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# Contributions to the Theory of Sequential Analysis, II, III

M. A. Girshick
The Annals of Mathematical Statistics
Vol. 17, No. 3 (Sep., 1946), pp. 282-298
Stable URL: http://www.jstor.org/stable/2236126
Page Count: 17
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## Abstract

This is a continuation of a paper Part I of which was published in the June, 1946 issue of the Annals of Mathematical Statistics. The present paper is divided into two parts, Parts II and III, which are summarized as follows: Part II. The Exact Power Curve and the Distribution of n for Sequential Tests Where z Takes on a Finite Number of Integral Values. Consider a sequential test defined by a decision function Zn = ∑n α = 1 zα with boundaries -b and a where a and b are positive integers and zα is the αth observation of a variate z which takes on a finite number of integral values ranging from the negative integer -r to the positive integer m with respective probabilities p-r, ⋯, pm. Let ξai = P[ Zn = (a + i)], (i = 1, 2, ⋯, m - 1), and ξbj = P[ Zn = -(b + j)], (j = 1, 2, ⋯, r - 1). Furthermore, let A be a square matrix of a + b - 1 rows and columns with elements defined by: aii = 1 - p0 for all i; ai,i + k = -pk for k = 1, 2, ⋯, m; ai,i- j = -p-j for j = 1, 2, ⋯, r; and aij = 0 otherwise. It is proved that \begin{equation*}\tag{i} \xi_{bj} = \sum^{r - j - 1}_{i=0} p_{i-r}A_{r-j-i,b},\quad(j = 0, 1, \cdots, r - 1) {(ii)} \xi_{aj} = \sum^{m - j - 1}_{i=0} p_{i+j +1}A_{a+b-i-1,b},\quad(j = 0, 1, \cdots, m - 1),\end{equation*} where Akb is the element of the kth row and bth column in A-1. Let Eajτ n and Ebjτ n be the conditional generating function of n under the restriction that Zn = (a + j) and Zn = -(b + j) respectively. Then ξbjEbjτ n is obtained by substituting τ pj for each pj occurring in equation (i) and ξajEajτ n is obtained by substituting τ pj for each pj occurring in equation (ii). The probability that Zn = a + j in exactly n steps is given by the coefficient of τn in the expansion of ξajEajτ n in a power series in τ. The probability that Zn = -(b + j) in exactly n steps is similarly obtained. This method is applied to the derivation of the exact power function and the distribution of n for the sequential binomial probability ratio test. Part III. On Conjugate Distributions. Consider a random variable X with a distribution density f(x, θ) which satisfies certain specified conditions. Let θ1 and θ2 be two values of θ and let z = log (f(x, θ2)/f(x, θ1)). For any hypothesis θ = θ', let φ(t ∣ θ') be the moment generating function of z and h the non-zero value of t for which φ(t ∣ θ') = 1. We set F(x) = ehzf(x, θ'). Then f and F are conjugate distributions. If F = f(x, θ"), then θ' and θ" are defined as conjugate pairs. A method is given for obtaining the totality of conjugate pairs for the general class of distributions which admit a sufficient statistic. It is then shown that the power of the sequential probability ratio test based on such distributions is given explicitly in terms of these pairs. It is proven that within the approximation obtained by neglecting the excess of ∣ Zn ∣ over a and b at a decision point the following relationship holds: Pb(n ∣ F) = e-hbPb(n ∣ f) Pa(n ∣ F) = ehaPa(n ∣ f) where Pb(n ∣ g) and Pa(n ∣ g) stand for the probability that Zn ≥ a and Zn ≥ -b respectively in exactly n steps under the hypothesis g.

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