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Schoenberg’s problem on positive definite functions, ( English translation in St
 Petersburg Math. J
, 1992
"... numbers β> 0 is the function exp(−‖x ‖ β q) positive definite on R ⋉? Here q> 2 and ‖x‖q = (x1  q + · · ·+xn  q) 1/q. Denote by Bn(q) the set of such numbers β. ..."
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numbers β> 0 is the function exp(−‖x ‖ β q) positive definite on R ⋉? Here q> 2 and ‖x‖q = (x1  q + · · ·+xn  q) 1/q. Denote by Bn(q) the set of such numbers β.
Influence function an maximum bias of projection depth based on estimators
"... Location estimators induced from depth functions increasingly have been pursued and studied in the literature. Among them are those induced from projection depth functions. These projection depth based estimators have favorable properties among their competitors. In particular, they possess the best ..."
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Location estimators induced from depth functions increasingly have been pursued and studied in the literature. Among them are those induced from projection depth functions. These projection depth based estimators have favorable properties among their competitors. In particular, they possess the best possible finite sample breakdown point robustness. However, robustness of estimators cannot be revealed by the finite sample breakdown point alone. The influence function, gross error sensitivity, maximum bias and contamination sensitivity are also important aspects of robustness. In this article, we study these other robustness aspects of two types of projection depth based estimators: projection medians and projection depth weighted means. The latter includes the Stahel–Donoho estimator as a special case. Exact maximum bias, the influence function, and contamination and gross error sensitivity are derived and studied for both types of estimators. Sharp upper bounds for the maximum bias and the influence functions are established. Comparisons based on these robustness criteria reveal that the projection depth based estimators enjoy desirable local as well as global robustness and are very competitive among their competitors.
CONVOLUTION ROOTS OF RADIAL POSITIVE DEFINITE FUNCTIONS WITH COMPACT SUPPORT
"... Abstract. A classical theorem of Boas, Kac, and Krein states that a characteristic function ϕ with ϕ(x) =0forx  ≥τ admits a representation of the form ϕ(x) = u(y)u(y + x)dy, x ∈ R, where the convolution root u ∈ L 2 (R) is complexvalued with u(x) =0for x  ≥τ/2. The result can be expressed equi ..."
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Abstract. A classical theorem of Boas, Kac, and Krein states that a characteristic function ϕ with ϕ(x) =0forx  ≥τ admits a representation of the form ϕ(x) = u(y)u(y + x)dy, x ∈ R, where the convolution root u ∈ L 2 (R) is complexvalued with u(x) =0for x  ≥τ/2. The result can be expressed equivalently as a factorization theorem for entire functions of finite exponential type. This paper examines the BoasKac representation under additional constraints: If ϕ is realvalued and even, can the convolution root u be chosen as a realvalued and/or even function? A complete answer in terms of the zeros of the Fourier transform of ϕ is obtained. Furthermore, the analogous problem for radially symmetric functions defined on R d is solved. Perhaps surprisingly, there are compactly supported, radial positive definite functions that do not admit a convolution root with halfsupport. However, under the additional assumption of nonnegativity, radially symmetric convolution roots with halfsupport exist. Further results in this paper include a characterization of extreme points, pointwise and integral bounds (Turán’s problem), and a unified solution to a minimization problem for compactly supported positive definite functions. Specifically, if f is a probability density on R d whose characteristic function ϕ vanishes outside the unit ball, then ∫ x  2 f(x)dx = −∆ϕ(0) ≥ 4 j 2 (d−2)/2 where jν denotes the first positive zero of the Bessel function Jν, andtheestimate is sharp. Applications to spatial moving average processes, geostatistical simulation, crystallography, optics, and phase retrieval are noted. In particular, a realvalued halfsupport convolution root of the spherical correlation function in R 2 does not exist. 1.
A note on positive definite norm dependent functions
 Proceedings of the Conference on High Dimensional Probability, Luminy
, 2008
"... Abstract. Let K be an origin symmetric star body in R n. We prove, under very mild conditions on the function f: [0, ∞) → R, that if the function f(‖x‖K) is positive definite on R n, then the space (R n, ‖ · ‖K) embeds isometrically in L0. This generalizes the solution to Schoenberg’s problem and ..."
