Zhang Zhengjun (张正军) is now a professor of the University of Chinese Academy of Sciences. His research covers a wide range of topics, including fundamental dependence theory, extreme value theory and risk analysis, high-dimensional statistical learning using max-linear regression, zero-inflated protective barrier regression, variable selection based on tail limits, variable screening based on tail dependence and sure explained variability and nonlinear causal inference.

Title: Necessary and Sufficient Estimation Is Needed in Accurate Statistical Inferences

Abstract: 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 order statistics and the simulated ones. Surprisingly, using relative errors results in much faster convergence rates than using 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. Thus, the NSE can be directly applied to many kinds of statistical distribution inference problems no matter whether existing estimation methods are applicable or not. Furthermore, the NSE provides not only point estimations but also interval estimations. 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 Bingyan Wang (Princeton) and Xinyang Hu (Yale).