Two stage sequential estimation procedures and a convex optimization problem.

This dissertation deals with sequential estimation and nonparametric function estimation with the convexity constraint. Among a variety of sequential or nonparametric methods, we focused on two-stage procedures for sequential estimation problems and smoothing splines for nonparametric estimation. The first part of the dissertation concerns two-stage procedures. Stein (1945) proposed a two-stage procedure for fixed-width confidence intervals. Let X1,···, X n, be i.i.d. normal random variables with mean mu and variance sigma 2. In order to construct a confidence interval for mu with fixed confidence 1-alpha and width 2h, the required sample size must be at least eta = c2sigma 2/h2. Since for unknown sigma 2, any fixed size sampling schemes cannot solve the problem, Stein developed a two-stage procedure. The basic idea of the two-stage procedure is as follows; that after taking an initial sample m, an estimate Nm of eta is computed and additional sample of size max{0, Nm-m} is taken based on the estimate for sigma from the initial sample. The choice of m , obviously made when there is no data yet, is subjective. So we propose an asymptotically minimax choice for the initial sample size m for the fixed-width confidence interval and extend the result to related problems such as one of well-known problems in sequential analysis, point estimation of a mean mu, from a normal distribution with loss | m&d4; - mu|p and cost of sampling. We also propose an asymptotic Bayesian solution for m in the point estimation problem with p = 2. Since Stein's did not address how to select m, the two-stage procedure does not have even first-order property. However, we can show that two-stage procedures can achieve second-order properties under the condition sigma > sigma*. Finally, a modification of Stein's two-stage procedure is suggested and its asymptotic properties are investigated. The second part of the dissertation develops an algorithm for the optimal convex smoothing spline. It turns out that the optimal convex smoothing spline is the solution to a differential equation with two boundary conditions. The difficulty arises in satisfying the conditions. Here we implement an algorithm for the calculation of the convex smoothing spline by using a nonlinear optimization method, the Nelder-Mead downhill simplex method.Ph.D.Pure SciencesStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/126940/2/3287560.pd