## Institute of Mathematical Statistics Lecture Notes - Monograph Series

- Lecture Notes--Monograph Series
- Number 50, 2006, 58-79

### Multivariate sequential analysis with linear boundaries

#### 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***Recent Developments in Nonparametric Inference and Probability: Festschrift for Michael Woodroofe* (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2006)

**Dates**

First available in Project Euclid: 28 November 2007

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/074921706000000608

**Mathematical Reviews number (MathSciNet)**

MR2409064

**Zentralblatt MATH identifier**

1268.62094

**Rights**

Copyright © 2006, Institute of Mathematical Statistics

#### Citation

Keener, Robert. Multivariate sequential analysis with linear boundaries. Recent Developments in Nonparametric Inference and Probability, 58--79, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2006. doi:10.1214/074921706000000608. https://projecteuclid.org/euclid.lnms/1196284053