Adaptive inference for small diffusion processes based on sampled data

We consider parametric estimation and tests for multi-dimensional diffusion processes with a small dispersion parameter $$\varepsilon $$ ε from discrete observations. For parametric estimation of diffusion processes, the main target is to estimate the drift parameter and the diffusion parameter. In this paper, we propose two types of adaptive estimators for both parameters and show their asymptotic properties under $$\varepsilon \rightarrow 0$$ ε → 0 , $$n\rightarrow \infty $$ n → ∞ and the balance condition that $$(\varepsilon n^\rho )^{-1} =O(1)$$ ( ε n ρ ) - 1 = O ( 1 ) for some $$\rho >0$$ ρ > 0 . Using these adaptive estimators, we also introduce consistent adaptive testing methods and prove that test statistics for adaptive tests have asymptotic distributions under null hypothesis. In simulation studies, we examine and compare asymptotic behaviors of the two kinds of adaptive estimators and test statistics. Moreover, we treat the SIR model which describes a simple epidemic spread for a biological application.