On the estimation of a harmonic component in a time series with stationary independent residuals

Abstract

Let {Xt} be a time series such that Xi = E(Xi) + Σ∞ u=o gu(θ)ɛt-w where E(Xt) is the sum of a finite number of simple harmonic terms of the form A cos (wt) + B sin (wt), the ɛt are independently and indetically distributed random variables each with mean zero and finite variance, and the gu(θ) are specified functions of a vector-valued parameter θ. Whittle (1952) proposed an approximate least squares method of simulataneously estimating θ and the angular freaquencies, sine and cosine coefficients, of each harmonic term from observations (X1, …, Xn) and derived heuristically the asymptotic distribution of the estimators. This paper presents rigorous proofs of Whittle's statements concerning the asymptotic distribution, formulated precisely as limit theorems, for the special case of independent residuals, where Xt = E(Xt) + ɛt, so that the parameter θ disappears. The arguments used here suggest how one can deal with the general case, and proofs for this will be given in a subsequent paper.