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January, 1988 Boundary Crossing Problems for Sample Means
Tze Leung Lai
Ann. Probab. 16(1): 375-396 (January, 1988). DOI: 10.1214/aop/1176991909


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 $|\lambda - \theta|^2/c \rightarrow \infty$.


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Tze Leung Lai. "Boundary Crossing Problems for Sample Means." Ann. Probab. 16 (1) 375 - 396, January, 1988.


Published: January, 1988
First available in Project Euclid: 19 April 2007

zbMATH: 0642.60018
MathSciNet: MR920279
Digital Object Identifier: 10.1214/aop/1176991909

Primary: 60F10
Secondary: 60G40 , 62L05 , 62L15

Keywords: boundary crossing probabilities , exponential family , Kullback-Leibler information , sequential analysis , stopping time , uniform strong laws

Rights: Copyright © 1988 Institute of Mathematical Statistics


Vol.16 • No. 1 • January, 1988
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