On the frequentist coverage of Bayesian credible intervals for lower bounded means

Abstract

For estimating a lower bounded location or mean parameter for a symmetric and logconcave density, we investigate the frequentist performance of the 100(1 − α)% Bayesian HPD credible set associated with priors which are truncations of flat priors onto the restricted parameter space. Various new properties are obtained. Namely, we identify precisely where the minimum coverage is obtained and we show that this minimum coverage is bounded between 1 − 3α 2 and 1 − 3α 2 + α 2 1+α ; with the lower bound 1 − 3α 2 improving (for α ≤ 1/3) on the previously established ([9]; [8]) lower bound 1−α 1+α . Several illustrative examples are given.AMS 2000 subject classifications: 62F10, 62F30, 62C10, 62C15, 35Q15, 45B05, 42A99.

Estimation of a non-negative location parameter with unknown scale

“…Applying Lemma 8, which is stated in the Appendix, with ξ ∼ ξ n+l h(ξ) 1 (0,∞) (ξ), r(ξ) = log(ξ), and s(ξ) = A y (ξ), we infer that I ≥ 0 since E[A y (ξ)] = 0 given the definition of g π l (y) in (10). There remains to establish (31), which we proceed to do separating the cases: (i) y ≤ 0 and (ii) y > 0.…”

“…Notice that ψ ρ (0, y) = B n (y, g π 0 (y)) = 0 for all y ∈ R by virtue of the definition of g π 0 in (10). Therefore, the risks of δ π 0 and δ 0 match at the boundary of Θ where µ = 0, σ > 0.…”

“…Part (b) is obvious given the convexity of ρ, while part (c) follows from the given representations (15) and (10). For establishing (a), suppose, in order to arrive at a contradiction, that B n+l (y + a, g π l (y)) ≤ 0.…”

“…The class includes the choice π 0 which is of intrinsic interest as it represents a plausible adaptation, or truncation onto Θ of the right Haar invariant measure π rh with the MRE estimator (also) being the generalized Bayes estimator δ π rh with respect to π rh . Moreover, the study of frequentist properties on the restricted parameter space of Bayesian procedures associated with π 0 or, more generally, truncations of the right Haar invariant prior measure has recently surfaced in interval estimation problems (Zhang and Woodroofe, 2003;Marchand and Strawderman, 2006;Marchand et al, 2008).…”

“…First, observe that by differentiating (10) for the normal case with f (u 2 +v 2 ) = φ(u)h(v) in (8), we have nonnegativity of g π l (·) (Lemma 5) (since c 0 = 0), and the convexity of ρ, we have for ǫ > 0: 7.3. Lemma used within the proof of Lemma 6.…”