Corrected Brownian Approximations and Hybrid Bootstrap Applications

布朗运动近似是用来估计离散时间边界跨越问题中出现的概率和期望的。本研究提出了修正项，以提高这些布朗近似在各种情况下的准确性。混合自举法是一种新的用于设置置信区间的重采样方法。当实验提供的有关感兴趣的参数的信息有限，但有关有害参数的信息大量时，这是特别相关和适当的。本文考虑了三种应用：估计高能物理中感兴趣的信号加噪声泊松模型中的信号；利用回交或交叉设计的数据估计DNA链上数量性状位点的位置；序列变化点检测后的新参数估计。这个项目考虑了两个不同的主题。第一个与布朗近似有关。具体目标是导出校正项，以改进离散时间下一类边界跨越问题的这些近似。布朗近似和扩散近似是随机建模的主要工具，应用于不同的领域，包括统计学中的顺序分析，工业和网络应用的排队论，以及金融中的期权定价。提高这些近似精度的新方法应该具有广泛的应用价值。另一个主题是关于区间估计的混合自举方法。统计学中的自举方法使用计算机模拟来帮助研究人员评估估计器的精度。混合自举是这种方法的现代变体，在实验提供有关感兴趣参数的有限信息的情况下特别相关。目前正在研究三种具体应用。第一类涉及物理学实验，在这些实验中，如果出现新的粒子或现象，将增加实验过程中计算的事件率。第二个应用涉及现代遗传学，试图估计与感兴趣的数量性状相关的DNA链上的位点的位置。最后一个应用程序出现在监视感兴趣的流程以检测更改的情况下。工业过程经常被跟踪，因为变化通常与生产问题有关，应该尽快注意和修复这些问题，以提高质量和产量。网络通常出于类似的原因进行监控，财务系列也可能出于各种原因进行监控。混合自举法对于描述变化点之后的过程演变的新参数是有用的。

Brownian motion approximations have been developed to estimate probabilities and expectations that arise in boundary crossing problems in discrete time. The proposed research derives correction terms to improve the accuracy of these Brownian approximations in various settings. The hybrid bootstrap is a new resampling method used to set confidence intervals. It is particularly relevant and appropriate when an experiment provides limited information about the parameter of interest, but substantial information about nuisance parameters. Three applications are considered: estimating the signal in a signal plus noise Poisson model of interest in high energy physics; estimating the location of a quantitative trait loci on a strand of DNA with data from a back-cross or inter-cross design; and estimating new parameters after sequential change point detection.Two different topics are considered for this project. The first concerns Brownian approximations. The specific goal is to derive correction terms to improve these approximations for a class of boundary crossing problems in discrete time. Brownian and diffusion approximations have been a major tool in stochastic modeling, with applications to diverse areas including sequential analysis in statistics, queuing theory for industrial and networking applications, and options pricing in finance. New methods to improve the accuracy these approximations should have broad value. The other topic concerns the hybrid bootstrap approach to interval estimation. Bootstrap methods in statistics use computer simulation to help a researcher assess the precision of an estimator. Hybrid bootstrapping is a modern variant of this approach which is particularly relevant in situations where the experiment provides limited information about the parameter of interest. Three specific applications are being studied. The first concerns experiments in physics in which a new particle or phenomenon, if present, will increase the rate of events, counted in the course of the experiment. A second application concerns modern genetics, trying to estimate locations for loci on a strand of DNA associated with a quantitative trait of interest. The final application arises in situations where a process of interest is monitored to detect changes. Industrial processes are often tracked since changes are often associated with production problems that should be noted and fixed as soon as possible to improve quality and output. Networks are often monitored for similar reasons, and financial series may be monitored for a variety of reasons. The hybrid bootstrap should be useful for new parameters describing process evolution after the change point.