AI 中文总结
该研究针对概率重心与柯尔莫哥洛夫矩，在固定、固有及估计的概率坐标图下开发了统计推断方法，推导了置信区间、渐近方差及联合协方差理论，分离了不同来源的不确定性。
AI 中文摘要
概率坐标图是一种连续严格递增的双射映射，可将观测值转换到开单位区间，在此区间内求平均始终有良好定义。对于重心推断，通过逆图返回第一坐标矩；更高阶的初始坐标矩通过拉回操作生成初始柯尔莫哥洛夫矩，而中心化坐标矩则保留在概率尺度上。我们针对固定、固有及估计的图，开发了这些量的推断方法。对于固定基准图，坐标均值是主要推断对象，在概率尺度上构造置信区间，再通过逆图转换，保留几何结构并允许观测尺度上的非对称性；Hoeffding不等式还提供了有限样本下无分布依赖的区间。固有情形需要不同处理：经验累积分布函数不是可接受的图，其自然的广义插件构造恰好退化为中间顺序统计量，因此可行的固有推断简化为中位数推断，该自诱导函数与冻结的 oracle 构造不同，后者具有不同的渐近方差。对于估计的位置-尺度图，渐近展开包含校准修正项，用于校正从同一样本学习图的一阶效应，这产生了影响函数方差估计量和一种在每次重采样中重新校准图的自助法。我们还针对初始和中心化坐标矩的向量开发了联合协方差理论，初始柯尔莫哥洛夫矩通过逆图拉回得到，该框架分离了概率尺度上的变异、逆图的放大效应以及图校准带来的不确定性。
英文摘要
A probability coordinate chart is a continuous strictly increasing bijection that transports observations to the open unit interval, where averaging is always well defined. For barycentric inference, the first coordinate moment is returned through the inverse chart. Higher initial coordinate moments similarly generate initial Kolmogorov moments by pullback, while centred coordinate moments remain on the probability scale. We develop inference for these quantities under fixed, intrinsic, and estimated charts. For a fixed benchmark chart, the coordinate mean is the primary inferential object. Confidence intervals are constructed on the probability scale and then transported through the inverse chart, preserving the geometry and allowing asymmetry on the observation scale. Hoeffding's inequality also provides finite-sample distribution-free intervals. The intrinsic case requires different treatment. The empirical cumulative distribution function is not an admissible chart, and its natural generalised plug-in construction collapses exactly to a middle order statistic. Feasible intrinsic inference therefore reduces to median inference. This self-induced functional is distinguished from the frozen oracle construction, which has a different asymptotic variance. For estimated location-scale charts, the asymptotic expansion contains a calibration correction for the first-order effect of learning the chart from the same sample. This yields an influence-function variance estimator and a bootstrap procedure that recalibrates the chart in every resample. Joint covariance theory is developed for vectors of initial and centred coordinate moments, with initial Kolmogorov moments obtained by inverse-chart pullback. The framework separates variation on the probability scale, amplification by the inverse chart, and uncertainty from chart calibration.
