Signal Processing - November 2017 - 142

Stefano Fortunati, Fulvio Gini, Maria S. Greco,
and Christ D. Richmond

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

nferring information from a set of acquired data is the main
objective of any signal processing (SP) method. 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 recent
decades, including wireless communications, radar and sonar,
biomedicine, image processing, and seismology.
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 behavior of the collected data/measurements. Experience
with real data, however, often exposes the limitations of any
assumed data model, since modeling 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 critical for any SP
practitioner. Furthermore, 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
article is to help bring attention to the main theoretical findDigital Object Identifier 10.1109/MSP.2017.2738017
Date of publication: 13 November 2017

142

ings 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 50 years.
Additionally, 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.

A formal theory of statistical
inference under misspecified models
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 [9].
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 existence of
some lower bounds on the performance of any estimator
designed for a specific inference task. Such performance
bounds, one of which is the celebrated Cramér-Rao bound
(CRB) [8], [33], are fundamentally important in practical
applications, as 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 performance

IEEE SIGNAL PROCESSING MAGAZINE

November 2017

Table of Contents for the Digital Edition of Signal Processing - November 2017