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

For estimating a lower bounded location or mean parameter for a symmetric and logconcave density, we investigate the frequentist performance of the $100(1-\alpha)%$ 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 $1-\frac{3\alpha}{2}$ and $1-\frac{3\alpha}{2}+\frac{\alpha^2}{1+\alpha}$; with the lower bound $1-\frac{3\alpha}{2}$ improving (for $\alpha \leq 1/3$) on the previously established ([9]; [8]) lower bound $\frac{1-\alpha}{1+\alpha}$. Several illustrative examples are given.

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.
Here is a glimpse of previous work pertaining to the credible interval I π * (X) and its frequentist coverage C(θ). For estimating a lower bounded normal mean θ (say θ ≥ 0), results due to (a) Roe and Woodroofe [9] (known variance), and (b) Zhang and Woodroofe [10] (unknown variance σ 2 ) establish the lower bound 1−α 1+α for the frequentist coverage of the 100 × (1 − α)% HPD Bayesian confidence interval with respect to the priors: More recently, Marchand and Strawderman [8] established, for a more general setting with underlying symmetry, the validity of the lower bound 1−α 1+α for the frequentist coverage of the 100×(1−α)% Bayesian HPD credible interval derived from the truncation onto the restricted parameter space of the Haar right invariant prior. We refer to their paper for details and similar developments for non-symmetric settings (also see [11]; [12]). The important starting point to keep in mind from the results of Marchand and Strawderman [8] is the applicability of the 1−α 1+α lower bound for symmetric and unimodal location models.
As mentioned above, the analysis which we have carried out and which is presented herein was motivated by the conservative nature of the previously established lower bound 1−α 1+α and other unknown and potentially useful aspects of the coverage C(θ). Finally, the results are cast here with the backdrop of a flurry of recent activity and debate, which is focused on the choice of methods for setting confidence bounds for restricted parameters as witnessed by the works referred to above as well at that of Mandelkern [7], Feldman and Cousins [5], Efron [4], among others.

