Local Asymptotic Normality for Shape and Periodicity in the Drift of a Time Inhomogeneous Diffusion

TL;DR

This paper establishes Local Asymptotic Normality for a diffusion process with unknown periodicity and shape, providing a foundation for efficient statistical inference on these parameters from continuous data.

Contribution

It proves LAN jointly for shape and periodicity parameters in a diffusion with periodic drift, extending existing results to the case where both are unknown.

Findings

LAN holds jointly for shape and periodicity parameters.

Local scales are $n^{-1/2}$ for shape and $n^{-3/2}$ for periodicity.

Generalizes known LAN results to more complex diffusion models.

Abstract

We consider a one-dimensional diffusion whose drift contains a deterministic periodic signal with unknown periodicity $T$ and carrying some unknown $d$-dimensional shape parameter $\theta$. We prove Local Asymptotic Normality (LAN) jointly in $\theta$ and $T$ for the statistical experiment arising from continuous observation of this diffusion. The local scale turns out to be $n^{-1/2}$ for the shape parameter and $n^{-3/2}$ for the periodicity which generalizes known results about LAN when either $\theta$ or $T$ is assumed to be known.