The Annals of Mathematical Statistics

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

R. A. Wijsman

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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 $P \varepsilon \mathscr{F}$.

Article information

Source
Ann. Math. Statist., Volume 38, Number 1 (1967), 8-24.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177699054

Digital Object Identifier
doi:10.1214/aoms/1177699054

Mathematical Reviews number (MathSciNet)
MR205415

Zentralblatt MATH identifier
0148.13901

JSTOR
links.jstor.org

Citation

Wijsman, R. A. General Proof of Termination with Probability One of Invariant Sequential Probability Ratio Tests Based on Multivariate Normal Observations. Ann. Math. Statist. 38 (1967), no. 1, 8--24. doi:10.1214/aoms/1177699054. https://projecteuclid.org/euclid.aoms/1177699054


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