Boundary Crossing Problems for Sample Means

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