Asymptotic Local Minimaxity in Sequential Point Estimation

Abstract

Let $X_1, X_2, \cdots$ be i.i.d. random variables with mean $\theta$ and finite, positive variance $\sigma^2,$ depending on unknown parameters $\omega\in\Omega.$ The problem addressed is that of finding a stopping time $t$ for which the risk $R_A(t, \omega) = E_\omega\{A \gamma^2_0(\omega)(\bar{X}_t - \theta)^2 + t\}$ is as small as possible (in a suitable sense), where $A > 0, \gamma_0$ is a positive function on $\Omega$, and $\bar{X}_t = (X_1 + \cdots + X_t)/t.$ For fixed (nonrandom) sample sizes, $2 \sqrt{A}(\gamma_0\sigma)$ is a lower bound for $R_A(n, \omega), n \geq 1$; and the regret of a stopping time $t$ is defined to be $r_A(t, \omega) = R_A(t, \omega) - 2\sqrt{A}(\gamma_0 \sigma).$ The main results determine an asymptotic lower bound, as $A \rightarrow\infty,$ for the minimax regret $M_A(\Omega_0) = \inf_t\sup_{\omega\in\Omega_0}r_A(t, \omega)$ for neighborhoods $\Omega_0$ of arbitrary parameter points $\omega_0 \in \Omega.$ The bound is obtained for multiparameter exponential families and the nonparametric case. The bound is attained asymptotically by an intuitive procedure in several special cases.