Wavelet-based estimation of power densities of size-biased data

TL;DR

This paper introduces a wavelet-based approach for density estimation of size-biased data, employing warped wavelet bases and estimating powers of the density to achieve more flexible and accurate results.

Contribution

It develops a novel wavelet method for size-biased density estimation using warped bases and power transformations, demonstrating theoretical consistency and practical advantages.

Findings

Warped wavelet estimators are consistent with optimal rates.

Estimating powers of the density improves estimation accuracy.

The method outperforms traditional approaches in simulations and real data.

Abstract

We propose a new wavelet-based method for density estimation when the data are size-biased. More specifically, we consider a power of the density of interest, where this power exceeds 1/2. Warped wavelet bases are employed, where warping is attained by some continuous cumulative distribution function. A special case is the conventional orthonormal wavelet estimation, where the warping distribution is the standard continuous uniform. We show that both linear and nonlinear wavelet estimators are consistent, with optimal and/or near-optimal rates. Monte Carlo simulations are performed to compare four special settings which are easy to interpret in practice. An application with a real dataset on fatal traffic accidents involving alcohol illustrates the method. We observe that warped bases provide more flexible and superior estimates for both simulated and real data. Moreover, we find that estimating the power of a density (for instance, its square root) further improves the results.