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We research adaptive maximum likelihood-type estimation for an ergodicdiffusion process where the observation is contaminated by noise. Thismethodology leads to the asymptotic independence of the estimators for thevariance of observation noise, the diffusion parameter and the drift one of thelatent diffusion process. Moreover, it can lessen the computational burdencompared to simultaneous maximum likelihood-type estimation. In addition toadaptive estimation, we propose a test to see if noise exists or not, andanalyse real data as the example such that data contains observation noise withstatistical significance.

Adaptive estimation and noise detection for an ergodic diffusion with observation noises