Non-parametric estimation of conditional densities: A new method

TL;DR

This paper introduces a new non-parametric method for estimating conditional densities using locally Gaussian approximations, effectively addressing the curse of dimensionality and demonstrating robustness and practical utility in time series analysis.

Contribution

The paper presents a simplified locally Gaussian-based estimator for conditional densities that is robust, computationally feasible in higher dimensions, and supported by large sample theory.

Findings

Estimator shows small approximation error in real and simulated data

Method is robust against noise from independent variables

Effective in time series analysis with dimensions up to p=6

Abstract

Let $\textbf{X} = (X_1,\ldots, X_p)$ be a stochastic vector having joint density function $f_{\textbf{X}}(x)$ with partitions $\textbf{X}_1 = (X_1,\ldots, X_k)$ and $\textbf{X}_2 = (X_{k+1},\ldots, X_p)$. A new method for estimating the conditional density function of $\textbf{X}_1$ given $\textbf{X}_2$ is presented. It is based on locally Gaussian approximations, but simplified in order to tackle the curse of dimensionality in multivariate applications, where both response and explanatory variables can be vectors. We compare our method to some available competitors, and the error of approximation is shown to be small in a series of examples using real and simulated data, and the estimator is shown to be particularly robust against noise caused by independent variables. We also present examples of practical applications of our conditional density estimator in the analysis of time series. Typical values for $k$ in our examples are 1 and 2, and we include simulation experiments with values of $p$ up to 6. Large sample theory is established under a strong mixing condition.