Asymptotic analysis of the jittering kernel density estimator

TL;DR

This paper provides a comprehensive asymptotic analysis of the jittering kernel density estimator for mixed data, demonstrating its strong consistency, asymptotic normality, and minimax-optimal convergence rates.

Contribution

It offers the first detailed theoretical analysis of jittering kernel density estimators, establishing their statistical properties and finite sample performance.

Findings

Estimator is strongly consistent and asymptotically normal.

Achieves minimax-optimal convergence rates.

Finite sample simulations show competitive performance.

Abstract

Jittering estimators are nonparametric function estimators for mixed data. They extend arbitrary estimators from the continuous setting by adding random noise to discrete variables. We give an in-depth analysis of the jittering kernel density estimator, which reveals several appealing properties. The estimator is strongly consistent, asymptotically normal, and unbiased for discrete variables. It converges at minimax-optimal rates, which are established as a by-product of our analysis. To understand the effect of adding noise, we further study its asymptotic efficiency and finite sample bias in the univariate discrete case. Simulations show that the estimator is competitive on finite samples. The analysis suggests that similar properties can be expected for other jittering estimators.