A Class of Bayesian Lower Bounds for Parameter Estimation Via Arbitrary Test-Point Transformation

In this article, a new class of global mean-squared-error (MSE) lower bound for Bayesian parameter estimation is derived. First, it is shown that under the non-Bayesian framework, the Hammersley-Chapman-Robbins (HCR) for the problem of single-source parameter estimation, is related to the corresponding ambiguity function. This result is achieved by judicious choice of signal test-point. This result implies that optimal shift test-points may be parameter-dependent, but this approach cannot be imitated in test-point based Bayesian bounds, where the test-points should be independent of the random parameters. Based on this observation, a new class Bayesian MSE lower bound is derived. In the proposed class, the shift test-points in the Cauchy-Schwarz based bounds are substituted with arbitrary transformations. The proposed class generalizes the Weiss-Weinstein bound (WWB) to arbitrary transformations. For the problem of single source parameter estimation, a scaling transformation for the signal test-point is proposed based on the structure of the optimal test-point in the HCR bound in the non-Bayesian framework. The test-point scale parameters are analytically optimized and a closed-form expression for the bound, which depends on the ambiguity function, is derived. For the problem of direction-of-arrival (DOA) estimation using a linear array, a nonlinear transformation for the parameter of interest is proposed. In the problems of frequency estimation and DOA estimation, the proposed bound accurately predicts the threshold phenomenon of the maximum a-posteriori probability estimator, and is much tighter than the WWB in both the threshold prediction and the asymptotic performance. In addition, an asymptotic version of the bound was derived and it is analytically shown to be tighter than the Bayesian Cramér-Rao bound.

I. Introduction

Performance bounds on parameter estimation are of utmost importance in statistical signal processing theory and applications. In addition to providing a benchmark for performance analysis of estimators [1], [2], [3], [4], they are frequently used for system design and feasibility study (see e.g. [5], [6], [7], [8], [9]) and in the recent years, they are employed as an optimization criterion in sequential cognitive systems [10], [11], [12], [13], [14], [15], [16], [17], in both the Bayesian and non-Bayesian (frequentist) approaches. The quality and relevance of the bounds are determined by the following factors: 1) tightness and attainability, 2) computational complexity, and 3) ability to provide insight into the considered problem.