具有保证必要性与充分性的间接统计推断
Indirect Statistical Inference with Guaranteed Necessity and Sufficiency
AI总结：
本文提出基于指数变换次序统计量相对误差的间接推断框架，证明其充要性，相比绝对误差方法收敛更快，适用于连续随机变量分布检验。
摘要：
本文为连续随机变量开发了一个新的间接统计推断框架，该框架具有保证的必要性与充分性。我们证明，当将假设分布中的指数变换次序统计量与模拟的单位指数样本中的相应统计量进行比较时，排序后的商呈现出不同的渐近行为：左段收敛于非退化分布，而中段和右段退化为一。这为两个连续随机变量序列服从同一分布提供了一个概率意义上的充要条件。在此基础上，我们提出了一种基于有序样本之间相对误差的优化准则。该准则达到最小值当且仅当假设分布与真实分布一致，从而提供了优化意义上的第二个充要条件。这些双重NS性质在文献中罕见，建立了一个比现有方法本质上更强的推断框架。与基于绝对误差的经典方法（如Kolmogorov-Smirnov）不同，NSE利用相对误差以确保更快的收敛速度，仅需累积分布函数的温和可逼近性，并提供点估计和区间估计。模拟和真实数据应用证实了NSE在传统方法失效的情况下保持分布假设方面的优越性能。
英文摘要：
This paper develops a new framework for indirect statistical inference with guaranteed necessity and sufficiency, applicable to continuous random variables. We prove that when comparing exponentially transformed order statistics from an assumed distribution with those from simulated unit exponential samples, the ranked quotients exhibit distinct asymptotics: the left segment converges to a non-degenerate distribution, while the middle and right segments degenerate to one. This yields a necessary and sufficient condition in probability for two sequences of continuous random variables to follow the same distribution. Building on this, we propose an optimization criterion based on relative errors between ordered samples. The criterion achieves its minimum if and only if the assumed and true distributions coincide, providing a second necessary and sufficient condition in optimization. These dual NS properties, rare in the literature, establish a fundamentally stronger inference framework than existing methods. Unlike classical approaches based on absolute errors (e.g., Kolmogorov-Smirnov), NSE exploits relative errors to ensure faster convergence, requires only mild approximability of the cumulative distribution function, and provides both point and interval estimates. Simulations and real-data applications confirm NSE's superior performance in preserving distributional assumptions where traditional methods fail.
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