Non-regular estimation theory for piecewise continuous spectral densities

For a class of Gaussian stationary processes, the spectral density f θ ( λ ) , θ = ( τ ′ , η ′ ) ′ , is assumed to be a piecewise continuous function, where τ describes the discontinuity points, and the piecewise spectral forms are smoothly parameterized by η . Although estimating the parameter θ is a very fundamental problem, there has been no systematic asymptotic estimation theory for this problem. This paper develops the systematic asymptotic estimation theory for piecewise continuous spectra based on the likelihood ratio for contiguous parameters. It is shown that the log-likelihood ratio is not locally asymptotic normal (LAN). Two estimators for θ , i.e., the maximum likelihood estimator θ ̂ ML and the Bayes estimator θ ̂ B , are introduced. Then the asymptotic distributions of θ ̂ ML and θ ̂ B are derived and shown to be non-normal. Furthermore we observe that θ ̂ B is asymptotically efficient, but θ ̂ ML is not so. Also various versions of step spectra are considered.