The Hidden Cost of Volatility Drag: Consequences of Non-Uniform Accuracy for Level 1 Confidence Intervals

The Hidden Cost of Volatility Drag

That said, here's a detailed analysis of Lecture 15 in the course material.

As we discussed earlier, certain optimality of confidence intervals is essential for evaluating confidence sets. In this lecture, we explored how to determine if a level-1 −α confidence set C(X) is Θ′-uniformly most accurate (UMA) and Θ′-unbiased.

Level 1 −α UMA-LCB

One way to evaluate a confidence set is by measuring its probability of false coverage, which is similar to a type II error probability in hypothesis testing. Let's consider the example of level-1 −α UMA-LCB C(X) = [l(X), u(X)]. In this case, l∗(X) and u∗(X) are level 1 −α UMA-LCBs for θ.

Theorem 1

We can define a theorem as follows: Consider non-randomized tests for H0 : θ = θ0 vs H1 : θ > θ0. Suppose d∗ θ0(x) = ( 1, if l∗(x) > θ0, 0, if l∗(x) ≤θ0, then l∗(X) is a level 1 −α UMA-LCB for θ.

Proof: Let l(X) be a level 1 −α LCB for θ. Then dθ0(x) = ( 1, if l(x) > θ0, 0, if l(x) ≤θ0 is a level-α test for H0 vs H1. Hence for θ > θ0, Pθ(l∗(X) ≤θ0) = 1 −β(θ, d∗ θ0) ≤1 −β(θ, dθ0) = Pθ(l(X) ≤θ0).

QED.

Theorem 2

Another theorem we can establish is: (i) Let l∗(X) and u∗(X) be level 1 −α UMA-LCBs for θ. Then for every θ and any level 1 −α LCB l(X), Eθ[θ −l∗(X)]+ ≤Eθ[θ −l(X)]+. (ii) Let [l∗(X), u∗(X)] be a level 1 −α UMAU-CI for θ. Then for every θ and any level 1 −α unbiased CI [l(X), u(X)], Eθ[u∗(X) −l∗(X)]+ ≤Eθ[u(X) −l(X)]+.