报告题目：必要充分计算机仿真试验打通精准统计推断之路

报告时间：2024-10-17  10:00-11:30

报 告 人：张正军  教授（中国科学院大学）

报告地点：数学院二楼报告厅

Abstract: Modern AI needs accurate statistical inference methods. The existing statistical estimations were often derived from some sufficient conditions but not necessary. We first establish a fundamental theorem that guarantees the transformed order statistics from the assumed distribution of a random variable (or an error term) to be arbitrarily close to the order statistics of a simulated sequence of the same distribution. The theorem leads to a necessary and sufficient condition for two series of random variables to follow the same distribution. Based on the condition, we propose a new necessary and sufficient estimation (NSE) method for preserving continuous distribution assumptions in various statistical studies. Unlike the Kolmogorov-Smirnov statistics and many other statistics based on absolute errors between the empirical distribution and the assumed distribution, the statistics proposed are based on relative errors between the transformed and simulated order statistics. Surprisingly, relative errors result in much faster convergence rates than absolute errors. Using the constructed statistic (or the pivotal quantity in estimation) to measure the relative distance between two ordered samples, we estimate parameters to minimize the distance. Furthermore, unlike many existing methods, which rely on some regularity conditions and/or the explicit forms of probability density functions, the NSE only assumes a mild condition that the cumulative distribution function can be approximated to a satisfying precision. The NSE can be directly applied to many kinds of statistical distribution inference problems regardless of whether existing estimation methods are applicable. Furthermore, the NSE provides not only point estimations but also interval estimations to first preserve the model assumption and then guarantee the significance of parameters. Using NSE, researchers and practitioners no longer need to assume any moment conditions to derive asymptotic results. There is no need to conduct the bootstrap method when the limiting distributions are not computable. This talk illustrates simulation examples and real applications to show NSE's superior performance for various commonly applied inference problems where existing estimation methods may fail to guarantee desired distributional assumptions. Joint work with BingyanWang (Princeton) and Xinyang Hu (Yale).