The Annals of Probability

Boundary Crossing Problems for Sample Means

Tze Leung Lai

Abstract

Motivated by several classical sequential decision problems, we study herein the following type of boundary crossing problems for certain nonlinear functions of sample means. Let $X_1, X_2,\ldots$ be i.i.d. random vectors whose common density belongs to the $k$-dimensional exponential family $h_\theta(x) = \exp\{\theta'x - \psi(\theta)\}$ with respect to some nondegenerate measure $\nu$. Let $\bar{X}_n = (X_1 + \cdots + X_n)/n, \hat\theta_n = (\nabla\psi)^{-1}(\bar{X}_n)$, and let $I(\theta, \lambda) = E_\theta\log\{h_\theta(X_1)/h_\lambda(X_1)\}$ ( = Kullback-Leibler information number). Consider stopping times of the form $T_c(\lambda) = \inf\{n: I(\hat\theta_n, \lambda) \geq n^{-1}g(cn)\}, c > 0$, where $g$ is a positive function such that $g(t) \sim \alpha \log t^{-1}$ as $t \rightarrow 0$. We obtain asymptotic approximations to the moments $E_\theta T^r_c(\lambda)$ as $c \rightarrow 0$ that are uniform in $\theta$ and $\lambda$ with $|\lambda - \theta|^2/c \rightarrow \infty$. We also study the probability that $\bar{X}_{Tc(\lambda)}$ lies in certain cones with vertex $\nabla\psi (\lambda)$. In particular, in the one-dimensional case with $\lambda > \theta$, we consider boundary crossing probabilities of the form $P_\theta\{\hat\theta_n \geq \lambda \text{and} I(\hat\theta_n, \lambda) \geq n^{-1} g(cn) \text{for some} n\}$. Asymptotic approximations (as $c \rightarrow 0$) to these boundary crossing probabilities are obtained that are uniform in $\theta$ and $\lambda$ with $|\lambda - \theta|^2/c \rightarrow \infty$.

Article information

Source
Ann. Probab., Volume 16, Number 1 (1988), 375-396.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991909

Digital Object Identifier
doi:10.1214/aop/1176991909

Mathematical Reviews number (MathSciNet)
MR920279

Zentralblatt MATH identifier
0642.60018

JSTOR