On Misspecifications in Regularity and Properties of Estimators

TL;DR

This paper investigates how misspecifying regularity conditions, such as using a change-point model instead of a smooth signal, affects the asymptotic properties of maximum likelihood estimators in Gaussian noise.

Contribution

It analyzes the impact of model misspecification on the asymptotic behavior of estimators in continuous-time Gaussian noise models.

Findings

Misspecification can lead to biased or inconsistent estimators.

Discontinuous models may not capture true signal properties.

Asymptotic properties are sensitive to regularity assumptions.

Abstract

The problem of parameter estimation by the continuous time observations of a deterministic signal in white gaussian noise is considered. The asymptotic properties of the maximul likelihood estimator are described in the asymptotics of small noise (large siglal-to-noise ratio). We are interested by the situation when there is a misspecification in the regularity conditions. In particular it is supposed that the statistician uses a discontinuous (change-point type) model of signal, when the true signal is continuously differentiable function of the unknown parameter.