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

Holbach S. Local asymptotic normality for shape and periodicity of a signal in the drift of a degenerate diffusion with internal variables. ELECTRONIC JOURNAL OF STATISTICS. 2019;13(2):4884-4915.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 theta and also has an unknown periodicity T, we prove Local Asymptotic Normality (LAN) jointly in theta 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 theta 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 theta 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