Local asymptotic normality for shape and periodicity of a signal in the drift of a degenerate diffusion with internal variables

Abstract

Taking a multidimensional time-homogeneous dynamical system and adding a randomly perturbed time-dependent deterministic signal to some of its components gives rise to a high-dimensional system of stochastic differential equations which is driven by possibly very low-dimensional noise. Equations of this type are commonly used in biology for modeling neurons or in statistical mechanics for certain Hamiltonian systems. Assuming that the signal depends on an unknown shape parameter $\vartheta $ and also has an unknown periodicity $T$, we prove Local Asymptotic Normality (LAN) jointly in $\vartheta $ and $T$ for the statistical experiment arising from (partial) observation of this diffusion in continuous time. The local scale turns out to be $n^{-1/2}$ for $\vartheta $ and $n^{-3/2}$ for $T$. Our approach unifies and extends existing results on LAN in variants of the signal in noise model where the parameters $\vartheta $ and $T$ are treated separately. Consequently, we can establish the same efficiency bounds in our more complex model and make use of efficient estimators known from these submodels.