marginal) likelihood method, $C_p$ statistics, and generalized crossvalidation method, and the optimality criteria are based on e.g., theinefficiency, the expectation inefficiency and the risk. In this paper, weconsider the regularized least squares regression problems with fixed number ofregression parameters, choose the optimality criterion based on the risk, andstudy the a.o. of several cross validation (CV) based hyper-parameterestimators including the leave $k$-out CV method, generalized CV method,$r$-fold CV method and hold out CV method. We find the former three methods canbe a.o. under mild assumptions, but not the last one, and we use Monte Carlosimulations to illustrate the efficacy of our findings.

On the Asymptotic Optimality of Cross-Validation based Hyper-parameter Estimators for Regularized Least Squares Regression Problems