Semiparametric Slepian-Bangs Formula for Complex Elliptically Symmetric Distributions.

This letter aims at deriving a Semiparametric Slepian-Bangs (SSB) formula for Complex Elliptically Symmetric (CES) distributed data vectors. The Semiparametric Cram\'{e}r-Rao Bound (SCRB), related to the proposed SSB formula, provides a lower bound on the Mean Square Error (MSE) of \textit{any} robust estimator of a parameter vector parameterizing the mean vector and the scatter matrix of the given CES-distributed vector in the presence of an unknown, nuisance, density generator.

[1]  G. Simons,et al.  On the theory of elliptically contoured distributions , 1981 .

[2]  P. Krishnaiah,et al.  Complex elliptically symmetric distributions , 1986 .

[3]  Kaare Brandt Petersen,et al.  The Matrix Cookbook , 2006 .

[4]  Frédéric Pascal,et al.  New Insights Into the Statistical Properties of $M$-Estimators , 2017, IEEE Transactions on Signal Processing.

[5]  V. Koivunen,et al.  On the Cramér-Rao bound for the constrained and unconstrained complex parameters , 2008, 2008 5th IEEE Sensor Array and Multichannel Signal Processing Workshop.

[6]  Visa Koivunen,et al.  Essential Statistics and Tools for Complex Random Variables , 2010, IEEE Transactions on Signal Processing.

[7]  Sofia C. Olhede,et al.  On probability density functions for complex variables , 2006, IEEE Transactions on Information Theory.

[8]  K. Do,et al.  Efficient and Adaptive Estimation for Semiparametric Models. , 1994 .

[9]  Bernard C. Picinbono,et al.  On circularity , 1994, IEEE Trans. Signal Process..

[10]  Visa Koivunen,et al.  Influence Function and Asymptotic Efficiency of Scatter Matrix Based Array Processors: Case MVDR Beamformer , 2009, IEEE Transactions on Signal Processing.

[11]  Olivier Besson,et al.  On the Fisher Information Matrix for Multivariate Elliptically Contoured Distributions , 2013, IEEE Signal Processing Letters.

[12]  Davy Paindaveine,et al.  A canonical definition of shape , 2008 .

[13]  Christ D. Richmond,et al.  Adaptive array signal processing and performance analysis in non-Gaussian environments , 1996 .

[14]  D. Paindaveine,et al.  SEMIPARAMETRICALLY EFFICIENT RANK-BASED INFERENCE FOR SHAPE I. OPTIMAL RANK-BASED TESTS FOR SPHERICITY , 2006, 0707.4621.

[15]  Abdelhak M. Zoubir,et al.  Semiparametric Inference and Lower Bounds for Real Elliptically Symmetric Distributions , 2018, IEEE Transactions on Signal Processing.

[16]  Bernard C. Picinbono,et al.  Second-order complex random vectors and normal distributions , 1996, IEEE Trans. Signal Process..

[17]  Priscilla E. Greenwood,et al.  An Introduction to Efficient Estimation for Semiparametric Time Series , 2004 .

[18]  Stefano Fortunati,et al.  Misspecified Cramér-rao bounds for complex unconstrained and constrained parameters , 2017, 2017 25th European Signal Processing Conference (EUSIPCO).

[19]  Bhaskar D. Rao,et al.  Cramer-Rao lower bound for constrained complex parameters , 2004, IEEE Signal Processing Letters.

[20]  A. Tsiatis Semiparametric Theory and Missing Data , 2006 .

[21]  H. Vincent Poor,et al.  Complex Elliptically Symmetric Distributions: Survey, New Results and Applications , 2012, IEEE Transactions on Signal Processing.

[22]  David S. Slepian,et al.  Estimation of signal parameters in the presence of noise , 1954, Trans. IRE Prof. Group Inf. Theory.

[23]  S. Kotz,et al.  Symmetric Multivariate and Related Distributions , 1989 .

[24]  Petre Stoica,et al.  MUSIC, maximum likelihood, and Cramer-Rao bound , 1989, IEEE Transactions on Acoustics, Speech, and Signal Processing.

[25]  C. Richmond A note on non-Gaussian adaptive array detection and signal parameter estimation , 1996, IEEE Signal Processing Letters.

[26]  Hualiang Li,et al.  Complex-Valued Adaptive Signal Processing Using Nonlinear Functions , 2008, EURASIP J. Adv. Signal Process..

[27]  Eric Chaumette,et al.  New Results on Deterministic Cramér–Rao Bounds for Real and Complex Parameters , 2012, IEEE Transactions on Signal Processing.

