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A note on estimation by least squares for harmonic component models
Author(s) -
Walker A. M.
Publication year - 2003
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/1467-9892.00325
Subject(s) - mathematics , least squares function approximation , harmonic , extension (predicate logic) , zero (linguistics) , minification , combinatorics , component (thermodynamics) , harmonic mean , non linear least squares , interval (graph theory) , mathematical analysis , estimation theory , statistics , mathematical optimization , thermodynamics , physics , linguistics , philosophy , quantum mechanics , estimator , computer science , programming language
. Let observations ( X 1 ,…, X n ) be generated by a harmonic model such that X t = A 0 cos ω 0 t + B 0 sin ω 0 t + ε t , where A 0 , B 0 , ω 0 are constants and ( ε t ) is a stationary process with zero mean and finite variance. The estimation of A 0 , B 0 , ω 0 by the method of least squares is considered. It is shown that, without any restriction on ω in the minimization procedure, the estimate is an n ‐consistent estimate of ω 0 , and hence () has the usual asymptotic distribution. The extension to a harmonic model with k >1 components is discussed. The case k =2 is considered in detail, but it was only found possible to establish the result under the restriction that both angular frequencies lie in the interval