General Proof of Termination with Probability One of Invariant Sequential Probability Ratio Tests Based on Multivariate Normal Observations

Abstract

$Z_1, Z_2, \cdots$ is a sequence of iid $k$-vectors with common distribution $P$. $G^\ast$ is a group of transformations $Z_n \rightarrow CZ_n + b, C \varepsilon G$, where $G$ is a Lie group of $k^2$ matrices, $\dim G \geqq 1, G$ closed in the group of all nonsingular $k^2$ matrices, and the totality of translation vectors $b$ is a subspace of $k$-space invariant under $G$. Let $\mathscr{N}$ be all $N(\mu, \Sigma)$ distributions, with $\Sigma k^2$ nonsingular. Let $U = (U_1, U_2, \cdots)$ be a maximal invariant under $G^\ast$ in the sample space, $\gamma = \gamma(\theta)$ a maximal invariant in $\mathscr{N}$, where $\theta = (\mu, \Sigma)$. For given $\theta_1, \theta_2 \varepsilon \mathscr{N}$ such that $\gamma(\theta_1) \neq \gamma(\theta_2)$ let $R_n$ be the probability ratio of $(U_1, \cdots, U_n)$. The limiting behavior of $R_n$ is studied under the assumption that the actual distribution $P$ belongs to a family $\mathscr{F} \supset \mathscr{N}$, defined as follows: the components of $Z_1$ have finite 4th moments, and there is no relation $Z'_1 AZ_1 + b'Z_1 =$ constant a.e. $P$, with $A$ symmetric, unless $A = 0, b = 0$. It is proved that $\mathscr{F}$ can be partitioned into 3 subfamilies, and for every $P$ in the first subfamily $\lim R_n = \infty$ a.e. $P$, in the second $\lim R_n = 0$ a.e. $P$, and in the third $\lim \sup R_n = \infty$ a.e. $P$ or $\lim \inf R_n = 0$ a.e. $P$. This implies that any SPRT based on $\{R_n\}$ terminates with probability one for every $P \varepsilon \mathscr{F}$.