Asymptotic distributions of robust shape matrices and scales

Abstract

a b s t r a c tIt has been frequently observed in the literature that many multivariate statistical methods require the covariance or dispersion matrix Σ of an elliptical distribution only up to some scaling constant. If the topic of interest is not the scale but only the shape of the elliptical distribution, it is not meaningful to focus on the asymptotic distribution of an estimator for Σ or another matrix Γ ∝ Σ. In the present work, robust estimators for the shape matrix and the associated scale are investigated. Explicit expressions for their joint asymptotic distributions are derived. It turns out that if the joint asymptotic distribution is normal, the estimators presented are asymptotically independent for one and only one specific choice of the scale function. If it is non-normal (this holds for example if the estimators for the shape matrix and scale are based on the minimum volume ellipsoid estimator) only the scale function presented leads to asymptotically uncorrelated estimators. This is a generalization of a result obtained by Paindaveine [D. Paindaveine, A canonical definition of shape, Statistics and Probability Letters 78 (2008) 2240-2247] in the context of local asymptotic normality theory.Since the seminal paper by Maronna [1], covariance matrix estimation has become a popular branch of robust statistics. Several techniques have been developed for calculating the asymptotic distributions of robust covariance matrix estimators such as the radial distribution approach of Tyler [2] and the approach based on influence functions [3]. Moreover, in recent years deep insights have been gained from the viewpoint of local asymptotic normality (LAN) theory [4][5][6].Let X be a d-dimensional random vector possessing an elliptically symmetric distribution, i.e. it can be represented by X = µ+ΛRU, where U is a k-dimensional random vector, uniformly distributed on the unit hypersphere, R is a nonnegative random variable that is stochastically independent of U, µ ∈ R d , and Λ ∈ R d×k [7,8, p. 42]. It is assumed that R and U are unobservable quantities. The positive-semidefinite matrix Σ := ΛΛ is called the dispersion matrix and R is the generating variate of X . If E(R 2 ) < ∞, the covariance matrix of X is given by Var(X ) = E(R 2 )/k · Σ, whereas if E(R 2 ) = ∞, the linear dependence structure of X can be further described using the dispersion matrix Σ although Var(X ) is not defined.In general I will assume that Σ is positive-definite, i.e. r(Λ) = d ≤ k. In the robust statistics literature [2,9, Ch. 13] and in the context of LAN theory [5,10] it is often supposed that the distribution of R is absolutely continuous. Then the density of X can be written aswhere the so-called density generator g : R + → R + 0 depends on x only through the quadratic form (x − µ) Σ −1 (x − µ). It can be shown [11, p. 9] that the density function of R is given by f (r) ∝ r d−1 g(r 2 ).Tatsuoka and Tyler [12] wrote that 'The assumption of an elliptically symmetric distribution is often made simply because of its mathematical trac...

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“…Here σ 2 : P d → R + is some predefined scale function. This means σ 2 is such that σ 2 (αΣ) = ασ 2 (Σ) > 0 for all α > 0 , σ 2 (I d ) = 1, and σ 2 is differentiable at any point Σ ∈ P d (Frahm, 2009;Paindaveine, 2008). The matrix Ω will be called the shape matrix of X (with respect to the scale function…”

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