Performance Bounds for Parameter Estimation under Misspecified Models: Fundamental findings and applications

📝 Abstract

Inferring information from a set of acquired data is the main objective of any signal processing (SP) method. In particular, the common problem of estimating the value of a vector of parameters from a set of noisy measurements is at the core of a plethora of scientific and technological advances in the last decades; for example, wireless communications, radar and sonar, biomedicine, image processing, and seismology, just to name a few. Developing an estimation algorithm often begins by assuming a statistical model for the measured data, i.e. a probability density function (pdf) which if correct, fully characterizes the behaviour of the collected data/measurements. Experience with real data, however, often exposes the limitations of any assumed data model since modelling errors at some level are always present. Consequently, the true data model and the model assumed to derive the estimation algorithm could differ. When this happens, the model is said to be mismatched or misspecified. Therefore, understanding the possible performance loss or regret that an estimation algorithm could experience under model misspecification is of crucial importance for any SP practitioner. Further, understanding the limits on the performance of any estimator subject to model misspecification is of practical interest. Motivated by the widespread and practical need to assess the performance of a mismatched estimator, the goal of this paper is to help to bring attention to the main theoretical findings on estimation theory, and in particular on lower bounds under model misspecification, that have been published in the statistical and econometrical literature in the last fifty years. Secondly, some applications are discussed to illustrate the broad range of areas and problems to which this framework extends, and consequently the numerous opportunities available for SP researchers.

💡 Analysis

Inferring information from a set of acquired data is the main objective of any signal processing (SP) method. In particular, the common problem of estimating the value of a vector of parameters from a set of noisy measurements is at the core of a plethora of scientific and technological advances in the last decades; for example, wireless communications, radar and sonar, biomedicine, image processing, and seismology, just to name a few. Developing an estimation algorithm often begins by assuming a statistical model for the measured data, i.e. a probability density function (pdf) which if correct, fully characterizes the behaviour of the collected data/measurements. Experience with real data, however, often exposes the limitations of any assumed data model since modelling errors at some level are always present. Consequently, the true data model and the model assumed to derive the estimation algorithm could differ. When this happens, the model is said to be mismatched or misspecified. Therefore, understanding the possible performance loss or regret that an estimation algorithm could experience under model misspecification is of crucial importance for any SP practitioner. Further, understanding the limits on the performance of any estimator subject to model misspecification is of practical interest. Motivated by the widespread and practical need to assess the performance of a mismatched estimator, the goal of this paper is to help to bring attention to the main theoretical findings on estimation theory, and in particular on lower bounds under model misspecification, that have been published in the statistical and econometrical literature in the last fifty years. Secondly, some applications are discussed to illustrate the broad range of areas and problems to which this framework extends, and consequently the numerous opportunities available for SP researchers.

📄 Content

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Performance Bounds for Parameter Estimation under Misspecified Models: Fundamental findings and applications S. Fortunati1, F. Gini1, M. S. Greco1, and C. D. Richmond2 1Dipartimento di Ingegneria dell’Informazione, University of Pisa, Italy 2Arizona State University, School of Electrical, Computer and Energy Engineering, Tempe, USA.

ABSTRACT Inferring information from a set of acquired data is the main objective of any signal proc- essing (SP) method. In particular, the common problem of estimating the value of a vector of parameters from a set of noisy measurements is at the core of a plethora of scientific and technological advances in the last decades; for example, wireless communications, radar and sonar, biomedicine, image processing, and seismology, just to name a few.
Developing an estimation algorithm often begins by assuming a statistical model for the measured data, i.e. a probability density function (pdf), which if correct, fully characterizes the behaviour of the collected data/measurements. Experience with real data, however, often exposes the limitations of any assumed data model since modelling errors at some level are always present. Consequently, the true data model and the model assumed to derive the esti- mation algorithm could differ. When this happens, the model is said to be mismatched or mis- specified. Therefore, understanding the possible performance loss or regret that an estimation algorithm could experience under model misspecification is of crucial importance for any SP practitioner. Further, understanding the limits on the performance of any estimator subject to model misspecification is of practical interest. Motivated by the widespread and practical need to assess the performance of a “mis- matched” estimator, the goal of this paper is to help to bring attention to the main theoretical 2

findings on estimation theory, and in particular on lower bounds under model misspecifica- tion, that have been published in the statistical and econometrical literature in the last fifty years. Secondly, some applications are discussed to illustrate the broad range of areas and problems to which this framework extends, and consequently the numerous opportunities available for SP researchers.

1. INTRODUCTION The mathematical basis for a formal theory of statistical inference was presented by Fisher, who introduced the Maximum Likelihood (ML) method along with its main properties [Fis25]. Since then, ML estimation has been widely used in a variety of applications. One of the main reasons for its popularity is its asymptotic efficiency, i.e. its ability to achieve a minimum value of the error variance as the number of available observations goes to infinity or as the noise power decreases to zero. The concept of efficiency is strictly related to the ex- istence of some lower bounds on the performance of any estimator designed for a specific in- ference task. Such performance bounds, one of which is the celebrated Cramér-Rao Bound (CRB) [Cra46] [Rao45], are of fundamental importance in practical applications since they provide a benchmark of comparison for the performance of any estimator. Specifically, given a particular estimation problem, if the performance of a certain algorithm achieves a relevant performance bound, then no other algorithm can do better. Moreover, evaluating a perfor- mance bound is often a prerequisite for any feasibility study. In particular, the availability of a lower bound for the estimation problem at hand makes the SP practitioner aware of the practi- cal impossibility to achieve better estimation accuracy than the one indicated by the bound itself. Another fundamental feature of a performance bound is its ability to capture and reveal the complex dependences amongst the various parameters of interest, thus offering the oppor- tunity to understand more deeply the estimation problem at hand and ultimately to identify an appropriate design choice of parameters and criterion for an estimator [Kay98].
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Before describing specific performance bounds, it is worth mentioning that estimation the- ory explores two different frameworks: one is deterministic and one is Bayesian. In the classi- cal deterministic approach, the parameters to be estimated are modelled as deterministic but unknown variables. This implies that no a-priori information is available that would suggest that one outcome is more or less likely than another. In the Bayesian framework instead, the parameters of interest are assumed to be random variables and the goal is to estimate their particular realizations. Unlike the classical deterministic approach, the Bayesian approach ex- ploits this random characterization of the unknown parameters by incorporating a-priori in- formation about the unknown parameters in the derivation of an estimation algorithm. In par- ticular, the joint pdf of the unknown parameters is assumed known, and therefore can be

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