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

On Harmonic Approximations of Inharmonic Signals

“…Definition 1 (Pseudo-true parameter [13]). Consider a signal sample y with probability density function f .…”

“…Here, it may be noted that the pseudo-true parameter θ 0 minimizes the Kullback-Leibler divergence between the distribution of the actual measurement, i.e., f , and the distribution of the model, encoded in the parametric likelihood L. Interestingly, it can be shown that the MLE corresponding to the misspecified model L converges to the pseudo-true parameter [13]. That is, as the number of samples from (1) tend to infinity, the maximumm likelihood estimator (MLE) derived under (4) tends to the pseudo-true parameter, making this an applicable definition of fundamental frequency for practical purposes.…”

Mismatched Estimation of Polynomially Damped Signals

“…Definition 1 (Pseudo-true parameter [15]). Consider a signal sample with likelihood L, parametrized by the parameter vector ψ.…”

“…where y is sampled from L, converges, under some regularity conditions, to the pseudo-true parameter θ 0 as the signal-to-noise ratio (SNR), or sample size, depending on the application, increases [15]. Herein, we are interested in estimating the parameters of purely sinusoidal and Lorentzian models when the actual measured signal may be Lorentzian or Voigt, respectively.…”

“…To find a remedy for this problem, we propose an estimation procedure that gradually increases the model complexity, such that one is restricting the parameter space over which the parameter search is performed. To alleviate this, we use the framework mismatched estimation (see, e.g., [15] for a recent overview). Specifically, we present a brief analysis on the expected behavior of approximate maximum likelihood estimators (MLEs) derived under simplified model assumptions, i.e., when using lower-order polynomials for describing the signal decay.…”