Efficient Nonparametric Estimation of Generalized Autocovariances

This paper provides a necessary and sufficient condition for asymptotic efficiency of a nonparametric estimator of the generalized autocovariance function of a stationary random process. The generalized autocovariance function is the inverse Fourier transform of a power transformation of the spectral density and encompasses the traditional and inverse autocovariance functions as particular cases. A nonparametric estimator is based on the inverse discrete Fourier transform of the power transformation of the pooled periodogram. The general result on the asymptotic efficiency is then applied to the class of Gaussian stationary ARMA processes and its implications are discussed. Finally, we illustrate that for a class of contrast functionals and spectral densities, the minimum contrast estimator of the spectral density satisfies a Yule-Walker system of equations in the generalized autocovariance estimator.