Canadian Journal of Statistics

An inclusion‐consistent solution to the problem of absurd confidence statements: 2. Consistent nonexact‐confidence‐interval estimation

Journal Article

Abstract

Two consistent nonexact‐confidence‐interval estimation methods, both derived from the consistency‐equivalence theorem in Plante (1991), are suggested for estimation of problematic parametric functions with no consistent exact solution and for which standard optimal confidence procedures are inadequate or even absurd, i.e., can provide confidence statements with a 95% empty or all‐inclusive confidence set. A belt C(·) from a consistent nonexact‐belt family, used with two confidence coefficients (γ = infθPθ [ θ ϵ C(X)] and γ+ = supθPθ[θ ϵ C(X)], is shown to provide a consistent nonexact‐belt solution for estimating μ2 ‐μ1 in the Behrens‐Fisher problem. A rule for consistent behaviour enables any confidence belt to be used consistently by providing each sample point with best upper and lower confidence levels [δ+(x) ≥ γ+, δ(x) ≤ γ], which give least‐conservative consistent confidence statements ranging from practically exact through informative to noninformative. The rule also provides a consistency correction L(x) = δ+(x)‐δ(X) enabling alternative confidence solutions to be compared on grounds of adequacy; this is demonstrated by comparing consistent conservative sample‐point‐wise solutions with inconsistent standard solutions for estimating μ21 (Creasy‐Fieller‐Neyman problem) and equation image, a distance‐estimation problem closely related to Stein's 1959 example

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