Confidence Sets

Abstract

Lecture 26: November 5 2. C 2 : Similarly we have that, P(θ ∈ C 2) = P([ θ ≤ θ − a 2 , θ ≥ θ − b 2) = P(θ ≤ θ − a 2) − P(θ ≤ θ − b 2) = θ − a 2 θ n − θ − b 2 θ n. Notice that now the coverage probability depends on the unknown parameter θ (which is undesirable). Furthermore, if we choose any constants (a 2 , b 2) (say depending only on the desired coverage probability α), then as θ → ∞ we have that the interval has coverage probability that tends to 0, i.e. the interval is not honest for any constants (a 2 , b 2). We will now discuss a few different ways of constructing confidence intervals. Although superficially different most of these methods are roughly the same. 26.1 Inverting a test We discussed this method in the last lecture. We suppose that we have a (family of) test(s) for the hypotheses: H 0 : θ = θ 0 H 1 : θ = θ 0. These tests have a rejection region and a corresponding acceptance region (where we fail to reject the null). Denote the acceptance region for the test of θ = θ 0 as A(θ 0). This is a subset of the sample space. Given observed data {X 1 ,. .. , X n } we consider the random set: C(X 1 ,. .. , X n) = {θ 0 : {X 1 ,. .. , X n } ∈ A(θ 0)}. Our confidence set is simply the set of parameters θ 0 that we would fail to reject using our family of tests. If our family of tests has level α then the set C(X 1 ,. .. , X n) is a 1 − α confidence set. To see this observe that since our test controls the Type I error we have that for any parameter θ 0 , P θ 0 ({X 1 ,. .. , X n } / ∈ A(θ 0)) ≤ α, so with probability at least 1 − α we have that, {X 1 ,. .. , X n } ∈ A(θ 0) and hence that θ 0 ∈ C(X 1 ,. .. , X n).