Normal, gamma and inverse-Gaussian are the only NEFs where the bilateral UMPU and GLR tests coincide

Shaul K. Bar-Lev, Daoud Bshouty, Gérard Letac

Research output: Contribution to journalArticlepeer-review

Abstract

Consider an NEF F on the real line parametrized by θ ∈ Θ. Also let θ0 be a specified value of θ. Consider the test of size α for a simple hypothesis H0:θ =θ 0 versus two sided alternative H1:θ≠θ 0 A UMPU test of size α then exists for any given α. Suppose that F is continuous. Therefore the UMPU test is nonrandomized and then becomes comparable with the generalized likelihood ratio test (GLR). Under mild conditions we show that the two tests coincide iff F is either a normal or inverse Gaussian or gamma family. This provides a new global characterization of this set of NEFs. The proof involves a differential equation obtained by the cancelling of a determinant of order 6.

Original languageEnglish
Pages (from-to)1524-1534
Number of pages11
JournalAnnals of Statistics
Volume30
Issue number5
DOIs
StatePublished - Oct 2002

Keywords

  • Generalized likelihood ratio test
  • Natural exponential families
  • Uniformly most powerful unbiased test
  • Variance functions

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Fingerprint

Dive into the research topics of 'Normal, gamma and inverse-Gaussian are the only NEFs where the bilateral UMPU and GLR tests coincide'. Together they form a unique fingerprint.

Cite this