TY - JOUR

T1 - On the uncertainty relation for positive-definite probability densities, II

AU - Ehm, Werner

AU - Gneiting, Tilmann

AU - Richards, Donald

N1 - Funding Information:
*Corresponding author. +~esearcshu pported, in part, by National Science Foundation, grant DMS-9703705.

PY - 1999

Y1 - 1999

N2 - 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.

AB - 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.

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U2 - 10.1080/02331889908802694

DO - 10.1080/02331889908802694

M3 - Article

AN - SCOPUS:0347036766

VL - 33

SP - 267

EP - 286

JO - Statistics

JF - Statistics

SN - 0233-1888

IS - 3

ER -