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Abstract. Let K be an origin symmetric star body in R n. We prove, under very mild conditions on the function f: [0, ∞) → R, that if the function f(‖x‖K) is positive definite on R n, then the space (R n, ‖ · ‖K) embeds isometrically in L0. This generalizes the solution to Schoenberg’s problem and leads to progress in characterization of ndimensional versions, i.e. random vectors X = (X1,...,Xn) in R n such that the random variables ∑ aiXi are identically distributed for all a ∈ R n, up to a constant depending on ‖a‖K only. 1.
© Institute of Mathematical Statistics, 2005 DEPTH WEIGHTED SCATTER ESTIMATORS
"... General depth weighted scatter estimators are introduced and investigated. For general depth functions, we find out that these affine equivariant scatter estimators are Fisher consistent and unbiased for a wide range of multivariate distributions, and show that the sample scatter estimators are st ..."
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General depth weighted scatter estimators are introduced and investigated. For general depth functions, we find out that these affine equivariant scatter estimators are Fisher consistent and unbiased for a wide range of multivariate distributions, and show that the sample scatter estimators are strong and nconsistent and asymptotically normal, and the influence functions of the estimators exist and are bounded in general. We then concentrate on a specific case of the general depth weighted scatter estimators, the projection depth weighted scatter estimators, which include as a special case the wellknown Stahel–Donoho scatter estimator whose limiting distribution has long been open until this paper. Large sample behavior, including consistency and asymptotic normality, and efficiency and finite sample behavior, including breakdown point and relative efficiency of the sample projection depth weighted scatter estimators, are thoroughly investigated. The influence function and the maximum bias of the projection depth weighted scatter estimators are derived and examined. Unlike typical highbreakdown competitors, the projection depth weighted scatter estimators can integrate high breakdown point and high efficiency while enjoying a boundedinfluence function and a moderate maximum bias curve. Comparisons with leading estimators on asymptotic relative efficiency and gross error sensitivity reveal that the projection depth weighted scatter estimators behave very well overall and, consequently, represent very favorable choices of affine equivariant multivariate scatter estimators. 1. Introduction. The
POSITIVE DEFINITE FUNCTIONS AND MULTIDIMENSIONAL VERSIONS OF RANDOM VARIABLES
, 903
"... Abstract. We say that a random vector X = (X1,..., Xn) in R n is an ndimensional version of a random variable Y if for any a ∈ R n the random variables ∑ aiXi and γ(a)Y are identically distributed, where γ: R n → [0, ∞) is called the standard of X. An old problem is to characterize those functions ..."
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Abstract. We say that a random vector X = (X1,..., Xn) in R n is an ndimensional version of a random variable Y if for any a ∈ R n the random variables ∑ aiXi and γ(a)Y are identically distributed, where γ: R n → [0, ∞) is called the standard of X. An old problem is to characterize those functions γ that can appear as the standard of an ndimensional version. In this paper, we prove the conjecture of Lisitsky that every standard must be the norm of a space that embeds in L0. This result is almost optimal, as the norm of any finite dimensional subspace of Lp with p ∈ (0, 2] is the standard of an ndimensional version (pstable random vector) by the classical result of P.Lèvy. An equivalent formulation is that if a function of the form f( ‖ · ‖K) is positive definite on R n, where K is an origin symmetric star body in R n and f: R → R is an even continuous function, then either the space (R n, ‖ · ‖K) embeds in L0 or f is a constant function. Combined with known facts about embedding in L0, this result leads to several generalizations of the solution of Schoenberg’s problem on positive definite functions. 1.
http://cowles.econ.yale.edu / Extreme Adverse Selection, Competitive Pricing, and Market Breakdown ∗
, 2006
"... Abstract: Extreme adverse selection arises when private information has unbounded support, and market breakdown occurs when no trade is the only equilibrium outcome. We study extreme adverse selection via the limit behavior of a financial market as the support of private information converges to an ..."
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Abstract: Extreme adverse selection arises when private information has unbounded support, and market breakdown occurs when no trade is the only equilibrium outcome. We study extreme adverse selection via the limit behavior of a financial market as the support of private information converges to an unbounded support. A necessary and sufficient condition for market breakdown is obtained. If the condition fails, then there exists competitive market behavior that converges to positive levels of trade whenever it is first best to have trade. When the condition fails, no feasible (competitive or not) market behavior converges to positive levels of trade.