有序双变量均值的经验似然置信区域
Empirical likelihood confidence regions for ordered bivariate means
浏览论文内容
中文总结 AI 辅助
本文研究满足$μ_1\leqμ_2$的双变量均值的经验似然置信区域，推导其在内点、边界点的极限分布，约简统计量并通过模拟验证，明确了与相关检验方法的实际差异。
中文摘要 AI 辅助
设$\boldsymbol{X}_i=(X_{1i},X_{2i})^\top$为独立同分布观测值，其均值$\boldsymbolμ=(μ_1,μ_2)^\top$满足$μ_1\leqμ_2$。本文研究固定均值向量的经验似然推断，并将其与已知的针对有序备择的相等性检验区分开来。在固定内点处，约束经验似然比具有常规的$χ^2_2$极限；在固定边界点$(m,m)^\top$处，其极限为卡方混合分布$\tfrac12χ^2_1+\tfrac12χ^2_2$。相比之下，在“相等性对有序性”检验中对未知公共均值进行轮廓处理，会得到$\tfrac12χ^2_0+\tfrac12χ^2_1$，这是El Barmi（1996）研究的$k=2$有序均值情形。本文将后一统计量精确约简为配对差值的经验似然，建立了固定边界展开所需的局部化步骤，并推导了近边界极限，表明内点校准在边界的$n^{-1/2}$邻域内并非一致。通过高斯分布、Student t₅分布和移位对数正态分布抽样下的蒙特卡洛实验，本文检验了固定、边界和局部情形，并明确记录了数值失效情况。配对数据的示例分析展示了固定候选置信区域、方向相等性检验以及截断到非负参数空间的普通标量经验似然区间之间的实际差异。
英文摘要
Let $\boldsymbol{X}_i=(X_{1i},X_{2i})^\top$ be independent and identically distributed observations with mean $\boldsymbolμ=(μ_1,μ_2)^\top$ constrained by $μ_1\leqμ_2$. We study empirical-likelihood inference for a fixed mean vector and distinguish it from the previously known test of equality against an ordered alternative. At a fixed interior point, the constrained empirical likelihood ratio has the usual $χ^2_2$ limit. At a fixed boundary point $(m,m)^\top$, its limit is the chi-bar-square distribution $\tfrac12χ^2_1+\tfrac12χ^2_2$. By contrast, profiling the unknown common mean in the equality-versus-order test yields $\tfrac12χ^2_0+\tfrac12χ^2_1$, the $k=2$ ordered-mean case of El Barmi (1996). We give an exact reduction of the latter statistic to the empirical likelihood of the paired differences, establish the localization step needed for the fixed-boundary expansion, and derive a local-to-boundary limit showing that interior calibration is not uniform over $n^{-1/2}$-neighborhoods of the boundary. Monte Carlo experiments under Gaussian, Student $t_5$, and shifted log-normal sampling examine fixed, boundary, and local regimes with explicit numerical-failure accounting. Illustrative paired-data analyses show the practical distinction between fixed-candidate confidence regions, directional equality tests, and ordinary scalar empirical-likelihood intervals truncated to the nonnegative parameter space.
