基于概率坐标图的G-指数族
G-Exponential Families through Probability Coordinates
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中文总结 AI 辅助
该研究提出基于概率坐标图构造G-指数族的概率几何方法，其充分统计量有界、估计器鲁棒，通过柯西坐标图示例和蒙特卡洛研究验证了其估计性能。
中文摘要 AI 辅助
我们开发了一种基于概率坐标图的广义指数族的概率几何构造方法。将单位区间上的经典指数族通过坐标图映射到值空间，从而得到我们所称的G-指数族。该构造改变了指数结构所表示的几何，而指数函数本身保持不变，这使其与Tsallis和Kaniadakis框架中使用的代数变形有所区别。被映射的族继承了概率坐标框架的结构，每个族成员的概率坐标恰好服从单位区间上的原始指数族。因此，初始Kolmogorov矩可通过将经典坐标矩拉回值空间得到。对于典范族，概率重心是均值参数的拉回，而Fisher信息与Kolmogorov方差一致。尾部行为由坐标图继承，每个族成员与坐标图密度尾部等价，因此重尾现象源于几何，且在概率坐标倾斜下阶数保持不变，这与经典指数倾斜相反。该构造的充分统计量有界，最大似然估计简化为概率坐标下的矩匹配，且最大似然估计器具有有界影响函数，因此鲁棒性源于几何紧化。我们还将G-指数族表征为在坐标矩约束下对坐标图律的相对熵的极小化者，并证明映射保留了族的信息几何。柯西坐标图示例和蒙特卡洛研究证实，即使每个族成员具有无限均值，仍能实现高效、校准且有界影响的估计。
英文摘要
We develop a probability geometric construction of generalized exponential families based on probability coordinate charts. A classical exponential family on the unit interval is transported to value space through a chart, producing what we call a G-exponential family. The construction changes the geometry in which exponential structure is represented while leaving the exponential function itself unchanged. This distinguishes it from the algebraic deformations used in the Tsallis and Kaniadakis frameworks. The transported families inherit the structure of the probability coordinate framework. The probability coordinate of each family member follows exactly the original exponential family on the unit interval. Initial Kolmogorov moments are therefore obtained by pulling classical coordinate moments back to value space. For the canonical family, the probability barycenter is the pullback of the mean parameter, while the Fisher information coincides with the Kolmogorov variance. Tail behaviour is inherited from the chart. Every family member is tail equivalent to the chart density, so heavy tails arise from geometry and remain unchanged in order under probability coordinate tilting. This is the opposite of classical exponential tilting. The sufficient statistic is bounded by construction, maximum likelihood reduces to moment matching in probability coordinates, and the maximum likelihood estimator has a bounded influence function. Thus robustness arises from geometric compactification. We also characterise G-exponential families as minimisers of relative entropy to the chart law under coordinate moment constraints and prove that transport preserves the information geometry of the family. A Cauchy chart example and a Monte Carlo study confirm efficient, calibrated, and bounded influence estimation even though every family member has an infinite mean.
