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 over(θ, ̂)_ML and the Bayes estimator over(θ, ̂)_B, are introduced. Then the asymptotic distributions of over(θ, ̂)_ML and over(θ, ̂)_B are derived and shown to be non-normal. Furthermore we observe that over(θ, ̂)_B is asymptotically efficient, but over(θ, ̂)_ML is not so. Also various versions of step spectra are considered.