Sequential Maximum Likelihood Estimation in Nonlinear Nonmarkov Diffusion Type Processes

Abstract

ABSTRACT:We obtain the strong consistency, uniform asymptotic normality and local asymptotic minimaxity (in the Hajek-LeCam sense) of the two stage sequential maximum likelihood estimator of a parameter appearing nonlinearly in the drift coefficient of a stochastic differential equation when the corresponding non-Markov diffusion type process is observed until the observed Fisher information of the process exceeds a predetermined level of precision and the level becomes large. Main results are illustrated by the exponential memory non-Markovian Ornstein-Uhlenbeck process.

On the Kolmogorov Distance for the Maximum Likelihood Estimator in the Explosive Ornstein-Uhlenbeck Process

“…With θ = 0, the process is nonstationary and satisfies the LABF (local asymptotic Brownian functional) property. For all θ ∈ R, the model satisfies the LABF property, see Bishwal [9] for the definitions of these LAN, LAMN and LABF properties. Bishwal [9] has shown that sequential sampling based on a stopping rule unifies the three properties and makes them LAN.…”

Parameter Estimation for SPDEs Driven by Cylindrical Stable Processes

“…While in Gaussian OU case, for different parts θ > 0, θ < 0 and θ = 0, LAN, LAMN and LABF hold respectively (see Bishwal [11]), in stable case entirely different phenomena occur.…”