LARGE-SAMPLE PROPERTIES OF LEAST-SQUARES ESTIMATORS OF HARMONIC COMPONENTS IN A TIME SERIES WITH STATIONARY RESIDUALS. I. INDEPENDENT RESIDUALS

Abstract

Let X sub t be a discrete parameter time series generated by a model such that E(X 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 - E(X 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 Whittle's 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.

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Document Details

Document Type
Technical Report
Publication Date
Jul 08, 1968
Accession Number
AD0672519

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  • A. M. Walker

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  • Stanford University

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  • Materials and Manufacturing Processes

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  • Classification
  • Data Science
  • Distribution Functions
  • Estimators
  • Frequency
  • Information Science
  • Least Squares Method
  • Maximum Likelihood Estimation
  • Military Research
  • Normal Distribution
  • Probability
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  • Security
  • Stationary
  • Statistical Algorithms
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  • Mathematics

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  • Analytical Mechanics
  • Approximation Theory.
  • Statistical inference.