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.
