Whittle estimator for finite-variance non-gaussian time series with long memory

摘要

We consider time series Y_1=G(X_t) where X_t is Gaussian with long memory and G is a polynomial. The series Y_t may or may not have long memory. The spectral density g_0(x) of Y_t is parameterized by a vector #theta# and we want to estimate its true value #theta#_0. We use a least-squares Whittle-type estimator #theta#_N for #theta#_0, based on observations Y_1,...,Y_N. If Y_t is Gaussian, then N~(1/2)(#theta#_N-#theta#_0) converges to a Gaussian distribution. We show that for non-Gaussian time series Y_t, this N~(1/2) consistency of the Whittle estimator does not always hold and that the limit is not necessarily Gaussian. This can happen even if Y_t has short memory.