Higher order asymptotic theory for minimum contrast estimators of spectral parameters of stationary processes

Let g(λ) be the spectral density of a stationary process and let f_θ(λ), θ ∈ Θ, be a fitted spectral model for g(λ). A minimum contrast estimator θ̂_n of θ is defined that minimizes a distance D(f_θ, ĝ_n) between f_θ and ĝ_n where ĝ_n is a nonparametric spectral density estimator based on n observations. It is known that θ̂_n is asymptotically Gaussian efficient if g(λ) = f_θ(λ). Because there are infinitely many candidates for the distance function D(f _θ, ĝ_n), this paper discusses higher order asymptotic theory for θ̂_n in relation to the choice of D. First, the second-order Edgeworth expansion for θ̂_n is derived. Then it is shown that the bias-adjusted version of θ̂ _n is not second-order asymptotically efficient in general. This is in sharp contrast with regular parametric estimation, where it is known that if an estimator is first-order asymptotically efficient, then it is automatically second-order asymptotically efficient after a suitable bias adjustment (e.g., Ghosh, 1994, Higher Order Asymptotics, p. 57). The paper establishes therefore that for semiparametric estimation it does not hold in general that "first-order efficiency implies second-order efficiency." The paper develops verifiable conditions on D that imply second-order efficiency.