Parameter bounds under misspecified models

Abstract

A class of parameter bounds emerges as a consequence of the covariance inequality, i.e. Cauchy-Schwarz inequality for expectations. The expectation operator forms an inner product space. Flexibility in the choice of expectation integrand and measure for integration exists, however, to establish a class of parameter bounds under a general form of model misspecification, i.e. distribution mismatch. The Cramér-Rao bound (CRB) primarily, and secondarily the Barankin, Hammersley-ChapmanRobbins, and Bhattacharyya bounds under misspecification are considered. Huber's sandwich covariance is easily established as the misspecified CRB, and a generalization of the Slepian-Bangs formula under misspecification is provided.

Performance bounds under misspecification model for MIMO radar application

“…Therefore, classical lower bounds on the MSE cannot be correctly applied. Consequently, in the following, we fill this lack by using the so-called misspecified lower bound on the MSE introduced in [6] and [11].…”

“…Moreover, due to the asymptotic efficiency of the MLE when the model is correctly specified, one can expect that a slight error on the model will moderately degrade the MLE performance [6]. Recently, lower bounds on the mean square error have been established in the case where the model is misspecified, i.e., when the true data distribution differs from the assumed observation model [11]. These bounds, called "misspecified lower bounds", are asymptotically achievable (under mild conditions) by the Mean Square Error (MSE) of the MLE and are applicable for a class of estimators including the MLE.…”

“…Unlike the work proposed in [3] in which the steering vector is assumed to be perturbed by a random vector with known probability density function (pdf), we consider here a more realistic situation where the MLE is based on an idealistic observation model (with no error) but the data used to perform estimation are driven by another pdf. Such methodology has been recently successfully applied in the context of array processing [11] and scattering matrix estimation in radar processing [6].…”

On the Cramér-Rao lower bound under model mismatch

“…In this work, we consider CRLB type lower bounds for deterministic parameter estimation under model mismatch conditions, where the assumed data model used in designing the estimator differs from the true model. Although the literature on CRLB under model-match conditions is vast, there are very few studies devoted to the model mismatch case [9,10]. The most relevant contribution to our work in the literature is the recent work by Richmond and Horowitz [10] where a CRLB type bound is computed for the MSE of the estimators having a specified bias with respect to (w.r.t.)…”

“…the true model. The fundamental difference between our approach and [10] is that, in our contribution, CRLBs are derived for estimators that are unbiased or that have a specified bias (gradient) w.r.t. the assumed model.…”