GENERALIZED CAUCHY DISTRIBUTION BASED ROBUST ESTIMATION Rafael E. Carrillo ⋆ , Tuncer C. Aysal † and Kenneth E. Barner ⋆ ⋆ University of Delaware,†McGill University ABSTRACT The generalized Cauchy distribution (GCD) family has the property that its pdf has closed form for the whole family and also has algebraic tails which makes it suitable to model many impulsive processes in real life. In this paper we pro- pose a robust M-type estimator based on the pdf of the GCD family. Robustness and properties of the new statistics are analyzed and it is noticed that this estimator provide desired characteristics in robust signal processing applications involv- ing non-Gaussian heavy-tailed models. Simulations of the filtering method are performed to evaluate and compare the proposed filtering structure performance to other classic and robust estimators. Index Terms— maximum likelihood estimation, nonlin- ear filters 1. INTRODUCTION Robust statistics is the stability theory of statistical proce- dures. It systematically investigates the effects of deviations from modelling assumptions on unknown procedures and, if necessary, develops new, better procedures. Robust nonlin- ear estimators are critical for applications in real situations involving impulsive processes (e.g. ocean acoustic noise, at- mospheric interference in LF and VLF communications and multiple access interference in wireless system communica- tions), where heavy-tailed non-Gaussian distributions model the signal [1]. M-estimators, which were developed in the theory of ro- bust statistics [2], have been of great importance in the de- velopment of robust signal processing techniques [3]. M- estimators can be described by a cost function ρ(u) (posing an optimization problem) or by its first derivative, ψ(u) (yield- ing an (set of) implicit equation(s)), which is proportional to the influence function. In the location case properties of ψ describe how robust the estimator is. Maximum likelihood location estimates form a special case of M-estimators, with the observations being independent and identically distributed and ρ(u)= - log f (u), where f (u) is the common density of the samples. The α-Stable density family has gained recent popular- ity in addressing heavytailed problems. Unfortunately, the This work was supported in part by NSF under grant 0728904. Cauchy distribution is the only algebraictailed α-Stable dis- tribution that possesses a closed form expression, limiting the flexibility and performance of methods derived from this fam- ily of distributions. In this paper the maximum likelihood (ML) estimate of location is derived for the GCD family and then extending the associated norm as an M-type estimator. The fact that M–GC estimator is likelihood–based guarantees that the estimate is, at least asymptotically, unbiased consis- tent and efficient in GCD statistics [2]. Robustness and prop- erties of the cost function are analyzed and it is noticed that this estimator provide desired characteristics in robust signal processing applications involving non-Gaussian heavy-tailed processes. 2. M–ESTIMATION AND GCD The generalized Cauchy distribution family was proposed by Miller and Thomas in 1972 and has been used in several stud- ies of impulsive radio noise [1]. The PDF of the GCD is given by f (x)= aσ(σ p + |x| p ) -2/p with a = pΓ(2/p)/2(Γ(1/p)) 2 . In this representation, σ is the scale parameter and p is the tail constant. The GCD family contains the Meridian [4] and Cauchy distributions as special cases with p =1 and p =2 respectively. For p< 2, the tail of the PDF decays slower than in the Cauchy distribution, resulting in a heavier-tailed PDF. In the M-estimation theory we want to estimate a deter- ministic but unknown parameter θ of a real-valued signal s(i; θ) corrupted by additive noise from a set of noisy observations {x(i)} N i=1 . M–estimate is given by the solution to an opti- mization problem ˆ θ = arg min θ∈Θ N  i=1 ρ(x(i) - s(i; θ)) or by an implicit equation N  i=1 ψ(x(i) - s(i; ˆ θ)) = 0 where ρ is an arbitrary cost function to be designed, Θ is the solution space, and ψ(x)=(∂/∂θ)ρ(x). 3389 1-4244-1484-9/08/$25.00 ©2008 IEEE ICASSP 2008 Authorized licensed use limited to: UNIVERSITY OF DELAWARE LIBRARY. Downloaded on March 06,2010 at 13:30:11 EST from IEEE Xplore. Restrictions apply.