On the estimation of a restricted location parameter for symmetric distributions

Éric Marchand^{a,}^{1}^{,}^{2} , Idir Ouassou^{b}, Amir T. Payandeh^{c}, François Perron^{d,2} a Département de mathématiques, Université de Sherbrooke, Sherbrooke, Qc, CANADA, J1K 2R1

b ÉNSA, Avenue Abdelkrim El Khattabi BP 575, Marrakech, Maroc

c Departement of Mathematics and Statistics, University of New Brunswick, Fredericton, N.B.

CANADA E3B 5A3

d Département de mathématiques et de statistique, Université de Montréal, C.P. 6128, Succursale Centre-Ville, Montréal, Qc, CANADA, H3C 3J7

Summary

For estimating the median θ of a spherically symmetric univariate distribution under squared error loss, when θ is known to be restricted to an interval [−m,m]; m known; we derive sufficient conditions for estimators δ to dominate the maximum likelihood estimator δmle. Namely: (i) we identify a large class of models where for sufficiently small m, all Bayesian estimators with respect to symmetric about 0 priors supported on [−m,m] dominate δmle, and (ii) we provide sufficient dominance conditions of the form m ≤ c_{π} which are applicable to various models and priors π. In terms of the models, applications include Cauchy and Student distributions, as well as logconcave densities with logconvex derivatives such as normal, logistic, double-exponential among others. In terms of priors π, applications include the uniform prior as well as priors which are symmetric about 0, continuous, and nondecreasing and logconcave on (0,m). AMS 2000 subject classifications: 62F10, 62F15, 62F30 Keywords and phrases: Maximum likelihood estimator, restricted parameter space, Bayes estimator, squared error loss, dominance, symmetric location families.

1. research supported by NSERC of Canada

2. Corresponding author; e-mail address: eric.marchand@usherbrooke.ca

1. Introduction

Consider the problem of estimating under squared-error loss, based on an observable X, the median θ of a spherically symmetric univariate model where θ is known to be restricted to an interval [a,b]. Without loss of generality, we set a = −m, b = m, with m > 0. Interesting questions pertain to the frequentist performance of Bayesian estimators, such as the determination or description of Bayesian estimators that improve upon the benchmark (but inadmissible; e.g., Charras and van Eeden, 1993) maximum likelihood estimator (mle). With respect to the last question, findings have been obtained, for normal models and among others, by Casella and Strawderman (1981); Gatsonis, MacGibbon and Strawderman (1987); and Marchand and Perron (2001); and for more general models by Moors (1981, 1985) and Marchand and Perron (2007).

The results of Marchand and Perron (2001), which are developed for multivariate versions with X ∼ N_{p}(θ,σ^{2}I_{p}), ∥θ∥ ≤ m, and known σ^{2}, will be of particular relevance to us. Namely, their results for p = 1 imply that:

(i) all Bayes estimators with respect to a symmetric about 0 prior dominates the mle as soon as m ≤ c_{0}σ with c_{0} ≈ 0,4837;

(ii) the Bayes estimator δ_{BU} with respect to a boundary uniform prior on {−m,m} dominates the

mle as soon as m ≤ σ; (iii) the Bayes estimator δ_{U} with respect to a fully uniform on [−m,m] prior dominates the mle as

soon as m ≤ c_{1}σ with c_{1} ≈ 0,5230.

The technical arguments relied upon to arrive at these findings are conditional risk decompositions of the type |X| ≤ m, |X| > m, and |X| = r, r > 0. Perron (2003) exploited these types of decompositions to obtain analogous dominance results for a estimating Binomial proportion p when |p − \frac{1}{2}| is constrained above. The main objectives of this paper are to make it apparent that these conditional risk techniques are not specific to a normal model, and to obtain explicit dominance results (e.g., Corollary 3) for symmetric location models analogous to (i) and (iii). With respect to (ii), Marchand and Perron (2007) have already obtained extensions for a large class of symmetric models.

Sections 3 and 4 contain various dominance results and illustrations. More specifically, Section 3 gives for various models, conditions on (m,δ) for which symmetric about 0 estimators δ dominate the mle. Section 4 makes connections between priors π and associated δ_{π}’s arriving at dominance conditions on π for Bayesian estimators δ_{π}. For several applications, such as Corollary 3, the results and the technical arguments leading to these beginning in Section 2, apply to logconcave densities with log convex derivatives (examples include Normal, Laplace, Logistic, Hyperbolic secant). However, not all the dominance results require such assumptions, as illustrated by the univariate Cauchy and Student distributions (Example 2).

