On the uncertainty relation for positive-definite probability densities, II

Werner Ehm, Tilmann Gneiting, Donald Richards

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

Let P denote the class of probability density functions on ℝ with a nonnegative and integrable characteristic function. To each p ε P, there is an adjoint density p̂, which is proportional to the characteristic function of p. The products λ(p) = Var(p) Var(p̂) have a greatest lower bound Λ, and it is known that 0.5276 < Λ < 0.8571. Several approaches to sharpen these bounds are discussed. In particular, a variational problem is considered, in which p is supposed to have a certain compactly supported convolution root, and which leads to an improved upper estimate, Λ < 0.8567... The paper closes with a proposal for a multivariate analogue of the uncertainty relation.

Original languageEnglish (US)
Pages (from-to)267-286
Number of pages20
JournalStatistics
Volume33
Issue number3
DOIs
StatePublished - Jan 1 1999

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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