On Positive Definiteness of Some Functions
AbstractLet [rho] be a nonnegative homogeneous function on n. General structure of the set of numerical pairs ([delta],Â [lambda]), for which the function (1-[rho][lambda](x))[delta]+ is positive definite on n is investigated; a criterion for positive definiteness of this function is given in terms of completely monotonic functions; a connection of this problem with the Schoenberg problem on positive definiteness of the function exp(-[rho][lambda](x)) is found. We also obtain a general sufficient condition of Polya type for a function f([rho](x)) to be positive definite on n.
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Bibliographic InfoArticle provided by Elsevier in its journal Journal of Multivariate Analysis.
Volume (Year): 73 (2000)
Issue (Month): 1 (April)
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Web page: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description
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- Richards, Donald St. P., 1985. "Positive definite symmetric functions on finite-dimensional spaces II," Statistics & Probability Letters, Elsevier, vol. 3(6), pages 325-329, October.
- Ma, Chunsheng, 2004. "Spatial autoregression and related spatio-temporal models," Journal of Multivariate Analysis, Elsevier, vol. 88(1), pages 152-162, January.
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- Dahl, Christian M. & Gonzalez-Rivera, Gloria, 2003. "Testing for neglected nonlinearity in regression models based on the theory of random fields," Journal of Econometrics, Elsevier, vol. 114(1), pages 141-164, May.
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