Asymptotically Optimum Properties of Certain Sequential Tests

Let X₁, X₂, ⋯ be independent and identically distributed random variables whose common distribution is of the one-parameter Koopman-Darmois type, i.e., the density function of X₁ relative to some σ-finite nondegenerate measure of F on the real line can be written as f(x, θ) = exp (θ x - b(θ)), where b(θ) is some real function of the parameter θ. Consider the hypotheses H₀ = \θ θ₀\ and H₁ = \θ θ₁\ where θ₀ < θ₁ and θ₀, θ₁ are in Ω, the natural parameter space. We want to decide sequentially between the two hypotheses. Suppose l(θ) is the loss for making a wrong decision when θ is the true parameter and assume 0 l(θ) 1 for all θ and l(θ) = 0 if θ is in (θ₀, θ₁), i.e., (θ₀, θ₁) is an indifference zone. Let c be the cost of each observation. It is sufficient to let the decision depend on the sequence (n, Sₙ), n 1, where Sₙ = X₁ + ⋯ + Xₙ. We shall consider the observed values of (n, Sₙ) as points in a $(u, v)$ plane. Then, for any test, the region in the $(u, v)$ plane where sampling does not stop is called the continuation region of the test. A test and its continuation region will be denoted by the same symbol. Schwarz [4] introduced an a priori distribution W and studied the asymptotic shape of the Bayes continuation region, say BW(c), as c → 0. He showed that BW(c)/ln c⁻¹ approaches, in a certain sense, a region BW that depends on W only through its support. Whereas Schwarz's work is concerned with Bayes tests, in this paper the main interest is in characteristics of sequential tests as a function of θ. In particular, it is desired to minimize the expected sample size (uniformly in θ if possible) subject to certain bounds on the error probabilities. Our approach, like Schwarz's, is asymptotic, as c → 0. It turns out that an asymptotically optimum test--in the sense indicated above, is BW ln c⁻¹ if W is a measure that dominates Lebesgue measure. Such a measure will be denoted by L (for Lebesgue dominating) from now on. Thus, Bayes tests, as a tool, will play a significant role in this paper. In order to prove the optimum characteristic of BL ln c⁻¹, some other results, of interest in their own right, are established. For any W satisfying certain conditions that will be given later, we show that the stopping variable $N(c)$ of BW(c) approaches ∞ a.e. P_θ for every θ in Ω. This result together with Schwarz's result that BW(c) ln c⁻¹ approaches a finite region, leads to the following results: (i) for BW(c), E_θ N(c)/ln c⁻¹ tends to a constant for each θ in Ω and (ii) the same is true for the stopping variable of BW ln c⁻¹. Furthermore, it is shown that for BL ln c⁻¹ the error probabilities tend to zero faster than c ln c⁻¹. Consequently, the contributions of the expected sample sizes of both BL ln c⁻¹ and BL(c) to their integrated risks, over any L-measure, approach 100%. Moreover BL ln c⁻¹ is asymptotically Bayes. The last result can be shown without (i) since it is sufficient to show (ii) and to apply the same argument used by Kiefer and Sacks [3] in the proof of their Theorem 1. But we show (i) because of its intrinsic interest and present a different proof using (i). Kiefer and Sacks assumed a more general distribution for X₁, constructed a procedure δ'Ic and showed that it is asymptotically Bayes. Our BL ln c⁻¹ is somewhat more explicit than their δc'I. We would also like to point out that an example of BL ln c⁻¹, when the distribution is normal, is very briefly discussed in their work. We shall restrict ourselves to a priori distribution W for which ( W)H₀ = θ₀, inf ( W)H₁ = θ₁ and 0 < W(H₀ ∪ H₁) < 1. The phrase "for any $W"$ or "for every W" is to be understood in that sense. Any Lebesgue dominating measure satisfies these conditions and also the following type of W that will be used: the support of W consists of θ₀, θ₁ and a third point θ^, θ₀ < θ^ < θ₁. Such a W will be called a θ^-measure, and the corresponding BW denoted by Bθ^. From Schwarz's equations for BW it follows readily that BL ⊂ BW for every W. In particular, BL ⊂ Bθ^. As a consequence, the statement about the error probabilities as well as others concerning BL ln c⁻¹ in the last paragraph, remain true when L is replaced by θ^ or any W. Those geometric characteristics will be dealt with in Section 2. We shall also show there that ∂ Bθ^, the boundary of Bθ^ (which consists of line segments), is tangent to ∂ BL at some point, and that if θ^ is such that b'(θ⁾ = (b(θ₁) - b(θ₀))/(θ₁ - θ₀) then max(u, ν)inBL u = maxu,v)inB_θ^ u. Let the ray through the origin and with slope equal to E_θ X₁ intersect ∂ BL at (m(θ), m(θ)E_θ X₁). In Section 3, after proving limc→ 0 N(c) = ∞ a.e. P_θ, we show limc→ 0 N(c)/ln c⁻¹ = m(θ) a.e. P_θ and limc→ 0 E_θ N(c)/ln c⁻¹ = m(θ). It is shown in Section 4 that θ in H₀ ∪ H₁ P_θ (error BL ln c⁻¹) = o(c ln c⁻¹). The main results are given in Section 5. We first show that after dividing by c ln c⁻¹, the difference of the integrated risks of BL ln c⁻¹ and BW(c), for any W, tends to zero. It follows from this result that BL ln c⁻¹ asymptotically minimizes the maximum (over θ in Ω) expected sample size in F(c), a family of tests whose error probabilities are bounded by maxi=0,1 Pθᵢ (error BL ln c⁻¹). The precise statement is given in Theorem 5.1. A sharper result under a stronger hypothesis is given in Theorem 5.2 which states that BL ln c⁻¹ asymptotically minimizes the expected sample size E_θ N for each θ, θ₀ < θ < θ₁, among all procedures of F(c) for which Eθ₀N/ln c⁻¹ and Eθ₁N/ln c⁻¹ are bounded in c.