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.
- 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