The Priestley-Chao Estimator of Conditional Density with Uniformly Distributed Random Design

Abstract

The present paper is focused on non-parametric estimation of conditional density. Conditional density can be regarded as a generalization of regression thus the kernel estimator of conditional density can be derived from the kernel estimator of the regression function. We concentrate on the Priestley-Chao estimator of conditional density with a random design presented by a uniformly distributed unconditional variable. The statistical properties of such an estimator are given. As the smoothing parameters have the most significant influence on the quality of the final estimate, the leave-one-out maximum likelihood method is proposed for their detection. Its performance is compared with the cross-validation method and with two alternatives of the reference rule method. The theoretical part is complemented by a simulation study.The present paper is focused on non-parametric estimation of conditional density. Conditional density can be regarded as a generalization of regression thus the kernel estimator of conditional density can be derived from the kernel estimator of the regression function. We concentrate on the Priestley-Chao estimator of conditional density with a random design presented by a uniformly distributed unconditional variable. The statistical properties of such an estimator are given. As the smoothing parameters have the most significant influence on the quality of the final estimate, the leave-one-out maximum likelihood method is proposed for their detection. Its performance is compared with the cross-validation method and with two alternatives of the reference rule method. The theoretical part is complemented by a simulation study.