The Efficiency of Sequential Estimates and Wald's Equation for Sequential Processes

Let n successive independent observations be made on the same chance variable whose distribution function f(x, θ) depends on a single parameter θ. The number n is a chance variable which depends upon the outcomes of successive observations; it is precisely defined in the text below. Let θ⁽x₁, ⋯, xₙ) be an estimate of θ whose bias is b(θ). Subject to certain regularity conditions stated below, it is proved that σ²(θ⁾ ≥ (1 + db/dθ)² EnE(∂log f/∂θ)²⁻¹. When f(x, θ) is the binomial distribution and θ^ is unbiased the lower bound given here specializes to one first announced by Girshick [3], obtained under no doubt different conditions of regularity. When the chance variable n is a constant the lower bound given above is the same as that obtained in [2], page 480, under different conditions of regularity. Let the parameter θ consist of l components θ₁, ⋯, θₗ for which there are given the respective unbiased estimates θ^₁(x₁, ⋯, xₙ), ⋯, θ^₁(x₁, ⋯, xₙ). Let \|λᵢⱼ\| be the non-singular covariance matrix of the latter, and \|λⁱʲ\| its inverse. The concentration ellipsoid in the space of (k₁, ⋯, kₗ) is defined as ∑i,j λⁱʲ(kᵢ - θᵢ)(kⱼ - θᵢ) = l + 2. (This valuable concept is due to Cramer). If a unit mass be uniformly distributed over the concentration ellipsoid, the matrix of its products of inertia will coincide with the covariance matrix \|λ)ᵢⱼ\|. In [4] Cramer proves that no matter what the unbiased estimates θ^₁, ⋯, θ^ₗ, (provided that certain regularity conditions are fulfilled), when n is constant their concentration ellipsoid always contains within itself the ellipsoid ∑i,j μᵢⱼ(kᵢ - θᵢ)(kⱼ - θⱼ) = l + 2 where μᵢⱼ = nE(∂log f/∂θᵢ∂log f/∂θᵢ). Consider now the sequential procedure of this paper. Let θ^₁, ⋯, θ^ₗ be, as before, unbiased estimates of θ₁, ⋯, θₗ, respectively, recalling, however, that the number of n of observations is a chance variable. It is proved that the concentration ellipsoid of θ^₁, ⋯, θ^ₗ always contains within itself the ellipsoid ∑i,j μ'ᵢⱼ(kᵢ - θᵢ)(kⱼ - θⱼ) = l + 2 where μ'ᵢⱼ = EnE(∂log f/∂θᵢ∂log f/∂θⱼ). When n is a constant this becomes Cramer's result (under different conditions of regularity). In section 7 is presented a number of results related to the equation EZₙ = EnEX, which is due to Wald [6] and is fundamental for sequential analysis.