Spline Smoothing for Estimation of Circular Probability Distributions via Spectral Isomorphism and its Spatial Adaptation

TL;DR

This paper introduces a Fourier Spline method for non-parametric density estimation on the circle, effectively capturing local features like discontinuities and edges, and handling outliers, with theoretical results, simulations, and real data application.

Contribution

It presents a novel Fourier Spline technique for circular density estimation that incorporates local features and outliers, improving over traditional kernel methods.

Findings

The method accurately detects local features such as discontinuities.

Simulation studies demonstrate superior performance over existing methods.

Real data example validates practical applicability.

Abstract

Consider the problem when $X_1,X_2,..., X_n$ are distributed on a circle following an unknown distribution $F$ on $S^1$. In this article we have consider the absolute general set-up where the density can have local features such as discontinuities and edges. Furthermore, there can be outlying data which can follow some discrete distributions. The traditional Kernel Density Estimation methods fail to identify such local features in the data. Here we device a non-parametric density estimate on $S^1$, by the use of a novel technique which we term as Fourier Spline. We have also tried to identify and incorporate local features such as support, discontinuity or edges in the final density estimate. Several new results are proved in this regard. Simulation studies have also been performed to see how our methodology works. Finally a real life example is also shown.