Nonparametric estimation on the circle based on Fej\'er polynomials

TL;DR

This paper develops nonparametric estimators for circular data using Fejér polynomials, establishing their theoretical properties, addressing origin dependence, and extending to measurement errors with practical applications.

Contribution

It introduces new Fejér polynomial-based estimators for circular density and distribution functions, with theoretical guarantees and methods to handle measurement errors and origin dependence.

Findings

Estimators are uniformly strongly consistent.

Extensions effectively handle measurement errors.

Simulation and real data demonstrate robustness.

Abstract

This paper presents a comprehensive study of nonparametric estimation techniques on the circle using Fej\'er polynomials, which are analogues of Bernstein polynomials for periodic functions. Building upon Fej\'er's uniform approximation theorem, the paper introduces circular density and distribution function estimators based on Fej\'er kernels. It establishes their theoretical properties, including uniform strong consistency and asymptotic expansions. Since the estimation of the distribution function on the circle depends on the choice of the origin, we propose a data-dependent method to address this issue. The proposed methods are extended to account for measurement errors by incorporating classical and Berkson error models, adjusting the Fej\'er estimator to mitigate their effects. Simulation studies analyze the finite-sample performance of these estimators under various scenarios, including mixtures of circular distributions and measurement error models. An application to rainfall data demonstrates the practical application of the proposed estimators, demonstrating their robustness and effectiveness in the presence of rounding-induced Berkson errors.