Corrected Brownian Approximations and Hybrid Bootstrap Applications

布朗运动近似被用来估计离散时间跨境问题中出现的概率和期望。拟议的研究推导了校正项，以提高这些布朗近似在不同设置下的精度。混合Bootstrap是一种新的重采样方法，用于设置置信度区间。当实验提供了有关感兴趣的参数的有限信息，但提供了关于有害参数的大量信息时，它尤其相关和合适。考虑了三个应用：在高能物理中感兴趣的信号加噪声泊松模型中估计信号；使用回交或交叉设计的数据估计DNA链上数量性状基因座的位置；以及在顺序变化点检测后估计新参数。第一个是关于布朗近似的。具体目标是推导出修正项，以改进离散时间内一类边界穿越问题的这些近似。布朗近似和扩散近似一直是随机建模的主要工具，应用于不同的领域，包括统计学中的序贯分析，工业和网络应用中的排队论，以及金融中的期权定价。提高这些逼近精度的新方法应该具有广泛的应用价值。另一个主题涉及区间估计的混合Bootstrap方法。统计学中的Bootstrap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，在实验提供的关于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的应用。第一个涉及物理学实验，在这个实验中，一个新的粒子或现象，如果存在，将增加事件的速度，在实验过程中计算在内。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链上的基因座位置。最后一个应用程序出现在监视感兴趣的进程以检测更改的情况下。工业过程经常被跟踪，因为变化往往与生产问题有关，应尽快注意并修复这些问题，以提高质量和产量。网络通常出于类似的原因被监控，而财务系列可能出于各种原因被监控。对于描述变化点后过程演化的新参数，混合自助法应该是有用的。

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.