TwinKernel Estimation for Point Process Intensity Functions: Adaptive Nonparametric Methods via Orbital Regularity

TL;DR

This paper introduces TwinKernel methods for nonparametric estimation of point process intensities, leveraging orbital regularity and martingale techniques to achieve adaptive, optimal convergence rates with practical applications in hazard rate estimation.

Contribution

The paper presents a novel TwinKernel framework that adaptively estimates intensity functions of point processes using orbital regularity, combining kernel transport, martingale methods, and model selection.

Findings

Achieves uniform consistency and optimal convergence rates.

Demonstrates 3-7x improvements over classical estimators.

Provides asymptotic confidence bands for hazard rates.

Abstract

We develop TwinKernel methods for nonparametric estimation of intensity functions of point processes. Building on the general TwinKernel framework and combining it with martingale techniques for counting processes, we construct estimators that adapt to orbital regularity of the intensity function. Given a point process $N$ with intensity $\lambda$ and a cyclic group $G = \langle\varphi\rangle$ acting on the time/space domain, we transport kernels along group orbits to create a hierarchy of smoothed Nelson-Aalen type estimators. Our main results establish: (i) uniform consistency via martingale concentration inequalities; (ii) optimal convergence rates for intensities in twin-H\"older classes, with rates depending on the effective dimension $d_{\mathrm{eff}}$; (iii) adaptation to unknown smoothness through penalized model selection; (iv) automatic boundary bias correction via local polynomial extensions in twin coordinates; (v) minimax lower bounds showing rate optimality. We apply the methodology to hazard rate estimation under random censoring, where periodicity or other orbital structure in the hazard may arise from circadian rhythms, seasonal effects, or treatment schedules. Martingale central limit theorems yield asymptotic confidence bands. Simulation studies demonstrate 3--7$\times$ improvements over classical kernel hazard estimators when the intensity exhibits orbital regularity.