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

Let g(lambda) be the spectral density of a stationary process and let f(theta)(lambda), theta is an element of Theta, be a fitted spectral model for g(lambda). A minimum contrast estimator (theta) over cap of theta is defined that minimizes a distance D (f(theta), (g) over cap (n)) between f(theta) and (g) over cap (n) where (g) over cap (n) is a nonparametric spectral density estimator based on n observations. It is known that (theta) over cap (n) is asymptotically Gaussian efficient if g(lambda) =f(theta)(lambda). Because there are infinitely many candidates for the distance function D (f(theta), (g) over cap (n)), this paper discusses higher order asymptotic theory for (theta) over cap (n) in relation to the choice of D. First, the second-order Edgeworth expansion for (theta) over cap (n) is derived. Then it is shown that the bias-adjusted version of (theta) over cap (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, 1094, Higher Order Asymptotics, p. 57). The paper establishes therefore that for semi-parametric 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.