Positive data kernel density estimation via the logKDE package for R

TL;DR

This paper introduces a log-transformation based kernel density estimation method for positive data, providing theoretical properties, bandwidth selection rules, and an R package implementation, improving density estimation accuracy for positive datasets.

Contribution

The paper develops a novel log-KDE approach for positive data, including theoretical bias, variance, MSE analysis, and a practical R package implementation.

Findings

Derived bias, variance, and MSE expressions for log-KDEs

Provided asymptotic MISE results and bandwidth selection rule

Demonstrated effectiveness through real data case studies

Abstract

Kernel density estimators (KDEs) are ubiquitous tools for nonparametric estimation of probability density functions (PDFs), when data are obtained from unknown data generating processes. The KDEs that are typically available in software packages are defined, and designed, to estimate real-valued data. When applied to positive data, these typical KDEs do not yield bona fide PDFs. A log-transformation methodology can be applied to produce a nonparametric estimator that is appropriate and yields proper PDFs over positive supports. We call the KDEs obtained via this transformation log-KDEs. We derive expressions for the pointwise biases, variances, and mean-squared errors of the log- KDEs that are obtained via various kernel functions. Mean integrated squared error (MISE) and asymptotic MISE results are also provided and a plug-in rule for log-KDE bandwidths is derived. We demonstrate the log-KDEs methodology via our R package, logKDE. Real data case studies are provided to demonstrate the log-KDE approach.