Stabilized Cross-Validation of Smoothness in Density Deconvolution

Abstract

We consider density estimation under measurement error with the Smoothness-Penalized Deconvolution (SPeD) estimator. The estimator has a tuning parameter regulating the smoothness of the estimate, and proper choice of this parameter is critical for forming good estimates. We derive the cross-validation choice of tuning parameter for the SPeD estimator, but it performs very poorly. We introduce a stabilized cross-validation (SCV) criterion which unbiasedly estimates the mean integrated squared error (MISE) for a smaller sample size, and use asymptotic arguments to obtain an appropriate tuning parameter from this stabilized criterion. We show that the SCV is a strongly consistent estimator of the MISE, and that it is the minimum variance unbiased estimator of the MISE. In a simulation study, we show that the SCV approach outperforms the previously recommended choice of tuning parameter in nearly all settings, and in a majority of the settings, SPeD with the SCV outperforms the classic deconvoluting kernel estimator with its recommended choice of tuning parameter.