A multivariate generalized C_p and surface estimation

We propose a new multivariate generalized C_p (MGC_p) criterion for tuning parameter selection in nonparametric regression, applicable when there are multiple covariates whose values may be irregularly spaced. Apart from an asymptotically negligible remainder, the MGC_p criterion has expected value equal to the sum of squared errors of a fitted derivative (rather than of a fitted mean response). Thus, unlike traditional criteria for tuning parameter selection, MGC_p is not prone to undersmoothed derivative estimation. We illustrate a scientific application in a case study that explores the relationship among three measures of liver function. Since recent technological developments hold promise for assessing two of these measures outside of medical and laboratory facilities, better understanding of the aforementioned relationship may allow enhanced monitoring of liver function, especially in developing countries and among persons for whom access to medical and laboratory facilities is limited.