2. Definitions and technical lemmas

Our work here relates to the estimation of the parameter θ, under squared error loss (d − θ)^{2}, for symmetric location families with densities f(·|θ) on ℝ (with respect to Lebesgue measure) of the form:

f(x| θ) = e^{−h(x−θ)}, θ ∈ Θ(m) = [−m,m], (m fixed).  (1)

The corresponding cdf’s will be denoted F_{θ,h}. In (1), h is a given continuous and even function. As our model and loss being invariant under sign changes, we will consider equivariant estimators only (i.e., odd functions); and corresponding risk functions will be symmetric functions on Θ(m). We will be concerned with the specification of such estimators that dominate δ_{0}, with δ_{0}(x) = (x ∧ m) for x > 0. For unimodal models in (1) or cases where h is increasing, δ_{0} coincides with the maximum likelihood estimator of θ (in such cases, we will refer to this target estimator as δmle; otherwise, we will write δ_{0}).

Our dominance results are based on the conditional risk decompositions : (i) |X| > m, (ii) |X| ≤ m, and (iii) |X| = r; and make use of the following preliminary results and definitions concerning various conditional expectations and how their behaviour depends on h. Definition 1. (a) For m > 0, set

H^{∗∗}(m) = {h : h increasing and convex on (0,2m)};
H^{∗}(m) = {h : h ∈ H^{∗∗}(m) and h′ is concave on (0,2m)};
H^{∗∗∗}(m) = {h : h increasing and h(θ + x) − h(θ − x) increasing in θ on (x,m), for all x ∈ (0,m)}.

(b) For fixed increasing h, set b_{h}^{∗}, b_{h}^{∗∗}, b_{h}^{∗∗∗} as

b_{h}^{∗} = sup{m ≥ 0 : H^{∗}(m) ∋ h};
b_{h}^{∗∗} = sup{m ≥ 0 : H^{∗∗}(m) ∋ h};
b_{h}^{∗∗∗} = sup{m ≥ 0 : H^{∗∗∗}(m) ∋ h}.

(note: if h is increasing and convex on (0,∞), we will say that h ∈ H^{∗∗}(∞) and b_{h}^{∗∗} = ∞. We adopt similar conventions for H^{∗}, H^{∗∗∗}, b_{h}^{∗}, and b_{h}^{∗∗∗}.)

Remark 1. It is immediate with the above definitions that: (i) for all m > 0, H^{∗}(m) ⊆ H^{∗∗}(m) ⊆ H^{∗∗∗}(m); and (ii) for all fixed increasing h, b_{h}^{∗} ≤ b_{h}^{∗∗} ≤ b_{h}^{∗∗∗}. From Marchand and Perron (2007), we also have that H^{∗∗∗}(m) ⊆ H^{∗∗}(\frac{m}{2}) for all m > 0. We also require the following quantities, defined for any h.

ρ_{h}(θ,x) = tanh(\frac{h(θ + x) − h(θ − x)}{2}), θ,x ∈ ℝ;
ρ̄_{h}(m,x) = sup{ρ_{h}(θ,x) : θ ∈ [0,m]} , m,x > 0;
α_{h}(m,θ) = E_{θ}[ρ_{h}(θ,|X|)||X| > m]; 0 ≤ θ ≤ m;
ᾱ_{h}(m) = sup{α_{h}(m,θ) ; θ ∈ [0,m]}, m > 0;
β_{h}(m,θ) = E_{θ}[\frac{θρ_{h}(θ,X)}{X}||X| ≤ m], 0 ≤ θ ≤ m;
β̄_{h}(m) = sup{β_{h}(m,θ) ; θ ∈ [0,m]}, m > 0;
G_{h}(m) = mh′(m), m > 0; for h differentiable at m.

Observe that for fixed h such that b_{h}^{∗∗} > 0, G_{h} necessarily increases on (0,b_{h}^{∗∗}). In such cases, the quantities G_{h}^{−1}(a) = inf{m : mh′(m) > a} will play a key role (e.g., Corollary 2, Remark 3, Table 1). Here are some useful properties concerning the quantities above. Lemma 1. For a fixed nondecreasing h, we have

(a) ρ_{h}(· , x) is nondecreasing on [0,m] for all x > 0 if and only if h ∈ H^{∗∗∗}(m);

(b) ρ_{h}(θ , ·) is nondecreasing on (0,∞) for all θ ∈ (0,m) if and only if h ∈ H^{∗∗}(∞);

(c) E_{θ}[sgn(X)||X| = r] = ρ_{h}(θ , r); r ≥ 0 ;

(d) for any fixed θ ≥ 0, lim_{x→0^{+}} \frac{ρ_{h}(θ , x)}{x} = h′(θ); (h differentiable at θ);

(e) for any fixed θ ∈ [0,m], \frac{ρ_{h}(θ , x)}{x} is decreasing in x on (0,m) whenever h ∈ H^{∗}(m); (f) whenever h ∈ H^{∗}(m), we have ρ̄_{h}(m,x) ≤ xh′(m) for all x ∈ (0,m).

