Kernel density estimation for directional-linear data

TL;DR

This paper introduces a new nonparametric kernel density estimator tailored for data with both directional and linear components, providing theoretical properties and explicit formulas for performance assessment.

Contribution

It presents a novel product kernel estimator for directional-linear data, with derived bias, variance, MISE expressions, and asymptotic normality results.

Findings

Explicit MISE formulas for specific distributions

Comparison between exact and asymptotic MISE

Bootstrap MISE expression derived

Abstract

A nonparametric kernel density estimator for directional-linear data is introduced. The proposal is based on a product kernel accounting for the different nature of both (directional and linear) components of the random vector. Expressions for bias, variance and Mean Integrated Squared Error (MISE) are derived, jointly with an asymptotic normality result for the proposed estimator. For some particular distributions, an explicit formula for the MISE is obtained and compared with its asymptotic version, both for directional and directional-linear kernel density estimators. In this same setting a closed expression for the bootstrap MISE is also derived.