Piecewise linear density estimation for sampled data

TL;DR

This paper demonstrates that histograms and frequency polygons can achieve optimal $L_2$-rates for density estimation in sampled continuous-time processes, with rates depending on sampling design and path regularity.

Contribution

It establishes that nonparametric density estimators attain optimal rates under various sampling schemes for continuous-time processes.

Findings

Histograms and frequency polygons reach optimal $L_2$-rates.

Rates depend on sampling design and path regularity.

High-frequency sampling minimizes observation time.

Abstract

Nonparametric density estimation is considered for a discretely observed stationary continuous-time process. For each of three given time sampling procedures either random or deterministic, we establish that histograms and frequency polygons can reach the same optimal $L_{2}$-rates as in the independent and identically distributed case. Moreover, thanks to a suitable "high frequency" sampling design, these rates are derived together with a minimized time of observation depending on the regularity of sample paths.