## Institute of Mathematical Statistics Lecture Notes - Monograph Series

### Multivariate sequential analysis with linear boundaries

Robert Keener

#### Abstract

Let $\{S_n=(X_n,W_n)\}_{n\ge0}$ be a random walk with $X_n\in\R$ and $W_n\in\R^m$. Let $\tau=\tau_a=\inf\{n:X_n>a\}$. The main results presented are two term asymptotic expansions for the joint distribution of $S_\tau$ and $\tau$ and the marginal distribution of $h(S_\tau/a,\tau/a)$ in the limit $a\to\infty$. These results are used to study the distribution of $t$-statistics in sequential experiments with sample size $\tau$, and to remove bias from confidence intervals based on Anscombe's theorem.

#### Chapter information

Source
Jiayang Sun, Anirban DasGupta, Vince Melfi, Connie Page, eds., Recent Developments in Nonparametric Inference and Probability: Festschrift for Michael Woodroofe (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2006), 58-79

Dates
First available in Project Euclid: 28 November 2007

https://projecteuclid.org/euclid.lnms/1196284053

Digital Object Identifier
doi:10.1214/074921706000000608

Mathematical Reviews number (MathSciNet)
MR2409064

Zentralblatt MATH identifier
1268.62094

Rights