On some properties of Buhmann functions

Zastavnyi, V. P.

doi:10.1007/s11253-006-0128-z

Cited by 28 publications

(29 citation statements)

References 24 publications

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“…it is simple to modify (8) to evaluate 20) and hence the problem of positivity reduces to the nonnegativity question on the 2 F 3 hypergeometric functions defined in (20). Our improvement of Theorem A reads as follows.…”

Section: Improved Results Of Misiewicz and Richardsmentioning

confidence: 99%

An extension of positivity for integrals of Bessel functions and Buhmann’s radial basis functions

Cho

Chung

Yun

2018

Proc. Amer. Math. Soc. Ser. B

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Section: Improved Results Of Misiewicz and Richardsmentioning

confidence: 99%

An extension of positivity for integrals of Bessel functions and Buhmann’s radial basis functions

Cho

Chung

Yun

2018

Proc. Amer. Math. Soc. Ser. B

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“…For two given non negative functions g 1 (x) and g 2 (x), with g 1 (x) g 2 (x) we mean that there exist two constants c and C such that 0 < c < C < ∞ and cg 2 (x) ≤ g 1 (x) ≤ Cg 2 (x) for each x. The next result follows from Zastavnyi (2006), Chernih and Hubbert (2014), and from standard properties of Fourier transforms. Their proofs are thus omitted.…”

mentioning

confidence: 95%

“…Inspired by this idea, we focus on a covariance model that offers the strength of the Matérn family and allows the use of sparse matrices. Specifically, we study estimation and prediction of Gaussian fields under fixed domain asymptotics, using the generalized Wendland (GW) class of covariance functions (Gneiting, 2002a;Zastavnyi, 2006), the members of which are compactly supported over balls of R d with arbitrary radii, and additionally allows for a continuous parameterization of differentiability at the origin, in a similar way to the Matérn family.…”

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confidence: 99%

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Estimation and prediction using generalized Wendland covariance functions under fixed domain asymptotics

et al. 2019

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We study estimation and prediction of Gaussian random fields with covariance models belonging to the generalized Wendland (GW) class, under fixed domain asymptotics. As for the Matérn case, this class allows for a continuous parameterization of smoothness of the underlying Gaussian random field, being additionally compactly supported. The paper is divided into three parts: first, we characterize the equivalence of two Gaussian measures with GW covariance function, and we provide sufficient conditions for the equivalence of two Gaussian measures with Matérn and GW covariance functions. In the second part, we establish strong consistency and asymptotic distribution of the maximum likelihood estimator of the microergodic parameter associated to GW covariance model, under fixed domain asymptotics. The third part elucidates the consequences of our results in terms of (misspecified) best linear unbiased predictor, under fixed domain asymptotics. Our findings are illustrated through a simulation study: the former compares the finite sample behavior of the maximum likelihood estimation of the microergodic parameter with the given asymptotic distribution. The latter compares the finitesample behavior of the prediction and its associated mean square error when using two equivalent Gaussian measures with Matérn and GW covariance models, using covariance tapering as benchmark.1 imsart-aos ver.

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“…Wendland functions. Arguments in Gneiting (2002) and Zastavnyi (2006) show that, for a given κ > 0, ϕ µ,κ,β,σ 2 ∈ Φ β d if and only if µ ≥ (d + 1) 2 + κ. In this special case κ = 0 the GW correlation function is defined as:…”

Section: Matérn Generalized Wendland and Generalized Cauchy Covarianmentioning

confidence: 99%

Estimation and prediction of Gaussian processes using generalized Cauchy covariance model under fixed domain asymptotics

Bevilacqua¹,

Faouzi²

2019

Electron. J. Statist.

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We study estimation and prediction of Gaussian processes with covariance model belonging to the generalized Cauchy (GC) family, under fixed domain asymptotics. Gaussian processes with this kind of covariance function provide separate characterization of fractal dimension and long range dependence, an appealing feature in many physical, biological or geological systems. The results of the paper are classified into three parts.In the first part, we characterize the equivalence of two Gaussian measures with GC covariance functions. Then we provide sufficient conditions for the equivalence of two Gaussian measures with Matérn (MT) and GC covariance functions and two Gaussian measures with Generalized Wendland (GW) and GC covariance functions.In the second part, we establish strong consistency and asymptotic distribution of the maximum likelihood estimator of the microergodic parameter associated to GC covariance model, under fixed domain asymptotics. The last part focuses on optimal prediction with GC model and specifically, we give conditions for asymptotic efficiency prediction and asymptotically correct estimation of mean square error using a misspecified GC, MT or GW model.Our findings are illustrated through a simulation study: the first compares the finite sample behavior of the maximum likelihood estimation of the microergodic parameter of the GC model with the given asymptotic distribution. We then compare the finite-sample behavior of the prediction and its associated mean square error when the true model is GC and the prediction is performed using the true model and a misspecified GW model.

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On some properties of Buhmann functions

Abstract: We study functions introduced by Buhmann. The exact exponent of smoothness of these functions is obtained and the problem of positivity of their Hankel transforms is analyzed.

Cited by 28 publications

References 24 publications

An extension of positivity for integrals of Bessel functions and Buhmann’s radial basis functions

An extension of positivity for integrals of Bessel functions and Buhmann’s radial basis functions

Estimation and prediction using generalized Wendland covariance functions under fixed domain asymptotics

Estimation and prediction of Gaussian processes using generalized Cauchy covariance model under fixed domain asymptotics

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