Figure 5: Comparison of profiles vectors and covariances.

Conditional mean of $t\left(Z\right)={\overline{Z}}_{1:3}$ as a function of θ for different covariances (panels) and profiles (colors). Threshold is c = 1. In all panels, we use the following profiles: s_Σ ∝ (1, 1, 1)′Σ in black, s_u = (1, 1, 1) in red, s₂ = (1.5, 0, 1.5) in blue, and s₁ = (3, 0, 0) in grey. In the top-left and bottom-right panels, s_u ≡ s_Σ). All covariances have unit variance, and the number of correlated variables decrease from Σ_A (ρ = 0.4 between every pair), through Σ_B (as before but ρ₁₃ = 0), Σ_C (ρ₁₂ = 0) and uncorrelated Σ_D. We observe method is not very sensitive to small differences in the profile, as s_Σ, s_u give almost identical curves for Σ_B and Σ_C. The figure shows that although s_Σ ensures E_Θs(θ)[t(Z)] strictly increases with θ (monotonicity, Lemma 2), monotonicity is not guaranteed for s₂ or s₁. For Σ_C, s₁ satisfies the conditions of Lemma 3 and displays monotonicity, whereas s₂ does not. When the covariance is iid, any non-negative profile satisfies Lemma 3. Under all covariances, the curves for s₁, s₂ are greater than s_Σ; this is a potential source for coverage error if s_Σ is used.