Non-regular estimation theory for piecewise continuous spectral densities

For a class of Gaussian stationary processes, the spectral density fθ(λ),θ=(τ′,η′)′fθ(λ),θ=(τ′,η′)′, is assumed to be a piecewise continuous function, where ττ describes the discontinuity points, and the piecewise spectral forms are smoothly parameterized by ηη. Although estimating the parameter θθ 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 θθ, i.e., the maximum likelihood estimator θ̂ML and the Bayes estimator θ̂B, are introduced. Then the asymptotic distributions of θ̂ML and θ̂B are derived and shown to be non-normal. Furthermore we observe that θ̂B is asymptotically efficient, but θ̂ML is not so. Also various versions of step spectra are considered.