Error bounds for stochastic estimation of signal parameters

This paper is concerned with stochastic-approximation algorithms for estimating signal parameters. Emphasis will be on the performance of the algorithm for a finite number of observations as opposed to the asymptotic convergence rate. We use as an upper bound a result due to Dvoretzky. A lower bound on the average mean-square error is derived. This new bound is based on the Cramér-Rao inequality. The conventional Cramér-Rao bound is not directly applicable, because it requires the knowledge of the bias function, which is difficult to find in a recursive estimation scheme. To avoid this difficulty, we introduce the concept of most favorable bias function and use the calculus of variations to derive the lower bound. The new bound also serves as a standard to judge the merits of the stochastic-estimation algorithm, since under some general conditions no estimate can yield smaller error. It is shown that under some conditions the two bounds are nearly equal, and hence the algorithm is near optimal. The asymptotic efficiency of the algorithm is compared with Sakrison´s result. A stochastic-estimation algorithm is derived for estimating Doppler frequency, and performance curves in terms of the error bounds are presented.