A Note on Estimation by Least Squares for Harmonic Component Models

Let observations (X1....Xn) be generated by a harmonic model such that Xt=A0 cos 0t+B0 sin 0t+t, where A0,B0,0 are constants and (t) is a stationary process with zero mean and finite variance. The estimation of A0,B0,0 by the method of least squares is considered. It is shown that, without any restriction on in the minimization procedure, the estimate is an n-consistent estimate of 0, and hence (Ao,Bo,)(o) 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[1/n, pi - 1/n]