Performance analysis of parameter estimation algorithms based on high‐order moments

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Abstract
Recently, there has been a considerable interest in parametric estimation of non‐Gaussian processes, based on high‐order moments. Several researchers have proposed algorithms for estimating the parameters of AR, MA and ARMA processes, based on the third‐order and fourth‐order cumulants. These algorithms are capable of handling non‐minimum phase processes, and some of them provide a good trade‐off between computational complexity and statistical efficiency.This paper presents some results about the performance of algorithms based on high‐order moments. A general lower bound is derived for the variance of estimates based on high‐order sample moments. This bound, which is shown to be asymptotically tight, is neither the Cramer‐Rao bound nor a trivial extension thereof. The performance of weighted least squares estimates of the type recently proposed in the literature is investigated. An expression for the variance of such estimates is derived and the existence of an optimal weight matrix is proven. The general formulae are specialized to MA and ARMA processes and used to analyse the performance of some algorithms in detail. The analytic results are verified by Monte Carlo simulations for some specific test cases.A by‐product of this paper is the derivation of asymptotic formulae for the variances and covariances of the sample third‐order moments of a certain class of processes.

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