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

S. Fortunati, F. Gini, M. S. Greco, C. D. Richmond

arXiv:1709.08210v1eess.SP

TL;DR

The paper examines estimation under misspecified models and reviews theoretical performance bounds for this setting. It presents the misspecified Cramér–Rao bound and discusses applications including DOA estimation and adaptive radar detection, while noting limits on the review’s claims and areas for future effort.

  • Problem

    Model misspecification complicates estimation-performance assessment, motivating bounds for estimators when the assumed and true models differ.

  • Method

    The paper reviews estimation theory under misspecified models, including the misspecified Cramér–Rao bound and covariance inequalities for MS-unbiased estimators.

  • Results

    The framework is applied to DOA estimation and disturbance covariance-matrix estimation for adaptive radar detection.

  • Takeaways & Limitations

    The review connects misspecified-model performance bounds with applications in array processing and adaptive radar detection.

  • Takeaways & Limitations

    The paper summarizes key results without claiming completeness, and identifies future effort concerning Bayesian bounds under misspecified models.

Abstract

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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.

1. INTRODUCTION

Performance bounds benchmark estimator accuracy, reveal parameter dependencies, and support feasibility and design decisions. Under misspecified models, the paper reviews how departures between assumed and true data models affect estimation limits and motivates bounds for deterministic and Bayesian settings.

  • Performance bounds benchmark estimator accuracy and can establish when no other algorithm can achieve better performance.
  • Lower bounds inform feasibility studies by identifying the practical accuracy limits of an estimation problem.
  • Bounds expose dependencies among parameters and can guide the choice of estimator parameters and criteria.
  • Deterministic estimation treats parameters as unknown fixed quantities, whereas Bayesian estimation models them as random and incorporates prior information.
  • Global bounds use separated parameter points and can characterize non-asymptotic performance, while local bounds primarily characterize asymptotic limits.
  • The Barankin Bound is a general global bound but is difficult to calculate and usually lacks a closed-form representation.
  • Model misspecification arises when the assumed data pdf differs from the true pdf, motivating bounds that assess its effect on estimation performance.
  • The paper reviews statistical literature on misspecified estimation theory and discusses applications in deterministic and Bayesian signal-processing problems.

2. DESCRIPTION OF A MISSPECIFIED MODEL PROBLEM

A misspecified estimation problem uses a parametric pdf for inference even though the true data pdf may differ for every parameter value. The paper formulates the resulting questions about bounds, estimator properties, and parameter- estimate meaning, then situates them in signal-processing applications.

  • The assumed parametric pdf can differ from the true data pdf, separating misspecified estimation from matched estimation.
  • A Gaussian assumed model is matched when it contains the true Gaussian distribution but mismatched when the true data follow a Laplace distribution.
  • Misspecification can reflect imperfect information about the data or the computational and hardware cost of implementing the true model.
  • Applications include direction-of-arrival estimation, covariance estimation, radar and communication systems, waveform estimation, and time-of-arrival estimation.
  • The framework collects M independent, identically distributed measurement vectors and forms true and assumed joint pdfs as products of marginal pdfs.
  • The paper asks whether error-covariance lower bounds remain possible and how unbiasedness, consistency, efficiency, and parameter- estimate meaning change under misspecification.

3. THE MISSPECIFIED CRAMÉR-RAO BOUND

The section introduces the misspecified Cramér–Rao bound (MCRB), a generalization of the classical CRB for estimators derived under possibly incorrect data models. It defines the pseudo-true parameter, regularity conditions, misspecified unbiasedness, and the covariance lower bound, then discusses its consistency with classical theory and practical uses.

  • The MCRB generalizes the classical Cramér–Rao bound to estimation under model misspecification.Under correct specification, it becomes the classical CRB.
  • 3.1 REGULAR MODELS: The framework requires regularity conditions, including a unique interior pseudo-true parameter and a nonsingular matrix A at that point.Additional conditions support the required interchange of integral and derivative operators.
  • 3.1 REGULAR MODELS: The pseudo-true parameter is the assumed-model parameter minimizing the KLD between the true and assumed pdfs.It serves as the counterpart of the true parameter in matched estimation theory.
  • 3.2 MISSPECIFIED-UNBIASED ESTIMATORS: An estimator is misspecified-unbiased when its expectation under the true pdf equals the pseudo-true parameter.This extends classical unbiasedness from the true parameter to the pseudo-true parameter.
  • 3.3 A COVARIANCE INEQUALITY IN THE PRESENCE OF MISSPECIFIED MODELS: The MCRB lower-bounds the error covariance of any misspecified-unbiased estimator, with error measured relative to the pseudo-true parameter.When the model is correctly specified, the bound reduces to the classical CRB.
  • 3.4 AN INTERESTING CASE: A LOWER BOUND ON THE MEAN SQUARE ERROR VIA THE MCRB: The MCRB can assess mismatched-estimator performance, support feasibility studies, quantify estimation loss, and predict potential weaknesses.It can also provide a useful surrogate for system analysis and design when the assumed model differs from the true data pdf.