Main results
Our main results apply to location models X ∼ g(x − θ) with densities g which are unimodal, symmetric about 0, and logconcave (i.e., log g is a concave function on its support). For such location families, the key assumption of logconcavity can equivalently be described as corresponding to those families with increasing monotone likelihood ratio densities, that is g(x−θ1) g(x−θ0) increases in x for all θ 1 , θ 0 such that θ 1 > θ 0 . Alternatively, the assumption of logconcavity is connected to an increasing hazard (or failure) rate (see Lemma 1). Before moving along, here is a useful checklist of notations and definitions used throughout.
Lemma 2. Let g be a unimodal, symmetric about 0 and logconcave density on R. Then, for all z ≥ 0, Proof. Part (a) implies (b) given the increasing hazard rate property of part (e) of Lemma 1, as For part (a), begin by observing that for all z > 0: so that (2) holds as soon as, for all z > 0, Finally, apply part (b) of Lemma 1 inside the above integral to infer that g(y) g(z) − g(y+z) g(2z) ≤ 0 for all y, z such that y ∈ (0, z), which yields (4) and the proof of (2).
We now recall previously established properties of I π * (X) and of its frequentist coverage before pursuing with the main analysis.
about 0, and the HPD credible set I π * (X), we have As shown in the following lemma, the lower bound l(·) is nondecreasing on (−∞, ∞), while the upper bound u(·) is nondecreasing on (d 0 , ∞). Furthermore, we show how the logconcavity of g forces u(·) to be nondecreasing on (−∞, d 0 ) as well. (It is easily verified that l(·) and u(·) are continuous functions on R.) Lemma 4. Consider I π * (X) as given in Lemma 3. Then, Proof. (a) Since l(x) = 0 for x ≤ d 0 , we only need to look at the behaviour of l(x) for x > d 0 (> 0). We have, for x > d 0 , .
given that g is unimodal with a mode at 0.
(b) Follows directly with G and G −1 being nondecreasing. 1 (c) Under the prior π * , the posterior density of θ|x is given by , proofs of Theorem 1 or Lemma 5). Now the logconcavity of g implies that the family of posterior densities of θ|x, with parameter x, has increasing monotone likelihood ratio in θ. Finally, the result follows since the quantiles u(x); x ≤ d 0 ; of such families are nondecreasing in x.
Remark 1. Above, only the properties relative to x 0 , as well as those of the coverage C(θ) for θ ≤ 2d 0 require the logconcavity of g. Analogously, some results below (e.g., Corollary 1) do not require the additional assumption of logconcavity. Remark 2. From (5) and the properties of x 0 , x 1 , and x 2 of Lemma 5, it follows that C(·) is a continuous function on [0, ∞), with the exception of a discontinuity at θ = a, and when a > 0. In this case, we have lim θ→a − x 0 (θ) = −∞, and lim θ→a + x 0 (θ) = u −1 (a) − a, which will lead to a drop of G(u −1 (a) − a) in the coverage at θ = a. An interesting example of such a discontinuity occurs for a Laplace model (see Example 3, Figure 2, and Remark 3). Remark 3. It is pertinent here to discuss the behaviour of u(x) as x → −∞.
In particular, we wish to single out cases where a > 0, which will imply that u(x) ≥ θ for any θ ∈ [0, a] (i.e., I π * (x) does not underestimate θ for such θ's, and coverage will occur as soon as underestimation does not occur; see (5)). As an example, consider a Laplace model with g(z) = G(z) = 1 2 e −z ; for z > 0; and which leads to u(x) = − ln(α) for all x < 0, hence a = − ln(α) > 0. Part (a) of Lemma 3 provides a way to verify this directly. Alternatively, observe that the posterior survivor function of θ is given by Thus, for x < 0, the posterior distribution does not vary and yields a constant I π * (x) = [0, − ln(α)]. Analogously, logconcave densities g with exponential tails will lead to a similar non-zero limit at −∞. A family of such densities, which will lead to u(x) → − ln(α) as x → −∞, is given by g(z) = P (|z|)e −|z| ; with P (·) nondecreasing and logconcave on (0, ∞), P ′ (0 + ) < P (0), and P ′ (z) P (z) → 0 as z → ∞. This may be verified by showing that the conditions on P force the density g to be logconcave, and that the posterior survivor function P (θ ≥ y|x) converges (for y > 0) as above to e −y when x → −∞. A simple example is given by P (z) = 1 4 (|z| + 1), that is g(z) = 1 4 (|z| + 1)e −|z| . On the other hand, if h ′ is unbounded where h ≡ − ln(g), then u(x) → 0 as x → −∞. To prove this, it is sufficient to show that P (θ ≥ y|x) → 0 as x → −∞ for all y > 0. But notice that (for y > 0) Finally, we emphasize that the assumption of logconcavity is indeed required for the upper bound u(·) to increase on (−∞, d 0 ). As for interesting counterexamples, we point out that u(x) → ∞ as x → −∞ for non-logconcave densities g such that lim x→∞ g(y+x) g(x) = 1, for all y > 0. This is the case for instance of Student densities with ν ≥ 1 degrees of freedom, as remarked upon previously by [10].
The next few results, culminating in Corollary 3, show that the minimum coverage is attained at θ = 2d 0 , and is bounded between 1− 3α 2 and 1− 3α 2 + α 2 1+α . The first of these results gives further information about the behaviour of C(θ) for θ ≤ d 1 .
Finally, the proof of (i) is complete since (3) implies the above chain of implications, as well as (8).
Proof of (ii). From (5) and the other properties of Lemma 5, we obtain in a straightforward manner: The last piece of the analysis consists in showing that the frequentist coverage C(θ) decreases on (d 1 , 2d 0 ). The proof of the next result relies in part on several lemmas which are stated and proven in the Appendix. Theorem 1. For X ∼ g(x − θ), θ ≥ 0, g unimodal, symmetric about 0, and logconcave, the frequentist coverage C(θ) of the HPD credible set I π * (X) decreases on (d 1 , 2d 0 ).
From this, we obtain that the property C ′ (θ) ≤ 0 for θ ∈ (d 1 , 2d 0 ) is equivalent to the inequality Using the fact that d 1 ≥ d 0 > 0 and the unimodality of g, we infer that the above condition is implied by either the condition or, given part (b) of Lemma 1, by the stronger condition Finally, the result follows with this very last inequality being equivalent to (2) given that α =Ḡ (d0) 1−Ḡ(d0) . Corollary 3. For X ∼ g(x − θ), θ ≥ 0, g unimodal, symmetric about 0, and logconcave, the frequentist coverage C(θ) attains its minimum at θ = 2d 0 , and is bounded below by 1 − 3α 2 . Proof. The result is a direct consequence of Theorem 1, Lemma 7, and Lemma 6.

Examples and final comments
We conclude with some comments and illustrative examples. We also refer to [3] and [6] for further examples and illustrations.
Remark 5. The new lower bound 1 − 3α 2 for the frequentist coverage is an improvement over the existing lower bound 1−α 1+α for α < 1/3, and a significant improvement for relatively smaller α. For instance, with a nominal coverage of 1 − α = 0.90, the lower bounds are 0.85 and 0.81 respectively. Also, as alluded to in the introduction and as a consequence of Lemma 7, the bound 1 − 3α 2 is, for α < 1/3, fairly sharp especially for relatively smaller α given that inf θ≥0 C(θ) ≤ C(2d 0 ) ≤ 1 − 3α 2 + α 2 1+α . For instance with 1 − α = 0.90, we obtain 0.85 ≤ inf θ≥0 C(θ) ≤ 0.8590. There is some numerical evidence to support the applicability of the lower bound 1 − 3α 2 for some models that are not logconcave location models. This is perhaps the case notably for estimating a lower bounded normal mean with unknown variance, (a model associated with a pivotal Student distribution which is not logconcave), based on the reported numerical evaluations in [10].
Finally, the findings in this paper does provide a sharper description of the frequentist coverage properties of the HPD credible interval I * π (X) with the improved lower bound on the minimal coverage mitigating in favour of desirable features (of course, added to the fact that the interval I * π (X) has exact credibility for a uniform prior on [0, ∞)). However, one can turn around the argument to point out the non-conformity of the frequentist and nominal coverage of I * π (X) in the worse case scenario θ = 2d 0 . For instance, if 1−α = 1/3, Lemma 7 implies that inf θ≥0 C(θ) is at most 7/12 (at least 1/2), in other words a departure of at least 1/12 between nominal and frequentist coverage at θ = 2d 0 .