Inference for the Mean

从个体样本中得出关于总体的结论的一个关键挑战是如何准确地描述总体平均的不确定性，因为人们只观察样本，而不是整个总体。标准方法依赖于用钟形分布来近似样本平均值的分布，钟形分布的分布可以通过样本观测的可变性来估计。当基本人口分布不是钟形时，这种近似在小样本中变得很差，导致对人口平均不确定性的错误表示。这项提议将开发一种新的方法来描述不确定性，即使对于非钟形人群也是如此。这是通过在不确定性估计的构造中对潜在的非钟形结构进行建模来实现的。更复杂的得出结论的问题可以归结为得出关于适当定义的人口的平均值的结论。因此，这项研究的结果可以在社会科学中有许多实际应用，并改善商业和政策决策。最终的结果是提高经济效益，加快经济增长。这项建议寻求开发一种替代标准t统计量的方法，用于从I.I.D.样本中推断平均值。更好地控制中等厚尾潜在种群大小的观察。它将k个最小观测值和k个最大观测值的极值理论与剩余中间n-2k个观测值的平均值的正态近似结合起来。均值漂移是控制极值分布的相同尾部参数的函数，以及极端观测值的实现值。由此得到了以尾部参数为扰参数的2k+1个观测值的近似参数模型，并利用数值技术对该近似参数模型进行了有效而有力的检验。主要理论结果表明，该方法是对现有方法的改进。对于方差有限但三阶矩无限的群体，新方法在拒绝概率方面的误差比通常的t统计量或百分位数t自助法要小，至少只要总体是这样的，极值理论提供了准确的近似。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

A key challenge of drawing conclusions about a population from a sample of individuals is how to accurately describe the uncertainty about the population average because one only observes a sample, rather than the whole population. The standard approach relies on approximating the distribution of the sample average by a bell-shaped distribution, whose spread can be estimated by the variability of the sample observations. When the underlying population distribution is not bell shaped, this approximation becomes poor in small samples, leading to a wrong representation of the uncertainty about the population average. This proposal will develop a new method for describing the uncertainty that remains accurate even for non-bell-shaped populations. This is done by modelling the potentially non-bell-shaped structure in the construction of the uncertainty estimate. More complicated problems of drawing conclusions can be cast into drawing conclusions about the average of a suitably defined population. The results of this research could thus have many practical applications in the social sciences and improve business and policy decision making. The end results are to increase economic efficiency and speed up economic growth. This proposal seeks to develop an alternative to standard t-statistic-based inference about the mean from a sample of i.i.d. observations that better controls size for moderately heavy-tailed underlying populations. It combines extreme value theory for the k smallest and k largest observations with a normal approximation for the average of the remaining middle n-2k observations. The mean shift is a function of the same tail parameters that govern the extreme value distributions, as well as the realized value of the extreme observations. One thus obtains an approximate parametric model for 2k+1 observation, with the tail parameters as nuisance parameters; one can apply numerical techniques to obtain valid and powerful test in this approximate parametric model. The main theoretical result shows that this approach represents an improvement over existing method. For populations with finite variance but infinite third moment, the new approach induces smaller errors in rejection probability compared to the usual t-statistic, or the percentile-t bootstrap, at least as long as the population is such that extreme value theory provides accurate approximations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.