A GENERAL RESAMPLING SCHEME FOR TRIANGULAR ARRAYS OF ALPHA-MIXING RANDOM-VARIABLES WITH APPLICATION TO THE PROBLEM OF SPECTRAL DENSITY-ESTIMATION

In 1989 Kunsch introduced a modified bootstrap and jackknife for a statistic which is used to estimate a parameter of the m-dimensional joint distribution of stationary and alpha-mixing observations. The modification amounts to resampling whole blocks of consecutive observations, or deleting whole blocks one at a time. Liu and Singh independently proposed (in 1988) the same technique for observations that are m-dependent. However, many time-series statistics, notably estimators of the spectral density function, involve parameters of the whole (infinite-dimensional)joint distribution and, hence, do not fit in this framework. In this report we generalize the ''moving blocks'' resampling scheme of Kunsch and Liu and Singh; a still modified version of the nonparametric bootstrap and jackknife is seen to be valid for general linear statistics that are asymptotically normal and consistent for a parameter of the whole joint distribution. We then apply this result to the problem of estimation of the spectral density.