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值的不确定性，因为人们只观察一个样本，而不是整个总体。标准的方法依赖于近似的样本平均值的分布的钟形分布，其蔓延可以估计样本观测的变异性。当底层人口分布不是钟形时，这种近似在小样本中变得很差，导致人口平均值的不确定性的错误表示。这项建议将开发一种新的方法来描述的不确定性，即使是非钟形人口仍然准确。这是通过在不确定性估计的构建中对潜在的非钟形结构进行建模来完成的。得出结论的更复杂的问题可以归结为对适当定义的总体的平均值得出结论。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。最终结果是提高经济效益，加快经济增长。该提案旨在开发一种替代标准的t统计为基础的推断平均从一个样本的i.i.d.观察，更好地控制大小为中度重尾的基础人口。它结合了k个最小和k个最大观测值的极值理论，以及剩余中间n-2k个观测值的平均值的正态近似。均值漂移是支配极值分布的相同尾部参数的函数，以及极值观测的实现值。因此，得到一个近似的参数模型为2k+1观察，尾部参数作为滋扰参数，可以应用数值技术，以获得有效的和强大的测试在这个近似的参数模型。主要的理论结果表明，这种方法是对现有方法的改进。对于方差有限但三阶矩无穷大的总体，与通常的t-统计量或t-bootstrap相比，新方法在拒绝概率上引起更小的误差，至少只要人口是这样的，极值理论提供准确的近似值。这个奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。

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.