Onα-Symmetric Multivariate Characteristic Functions

Gneiting, Tilmann

doi:10.1006/jmva.1997.1713

Cited by 42 publications

(32 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In the case of (A1) and (B1) this easily follows from a result due to R. Askey, see Theorem 4.1 in Gneiting, 1998. In the case of assumptions (A2) and (B2), symmetry and unimodality…”

Section: Consequently Ifmentioning

confidence: 65%

EFFICIENCY OF LINEAR ESTIMATORS UNDER HEAVY-TAILEDNESS: CONVOLUTIONS OF [alpha]-SYMMETRIC DISTRIBUTIONS

Ибрагимов

2007

Econ. Theory

View full text Add to dashboard Cite

The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. Among other results, the paper shows that the sample mean is the best linear unbiased estimator of the population mean for not extremely heavy-tailed populations in the sense of its peakedness properties. In addition, in such a case, the sample mean exhibits the property of monotone consistency and, thus, an increase in the sample size always improves its performance. However, efficiency of the sample mean in the sense of its peakedness decreases with the sample size if the sample mean is used to estimate the population center under extreme heavy-tailedness. The paper also provides applications of the main efficiency and majorization comparison results in the study of concentration inequalities for linear estimators.

show abstract

“…In the case of (A1) and (B1) this easily follows from a result due to R. Askey, see Theorem 4.1 in Gneiting, 1998. In the case of assumptions (A2) and (B2), symmetry and unimodality…”

Section: Consequently Ifmentioning

confidence: 65%

EFFICIENCY OF LINEAR ESTIMATORS UNDER HEAVY-TAILEDNESS: CONVOLUTIONS OF [alpha]-SYMMETRIC DISTRIBUTIONS

Ибрагимов

2007

Econ. Theory

View full text Add to dashboard Cite

show abstract

“…for t > 0, where ϕ k+l,k+l is the Euclid's hat function (5), and G is nondecreasing with G(0+) = 0 but not necessarily bounded. The interchange in the order of the differentiation and integration is justified by the compact support and boundedness of Wu's functions.…”

Section: Proof Of the Criterionmentioning

confidence: 99%

“…Thus, Φ 1 (α) = Φ 1 , independently of α, and Φ n (2) is the class Φ n defined in Section 1. For recent reviews of · α -dependent positive definite functions and connections to the isometric theory of Banach spaces, we refer to Koldobsky [13], Misiewicz [19], and Gneiting [5].…”

Section: · α -Dependent Positive Definite Functionsmentioning

confidence: 99%

“…Clearly k 1,α (λ) is Kuttner's function k 1 (λ), independently of α, and k n,2 (λ) is the Kuttner-Golubov function k n (λ). If we fix λ = 1, then k n,α (1) is the Richards-Askey function which the author introduced in [5]. Zastavnyi [35] established an interesting connection to Schoenberg's [28] classical question whether ϕ(t) = exp(−t λ ) is an element of Φ n (α).…”

Section: · α -Dependent Positive Definite Functionsmentioning

confidence: 99%

“…Using computer algebra systems, this lends itself to simple and powerful tests for positive definiteness. Finally, the criterion can also be used to check whether a radial function has a unimodal Fourier transform (compare Askey [2] and Section 4 of Gneiting [5]). …”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Criteria of Pólya type for radial positive definite functions

Gneiting

2001

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. This article presents sufficient conditions for the positive definiteness of radial functions f (x) = ϕ( x ), x ∈ R n , in terms of the derivatives of ϕ. The criterion extends and unifies the previous analogues of Pólya's theorem and applies to arbitrarily smooth functions. In particular, it provides upper bounds on the Kuttner-Golubov function kn(λ) which gives the minimal value of κ such that the truncated power function (1 − x λ ) κ + , x ∈ R n , is positive definite. Analogous problems and criteria of Pólya type for · α-dependent functions, α > 0, are also considered.

show abstract