AI 中文总结
该研究探讨自然指数族中精确双侧p值的非唯一性，分析不同构造p值的重合条件，结合相关定理刻画高斯族等特殊分布，并通过示例量化p值差异与决策分歧。
AI 中文摘要
我们研究连续单参数自然指数族（NEF）中精确双侧p值的非唯一性问题。对于定向单侧问题，尾部p值与使用UMP、UMPU及似然比（LR）检验得到的p值一致。对于双侧简单原假设，我们区分四种构造：等尾p值、密度有序p值、UMPU p值及LR p值。在固定原假设参数下，当且仅当原分布关于其均值对称时，UMPU p值与等尾p值重合；在正则双分支密度水平条件下，相同的固定原假设对称性特征也适用于UMPU与密度有序p值、等尾与密度有序p值的比较。要求NEF中存在此类重合关系可刻画高斯族。我们将这些结果与Bar-Lev、Bshouty和Letac的定理结合，该定理指出连续NEF中UMPU与LR p值重合的情况仅发生在正态、伽马及逆高斯族中。我们还研究了未被上述结果覆盖的两类LR配对：若满足常规正则性假设的NEF中，等尾p值与LR p值全程重合，则均值域上(V²/3)'''=0；此类重合还会强制出现明确的均值处密度恒等式。对于具有典范充分统计量Tₙ=Σⁿᵢ₌₁Xᵢ的独立同分布样本，沿无界样本量序列的等尾-LR重合性在族中持续存在可强制高斯性。基于明确给出的微分局部Edgeworth展开，给出了对应的密度-LR结论。最后，逆高斯与双曲正割示例量化了数值p值差异、拒绝决策分歧及功效差异。
英文摘要
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 $(V^{2/3})^{\prime \prime \prime }=0$ 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 $T_n=\sum_{i=1}^nX_i$, 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.
