Unrestricted one-observation classification of LR coincidence

Classify all continuous one-parameter natural exponential families for which, at sample size n = 1, equal-tail and likelihood-ratio two-sided p-values coincide throughout the family, and likewise classify all such families for which density-ordered and likelihood-ratio two-sided p-values coincide, without restricting to power-variance families or imposing sample-size stability.

Background

The paper studies four exact two-sided p-value constructions in continuous one-parameter natural exponential families: equal-tail, density-ordered, UMPU, and likelihood-ratio p-values. It establishes several global coincidence characterizations, including Gaussian characterizations for the symmetry-based pairings and the normal–gamma–inverse-Gaussian classification for UMPU–likelihood-ratio coincidence.

The equal-tail–likelihood-ratio and density–likelihood-ratio pairings remain incompletely characterized at one observation. For equal-tail–likelihood-ratio coincidence, the paper derives necessary conditions: the variance function must satisfy that its two-thirds power has vanishing third derivative, and the density evaluated at its own mean must be proportional to the inverse one-third power of the variance function. These conditions yield a Gaussian conclusion within the power-variance class, but the paper does not establish sufficiency or a full classification for arbitrary natural exponential families. The density–likelihood-ratio case likewise receives a conditional sample-size-stability result rather than an unrestricted n = 1 theorem.