LARGE-SAMPLE PROPERTIES OF LEAST-SQUARES ESTIMATORS OF HARMONIC COMPONENTS IN A TIME SERIES WITH STATIONARY RESIDUALS. I. INDEPENDENT RESIDUALS
STANFORD UNIV CA DEPT OF STATISTICS
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Let X sub t be a discrete parameter time series generated by a model such that EX sub t is the sum of a finite number of simple harmonic terms of the form A cos omega t B sin omega t, and X sub t - EX sub t is a moving average of independently identically distributed random variables epsilon sub t, whose weights g sub u theta are specified functions of a vector-valued parameter theta. In 1952 P. Whittle proposed an approximate least-squares method of simultaneously estimating theta and the angular frequencies, sine and cosine coefficients of each harmonic term from observations X sub 1, X sub 2,... ,X sub n, and derived heuristically the asymptotic n approaching infinity distribution of the estimators. This paper presents rigorous proofs of Whittles statements concerning the asymptotic distribution, formulated precisely as limit theorems, for the simpler but important case where the moving average reduces to epsilon sub t, so that the parameter theta disappears. Proofs for the general case will be given in a subsequent paper.
- Statistics and Probability