Hypergeometric Functions in Sequential Analysis

Abstract

In several sequential probability ratio tests [9] [12], density ratios may be expressed in terms of hypergeometric functions whose asymptotic behavior is indirectly available in the literature, and is useful in establishing the almost sure termination of these tests [6] [7] [8] [10]. The results of this paper are new for the sequential ordinary and multiple correlation coefficient tests [4] [6]. In addition, they complete the results of [8] and [10] for the sequential $F$-test [2] [6] as well as those of [7] for the sequential $\chi^2$- and $T^2$-tests [5] [6]. The generalized hypergeometric function $_pF_q$ is defined by: \begin{equation*} \tag{1.1}_pF_q(a_1, \cdots, a_p; c_1, \cdots, c_q; z) = 1 + (a_1 \cdots a_p/c_1 \cdots c_q)z/1! \end{equation*} $ + \lbrack a_1(a_1 + 1) \cdots a_p(a_p + 1)/c_1(c_1 + 1) \cdots c_q(c_q + 1) \rbrack z^2/2! + \cdots $ for $p, q \geqq 0$ and $c_i > 0, i = 1, \cdots, q$. We shall need in the sequel three such functions: $_2F_1(a, b; c; z)$, which is convergent for $|z| < 1, _1F_1(a; c; z)$, and $_0F_1(\quad ; c; z)$, which are convergent for all $z$.