Dirichlet kernel density estimation for strongly mixing sequences on the simplex

TL;DR

This paper extends Dirichlet kernel density estimation to strongly mixing time-dependent compositional data on the simplex, providing theoretical guarantees and demonstrating practical utility with vehicle market-share data.

Contribution

It introduces the first theoretical analysis of Dirichlet kernel estimators for strongly mixing sequences, including asymptotic properties and error bounds.

Findings

Establishes asymptotic normality for the estimators.

Demonstrates effective application to real-world vehicle data.

Shows robustness of Dirichlet kernels with dependent data.

Abstract

This paper investigates the theoretical properties of Dirichlet kernel density estimators for compositional data supported on simplices, for the first time addressing scenarios involving time-dependent observations characterized by strong mixing conditions. We establish rigorous results for the asymptotic normality and mean squared error of these estimators, extending previous findings from the independent and identically distributed (iid) context to the more general setting of strongly mixing processes. To demonstrate its practical utility, the estimator is applied to monthly market-share compositions of several Renault vehicle classes over a twelve-year period, with bandwidth selection performed via leave-one-out least squares cross-validation. Our findings underscore the reliability and strength of Dirichlet kernel techniques when applied to temporally dependent compositional data.