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

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.

The Gaussian Assumption in Second-Order Estimation Problems in Digital Communications

“…4.3], [30,Result R8]. In both cases, it follows that (22) In Fig. 3, it is shown that the BQUE attains this asymptotic performance-for practical values of the SNR-in case of constant amplitude nuisance parameters even if there are multiple users and the number of antennas is very small .…”

“…4 and 5). It can be seen that the UCRB converges to (22) as the number of antennas increases. On the other hand, when the nuisance parameters are constant modulus , the optimal second-order estimator attains (22) for any value of , except for an intermediate interval in which the estimator converges to the UCRB.…”

“…In that case, the optimum secondorder estimator, which was originally named the best quadratic unbiased estimator (BQUE) [20], is computed by means of the following recursion: (9) where are the gradient vector and Fisher information matrix in the non-Gaussian case. Consequently, the covariance of any second order estimator is lower bounded by (10) The last lower bound was already proposed by Porat et al in [21], [22] and extended to noncircular data by Delmas in [23]. In the latter work, a "covariance matching" method is formulated in terms of to attain asymptotically in case of non-Gaussian data (see also [21], [22], and [24]).…”

“…Consequently, the covariance of any second order estimator is lower bounded by (10) The last lower bound was already proposed by Porat et al in [21], [22] and extended to noncircular data by Delmas in [23]. In the latter work, a "covariance matching" method is formulated in terms of to attain asymptotically in case of non-Gaussian data (see also [21], [22], and [24]). In the context of DOA estimation, other closed-form expressions have been obtained in case of constant-modulus sources [14] and for minimum-shift keying (MSK) and binay/quaternary PSK transmitters [25].…”

Performance Analysis of Optimal Blind Fusion of Bits

“…We note that the Jacobian and the Hessian matrices associated with a one-to-one mapping "alg" and its inverse mapping "alg " are connected by the following relations (see, e.g., [10] From the works of [8] and [9], the asymptotic covariance of a consistent estimator of is lower bounded by the symmetric positive-definite matrix , where is the matrix defined as (note that the particular ordering of the row of is irrelevant in the expression if this order is consistent with the ordering of the terms of defined in (4.1)). Finally, to assess the relatively efficiency of the statistic , this asympotically minimum variance lower bound is compared to the Cramér-Rao bound.…”