On the Estimation of Spectra by Means of Non-linear Functions of the Covariance Matrix

Abstract

The estimation of spectra of random stationary processes is an important part of the statistics of random processes. There are several books on spectral analysis, e.g. Blackman & Tukey, Hannan, and Jenkins & Watts. As a rule, spectral estimators are quadratic functions of the realizations. Recently Capon suggested a new method for estimation of spectra of random fields, in which a non-quadratic function of the realization is used: he considered a homogeneous random field ξ(t, x1, x2), i.e. one which is stationary in time and space and a random function of the time and space co-ordinates t, x1, x2. For the sake of expository convenience we shall consider ordinary stationary processes of time only, ξ(t); the generalization of our results to the case of random fields is easy. Comparison of the conventional spectral estimator and the ‘high-resolution’ estimator for an artificial example showed that the latter has less smoothing effect on the true spectrum (Capon). This was later confirmed by examples using real data (Capon). However, it was not clear whether for a finite realization the high-resolution estimator distorted the true spectrum, i.e. whether it behaved for example like a conventional estimator raised to some power. In the present paper we introduce and study a new class of spectral estimators which are generally non-linear and non-quadratic functionals of the realizations. These estimators include the conventional and high-resolution ones, for which we shall give the approximate distributions. We derive under rather general conditions the limiting distribution of the new class of estimators, and illustrate them with several examples. As a matter of fact, these new estimators are weighted means of the eigenvalues of the covariance matrix, e.g. the arithmetic mean, geometric mean, and so on.