Exact two-sided p-values in natural exponential families: coincidence, non-uniqueness, and sample-size stability
Abstract
We study the non-uniqueness of exact two-sided p-values in continuous one-parameter natural exponential families (NEFs). For directed one-sided problems, the tail p-value agrees with the p-values using UMP, UMPU, and likelihood-ratio (LR) tests. For a two-sided simple null, we distinguish four constructions: equal-tail, density-ordered, UMPU, and LR p-values. At a fixed null parameter, UMPU and equal-tail p-values coincide if and only if the null law is symmetric about its mean; under a regular two-branch density-level condition, the same fixed-null symmetry characterization holds for UMPU versus density ordering and equal-tail versus density ordering. Requiring any of these coincidences throughout the NEF characterizes the Gaussian family. We combine these results with the theorem of Bar-Lev, Bshouty and Letac that UMPU and LR p-values coincide throughout a continuous NEF precisely for the normal, gamma and inverse-Gaussian families. We also investigate the two LR pairings not covered by those results. If equal-tail and LR p-values coincide throughout a NEF satisfying our standing regularity assumptions, then (V2/3)′ ′ ′ =0 on the mean domain. The same coincidence also forces an explicit density-at-the-mean identity. For an i.i.d.\ sample with canonical sufficient statistic Tn=∑i=1nXi, persistence of equal-tail-LR coincidence throughout the family along an unbounded sequence of sample sizes forces Gaussianity. A corresponding density-LR statement is given conditionally on an explicitly stated differentiated local Edgeworth expansion. Finally, inverse-Gaussian and hyperbolic-secant examples quantify numerical p-value differences, disagreement of rejection decisions, and differences in power.
Related
- On Natural Exponential Families with Log-Affine Generalized Variance
- Kullback–Leibler Asymmetry and Information Homogeneity in Natural Exponential Families
- From Counting to Continuous Natural Exponential Families
- Equivalence on exponential families as Hessian manifolds and classification of exponential families of order 1 with constant Hessian sectional curvature
- Cramer-Rao Type Bounds for Generalized Exponential Families
BibTeX
@misc{barlev2026exact,
title = {{Exact two-sided p-values in natural exponential families: coincidence, non-uniqueness, and sample-size stability}},
author = {Shaul K. Bar-Lev and Linard Hoessly},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28221},
eprint = {2608.28221},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28221},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact two-sided p-values in natural exponential families: coincidence, non-uniqueness, and sample-size stability
AU - Shaul K. Bar-Lev
AU - Linard Hoessly
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28221
UR - https://arxiv.org/abs/2608.28221
AB - We study the non-uniqueness of exact two-sided p-values in continuous one-parameter natural exponential families (NEFs). For directed one-sided problems, the tail p-value agrees with the p-values using UMP, UMPU, and likelihood-ratio (LR) tests. For a two-sided simple null, we distinguish four constructions: equal-tail, density-ordered, UMPU, and LR p-values. At a fixed null parameter, UMPU and equal-tail p-values coincide if and only if the null law is symmetric about its mean; under a regular two-branch density-level condition, the same fixed-null symmetry characterization holds for UMPU versus density ordering and equal-tail versus density ordering. Requiring any of these coincidences throughout the NEF characterizes the Gaussian family. We combine these results with the theorem of Bar-Lev, Bshouty and Letac that UMPU and LR p-values coincide throughout a continuous NEF precisely for the normal, gamma and inverse-Gaussian families. We also investigate the two LR pairings not covered by those results. If equal-tail and LR p-values coincide throughout a NEF satisfying our standing regularity assumptions, then (V^2/3)^′ ′ ′ =0 on the mean domain. The same coincidence also forces an explicit density-at-the-mean identity. For an i.i.d.\ sample with canonical sufficient statistic T_n=∑_i=1^nX_i, persistence of equal-tail-LR coincidence throughout the family along an unbounded sequence of sample sizes forces Gaussianity. A corresponding density-LR statement is given conditionally on an explicitly stated differentiated local Edgeworth expansion. Finally, inverse-Gaussian and hyperbolic-secant examples quantify numerical p-value differences, disagreement of rejection decisions, and differences in power.
ER -