[28]  Michael Muma,et al.  Robust Statistics for Signal Processing , 2018 .

[29]  Steven Kay,et al.  Fundamentals Of Statistical Signal Processing , 2001 .

[30]  Adriaan van den Bos,et al.  A Cramer-Rao lower bound for complex parameters , 1994, IEEE Trans. Signal Process..

[31]  D. Paindaveine,et al.  SEMIPARAMETRICALLY EFFICIENT RANK-BASED INFERENCE FOR SHAPE II. OPTIMAL R-ESTIMATION OF SHAPE , 2006, 0708.0079.

[32]  Anthony J. Weiss,et al.  On the Cramer-Rao Bound for Direction Finding of Correlated Signals , 1993, IEEE Trans. Signal Process..

[33]  Fulvio Gini,et al.  Cramér-Rao Lower Bounds on Covariance Matrix Estimation for Complex Elliptically Symmetric Distributions , 2013, IEEE Transactions on Signal Processing.

[34]  P. Stoica,et al.  The stochastic CRB for array processing: a textbook derivation , 2001, IEEE Signal Processing Letters.

[35]  Fulvio Gini,et al.  Performance Bounds for Parameter Estimation under Misspecified Models: Fundamental Findings and Applications , 2017, IEEE Signal Processing Magazine.

[36]  Christ D. Richmond,et al.  Parameter Bounds on Estimation Accuracy Under Model Misspecification , 2015, IEEE Transactions on Signal Processing.

[37]  Abdelhak M. Zoubir,et al.  A Fresh Look at the Semiparametric Cramér-Rao Bound , 2018, 2018 26th European Signal Processing Conference (EUSIPCO).

[38]  L. Scharf,et al.  Statistical Signal Processing of Complex-Valued Data: The Theory of Improper and Noncircular Signals , 2010 .

[39]  P. Bickel On Adaptive Estimation , 1982 .

[40]  Petre Stoica,et al.  Performance study of conditional and unconditional direction-of-arrival estimation , 1990, IEEE Trans. Acoust. Speech Signal Process..

[41]  Bas J. M. Werker,et al.  Semiparametric efficiency, distribution-freeness and invariance. , 2003 .

[42]  B. A. D. H. Brandwood A complex gradient operator and its applica-tion in adaptive array theory , 1983 .

[43]  Jan R. Magnus,et al.  The Elimination Matrix: Some Lemmas and Applications , 1980, SIAM J. Algebraic Discret. Methods.

[44]  Abdelhak M. Zoubir,et al.  Slepian-Bangs-type formulas and the related Misspecified Cramér-Rao Bounds for Complex Elliptically Symmetric distributions , 2018, Signal Process..

[45]  J. Magnus,et al.  The Commutation Matrix: Some Properties and Applications , 1979 .

[46]  Steeve Zozor,et al.  Some Results on the Denoising Problem in the Elliptically Distributed Context , 2010, IEEE Transactions on Signal Processing.

[47]  Are Hjrungnes,et al.  Complex-Valued Matrix Derivatives: With Applications in Signal Processing and Communications , 2011 .

[48]  Erik G. Larsson,et al.  Stochastic Cramer-Rao bound for direction estimation in unknown noise fields , 2002 .

[49]  W. J. Hall,et al.  Information and Asymptotic Efficiency in Parametric-Nonparametric Models , 1983 .

[50]  Ken Kreutz-Delgado,et al.  The Complex Gradient Operator and the CR-Calculus ECE275A - Lecture Supplement - Fall 2005 , 2009, 0906.4835.

[51]  Björn E. Ottersten,et al.  Analysis of subspace fitting and ML techniques for parameter estimation from sensor array data , 1992, IEEE Trans. Signal Process..

[52]  A. Bos Complex gradient and Hessian , 1994 .

[53]  W. Newey,et al.  Semiparametric Efficiency Bounds , 1990 .

[54]  S. Kay Fundamentals of statistical signal processing: estimation theory , 1993 .

[55]  Marc Hallin,et al.  Parametric and semiparametric inference for shape: the role of the scale functional , 2006 .

[56]  Abdelhak M. Zoubir,et al.  The Semiparametric Cramér-Rao Bound for Complex Elliptically Symmetric Distributions , 2018, 1807.08505.

[57]  Tülay Adali,et al.  Complex-Valued Signal Processing: The Proper Way to Deal With Impropriety , 2011, IEEE Transactions on Signal Processing.