4. THE MISMATCHED MAXIMUM LIKELIHOOD (MML) ESTIMATOR

Under suitable regularity conditions, the MML estimator converges almost surely to the pseudo-true parameter minimizing the KLD between true and assumed pdfs. It is asymptotically efficient relative to the MCRB, which can also be consistently estimated from data.

  • MML maximizes the misspecified log-likelihood and is consistent with the classical matched ML estimator.
  • The MML estimator converges almost surely to the pseudo-true parameter minimizing the KLD between the true and assumed pdfs.
  • MML is asymptotically MS-unbiased and has the lowest possible asymptotic error covariance among MS-unbiased mismatched estimators.
  • Under correct specification, the MML estimator converges to the true parameter and its asymptotic covariance becomes the classical CRB.
  • Evaluating sample counterparts of A(θ) and B(θ) at the MML estimate yields a strongly consistent MCRB estimate without prior knowledge of the true pdf.
  • In Gaussian variance estimation with a misspecified mean, the MCRB exceeds the classical CRB unless the true and assumed means agree.

5. GENERALIZATION TO THE BAYESIAN SETTING

The Bayesian extension studies posterior and estimator behavior when the assumed joint model, including the prior or likelihood, is misspecified. Under regularity conditions, mismatched Bayesian and maximum-likelihood estimators converge to the same KLD-minimizing point, while the paper notes that the Bayesian summary is not exhaustive.

  • Bayesian estimation uses the joint pdf, combining the likelihood with a prior, and derives estimators from the posterior distribution.
  • An incorrect prior can require significantly more observations or higher SNR before the Bayes estimator becomes prior-independent.
  • Under misspecification, the assumed joint pdf may be wrong in either its prior or conditional data model.
  • For a unique KLD-minimizing point, the posterior distribution concentrates asymptotically at that point as the number of observations increases.
  • Under suitable regularity conditions, mismatched Bayesian and MML estimators converge almost surely to the KLD-minimizing point and are asymptotically normal.
  • The paper introduces a misspecified Bayesian Cramér–Rao bound whose form resembles the non-Bayesian bound.

6. EXAMPLES OF APPLICATIONS

Applications show how misspecified bounds and estimators quantify performance loss in direction-of-arrival and scatter-matrix estimation. In both examples, mismatch-aware bounds predict estimator behavior and reveal how modelling errors affect attainable performance.

  • 6.1 DOA ESTIMATION UNDER MODEL MISSPECIFICATION: In DOA estimation, unknown sensor-position errors make the assumed steering-vector model misspecified and motivate the MCRB.
  • 6.1 DOA ESTIMATION UNDER MODEL MISSPECIFICATION: The MCRB accurately predicts MML performance in the DOA example and quantifies the impact of imperfect sensor-position knowledge.
  • 6.1 DOA ESTIMATION UNDER MODEL MISSPECIFICATION: 10-to-1 beamsplit resolution requires 9.28dB SNR with known geometry but approximately 19.4dB with sensor-position uncertainty.
  • 6.1 DOA ESTIMATION UNDER MODEL MISSPECIFICATION: At approximately 9.3dB SNR with array errors, the minimum achievable beamsplit ratio is 3-to-1 rather than 10-to-1.
  • 6.2 SCATTER MATRIX ESTIMATION UNDER MODEL MISSPECIFICATION: For heavy-tailed data, the distance between CCRB and CMCRB measures mismatch loss, which increases as the shape parameter λ approaches zero.
  • 6.2 SCATTER MATRIX ESTIMATION UNDER MODEL MISSPECIFICATION: As λ→∞ and the t-distribution approaches Gaussianity, CCRB and CMCRB coincide, while constrained MML is efficient relative to CMCRB.

7. CONCLUDING REMARKS

The paper reviews mismatched estimation theory for a broad SP audience, covering deterministic and Bayesian frameworks, misspecified bounds, estimator behavior, and applications. It also identifies open problems involving more general bounds, Bayesian lower bounds, and decision theory under model misspecification.

  • Motivation: Model mismatch is inevitable in practical applications, yet performance bounds under misspecification have received limited attention in the SP community.The paper contrasts this limited SP attention with deeper investigation in statistics.
  • Scope and contributions: The tutorial reviews mismatched estimation theory for a wide SP audience across deterministic and Bayesian frameworks.It presents a comprehensive review of major contributions to mismatched estimation theory.
  • Scope and contributions: The paper introduces the MCRB, investigates the MML estimator, and discusses existence and asymptotic properties of a mismatched Bayesian estimator.It also outlines general ideas for misspecified Bayesian Cramér-Rao bounds.
  • Applications: The theoretical findings are applied to DOA estimation in array processing and disturbance covariance estimation for adaptive radar detection.These applications illustrate the framework in two established SP problems.
  • Open problems: Open problems include more general misspecified bounds, Bayesian lower bounds without current constraints, and a systematic decision theory under model misspecification.The paper points to extensions of the Bhattacharyya, Barankin, and Bobrovsky-Mayer-Wolf-Zakai bounds as one direction.
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