Minimum Hellinger distance estimation for bivariate samples and time series with applications to nonlinear regression and copula-based models

We study minimum Hellinger distance estimation (MHDE) based on kernel density estimators for bivariate time series, such that various commonly used regression models and parametric time series such as nonlinear regressions with conditionally heteroscedastic errors and copula-based Markov processes, where copula densities are used to model the conditional densities, can be treated. It is shown that consistency and asymptotic normality of the MHDE basically follow from the uniform consistency of the density estimate and the validity of the central limit theorem for its integrated version. We also provide explicit sufficient conditions both for the i.i.d. case and the case of strong mixing series. In addition, for the case of i.i.d. data, we briefly discuss the asymptotics under local alternatives and relate the results to maximum likelihood estimation.