Asymptotic expansions for first passage times

Let F be a strongly non-lattice distribution function with a positive mean, a positive variance, and a finite third moment. Let X1, X2,... be i.i.d. with common distribution function F; and let Sn=X1+...+Xn, and ta = inf{n[ges]1:Sn > a} for n [ges] 1 and a >0. The main result reported here is a two term asymptotic expansion for Ha(n, z) = P{ta n, Sn - a [les] z } as a --> [infinity]. Assuming higher moments, a three term expansion for P{ta [les] n} and refined estimates for the probability of ruin in finite time are obtained as simple corollaries. A key tool is an asymptotic expansion in Stone's formulation of the local limit theorem.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/27279/1/0000295.pd

Singh's theorem in the lattice case

The asymptotic behavior of the parametric bootstrap estimator of the sampling distribution of a maximum likelihood estimator is investigated in a simple lattice case, integer valued random variables whose distributions form an exponential family. The expected value of the bootstrap estimator is compared with an Edgeworth expansion, less the continuity correction.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/27029/1/0000017.pd

Isotonic smoothing splines under sequential designs

Isotonic smoothing splines are introduced as a natural extension of ordinary isotonic estimates in the estimation of a regression function [mu](x)=E{Y|X=x}. A constructive characterization for the isotonic smoothing splines is given. Conditions are given for consistency under sequential designs, where the observation points are random and the experimenter after observing Xk may choose to observe or to skip observing Yk.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/31875/1/0000825.pd

Fixed width interval estimation for the reciprocal drift of Brownian motion

Let b(t), 0t[mu]. The problem of setting a fixed width confidence interval for [theta]=1[+45 degree rule][mu] is considered. The intervals studied are of the form , where [zeta] is a stopping time and . Stopping times [tau]h are derived s o that these intervals have coverage probabilities converging to a set value [gamma] as h-->0. This convergence is uniform for [mu] near 0. Asymptotic optimality of [tau]h is also addressed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30248/1/0000643.pd

Improved Probability Method for Estimating Signal in the Presence of Background

A suggestion is made for improving the Feldman Cousins method of estimating signal counts in the presence of background. The method concentrates on finding essential information about the signal and ignoring extraneous information about background. An appropriate method is found which uses the condition that the number of background events obtained does not exceed the total number of events obtained. Several alternative approaches are explored.Comment: Modified 12/21 for singlespace to save trees, 9 pages, 1 figure. Modified 8/11/99 to add small modifications made for the Phys. Rev. articl