Bounds on the Sample Size Distribution for a Class of Invariant Sequential Probability Ratio Tests

Abstract

In a previous paper [8] the asymptotic behavior of a class of invariant sequential probability ratio tests was studied to the extent of establishing termination with probability one. In this paper, under somewhat stronger conditions, certain bounds on the distribution of sample size will be obtained. The statistical framework is as follows: $Z_1, Z_2, \cdots$ is a sequence of independent and identically distributed (iid) random vectors with values in Euclidean $k$-space $E^k$. The common distribution is at first assumed to belong to the family $\mathscr{N}$ of all nondegenerate $k$-variate normal distributions. On $E^k$ there acts a group $G^\ast$ of affine transformations (precise assumptions on $G^\ast$ are given in Section 2). Let $V_1, V_2, \cdots$ be a maximal invariant sequence obtained from $Z_1, Z_2, \cdots$ under the application of $G^\ast$. Since every transformation in $G^\ast$ sends a member of $\mathscr{N}$ into a member of $\mathscr{N}, G^\ast$ also acts on $\mathscr{N}$. Let $\gamma$ be a maximal invariant function on $\mathscr{N}$, then the joint distribution of $V_1, V_2, \cdots$ depends only on $\gamma$. Denote the distribution of $(V_1, \cdots, V_n)$ by $P_{n\gamma}$. Let $\gamma_1, \gamma_2$ be two distinct values of $\gamma$, let $r_n = dP_{n\gamma_2}/dP_{n\gamma_1}$ (set equal to $\infty$ wherever $P_{n\gamma_2}$ is not absolutely continuous with respect to $P_{n\gamma_1}$) and put $R_n = r_n(V_1, \cdots, V_n)$. Then an invariant sequential probability ratio test, based on the sequence $\{R_n\}$, is defined by choosing stopping bounds $B < A$, letting the stopping variable $N$ be the smallest $n$ such that $R_n \leqq B$ or $\geqq A$, and accepting $\gamma_1$ or $\gamma_2$ according as $R_N \leqq B$ or $\geqq A$. Once the sequence $\{R_n\}$ has been defined we are at liberty to study its behavior when the actual common distribution of the $Z_n$ is not necessarily on the orbit of $\gamma_1$ or of $\gamma_2$ or, for that matter, is not even a member of $\mathscr{M}$. Still assuming $Z_1, Z_2, \cdots$ to be iid, the common distribution $P$ will be assumed to be a member of $\mathscr{P}$, to be defined later, where $\mathscr{P} \supset \mathscr{M}$. The joint distribution of $Z_1, Z_2, \cdots$ will also be denoted $P$. The object is to establish a bound on $P(N > n)$ as a function of $n$, for each $P \varepsilon \mathscr{P}$. Results on sample size distribution of invariant sequential probability ratio tests are very scarce. Ifram [5] considered a certain class of problems and obtained an exponential bound on $P(N > n)$, where $P$ is a member of the original model and should not belong to a certain exceptional set of distributions for which no results could be obtained. Sacks [6] also obtained an exponential bound in the case of the sequential $t$-test (as a by-product of other results), again excluding certain exceptional $P$'s. Savage and Sethuraman [7] obtained an exponential bound in a nonparametric problem, a sequential rank test. They allowed $P$ to be outside the original model, but again for a certain set of exceptional $P$'s no results could be obtained. In the present paper we shall establish, under Assumptions A and B (Section 2), an exponential bound of the form $P(N > n) < c\rho^n$ for some $\rho < 1$, except, again, for $P$'s in a certain set. For these exceptional $P$'s, however, we also obtain results, even though weaker, of the form $P(N > n) < c\rho^{n^{1/3}}$ (under an additional assumption on the function $\Phi$ of Section 2). This result is still strong enough to assert the existence of all moments of $N$, although not the existence of a moment generating function as is the case if $P(N > n)$ has an exponential bound. Thus, our results are more general and stronger in some respects than those of Ifram [5], but Ifram does give a precise value of the smallest possible $\rho$ in the exponential bound, whereas we have nothing comparable to offer.