Mathematical Sciences: Expanison and Likelihood Methods forMartingales and Martingale Inference

本研究涉及对纵向数据的推断，如生存分析和时间序列类型。这类问题的检验和置信区间通常基于鞅的渐近行为。本项目将重点研究在这种推理中更准确地设置临界点的方法。解决这个问题的一种方法是对鞅进行变换，以创建更精确地近似于渐近分布的统计量。另一种方法是寻找鞅定律的更好的近似，通过埃奇沃斯或鞍点展开，或自举。这项研究的目的是改进统计学家从随时间演变的过程中解释数据的方式。主要案例有：(1)长期医疗数据，即对患者进行一段时间的跟踪；(2)经济数据。这些不同类型的设置可以用一种相对统一的技术来研究，这与公平博弈理论有关。通过进一步利用这种联系，可以在如何解释数据方面取得实质性进展。

This research is concerned with inference for longitudinal data, such as of survival analysis and time series type. Tests and confidence intervals in such problems are typically based on the asymptotic behavior of martingales. This project will focus on more accurate ways of setting critical points in such inference. One approach to this problem is to transform the martingales to create statistics more accurately approximated by the asymptotic distribution. Another approach is to look for better approximations to the law of the martingale, via Edgeworth or saddlepoint expansions, or bootstrapping. The research is intended to improve the way statisticians interpret data from processes that evolve over time. Main cases are (1) long term medical data, where patients are followed over a period of time, and (2) economic data. These quite different types of settings can be studied with a relatively unified technology, which relates to the theory of fair games. Substantial gains in how data are interpreted can be obtained by further exploiting this connection.