Proof. Parts (a,b,c) are given by Marchand and Perron (2007, lemmas 1 and 2), while part (d) follows by a direct evaluation. Part (f) is a direct consequence of (d,e). In order to establish the result in (e), it will suffice to show that, for any fixed θ ∈ [0,m], the function ρ_{h}(θ , x) is concave in x on (0,m), given that ρ_{h}(θ,0) = 0 for all θ. Since the hyperbolic tangent function is concave on [0,∞), and that a composition tanh ∘φ is necessarily concave for nondecreasing and concave φ, it will suffice to show that φ(x) = h(θ + x) − h(θ − x) is nondecreasing and concave in x on (0,m) and takes its values on [0,∞). Since h is nondecreasing on (0,m), even and convex on (−2m,2m), we have φ(0) = 0 and φ nondecreasing on (0,m). Also, we have for (x_{0},x_{1}) such that 0 < x_{0} < x_{1} < m:

(i) φ′(x_{1}) − φ′(x_{0}) = [{h′(θ + x_{1}) − h′(θ + x_{0})} − {h′(θ − x_{0}) − h′(θ − x_{1})}] ≤ 0 if x_{1} ≤ θ

and

(ii) φ′(x_{1}) − φ′(x_{0}) = [{h′(x_{1} + θ) − h′(x_{1} − θ)} − {h′(x_{0} + θ) − h′(x_{0} − θ)}] ≤ 0 if θ ≤ x_{0},

because h′ is concave on (0,2m). Therefore, φ is concave on both the intervals (0,θ) and (θ,m). Finally, φ is continuous at θ and φ′(θ−) = h′(2θ) − h′(0−) ≥ h′(2θ) − h′(0+) = φ′(θ+), implying that φ is concave on (0,m). Lemma 2. For any fixed h,

(a) the function α_{h}(m,·) increases on [0,m] with ᾱ_{h}(m) = α_{h}(m,m) = \frac{1−2F_{0,h}(−2m)}{1+2F_{0,h}(−2m)}, lim_{m→0^{+}} ᾱ_{h}(m) = 0, lim_{m→∞} ᾱ_{h}(m) = 1, and ᾱ_{h}^{−1}(1/2) = \frac{1}{2}F_{0,h}^{−1}(\frac{5}{6});

(b) β̄_{h}(m) ≤ G_{h}(m) for any fixed (m,h) such that h ∈ H^{∗}(m).

Proof. For part (a), a direct evaluation gives us for θ ∈ [0,m]:

α_{h}(m,θ) = E_{θ}[tanh(\frac{h(θ + |X|) − h(θ − |X|)}{2})||X| > m]
= \frac{∫_{m}^{∞}(e^{−h(θ−x)} − e^{−h(θ+x)})dx}{∫_{m}^{∞}(e^{−h(θ−x)} + e^{−h(θ+x)})dx}
= \frac{F_{0,h}(m + θ) − F_{0,h}(m − θ)}{2 − F_{0,h}(m + θ) − F_{0,h}(m − θ)}
= 1 − 2\frac{1}{1 + \frac{1−F_{0,h}(m−θ)}{1−F_{0,h}(m+θ)}} .

From the above, the increasing property of α_{h}(m,·) follows at once, with ᾱ_{h}(m) = α_{h}(m,m) as given. The other given properties of (a) follow in a straightforward manner. For (b), we have indeed from parts (d,e) of Lemma 1, and for all θ ∈ [0,m]: β_{h}(m,θ) ≤ θh′(θ) ≤ mh′(m) = G_{h}(m), since h ∈ H^{∗}(m).

Lemma 3. For h ∈ H^{∗}(∞), the density of R = |X| has increasing monotone likelihood ratio in R, with λ = |θ| taken as the parameter.

Proof. It is sufficient (and necessary) to show that the mixed derivative \frac{∂^{2}}{∂λ∂r} log g_{h}(λ,r) ≥ 0 for all λ,r > 0; g_{h}(λ,·) being the density of R under (1). Now set z_{1} = \frac{1}{2}h(λ + r), z_{2} = \frac{1}{2}h(λ − r). Under (1), we have directly g_{h}(λ,r) = \frac{1}{2}(e^{−2z_{1}} + e^{−2z_{2}}), which we can also write as

g_{h}(λ,r) = cosh(z_{1} − z_{2})e^{−(z_{1}+z_{2})}.

Working with this last representation, it is easy to establish that:

\frac{∂^{2}}{∂λ∂r} log g_{h}(λ,r) = (1−tanh^{2}(z_{1}−z_{2}))\frac{∂}{∂λ}(z_{1}−z_{2})\frac{∂}{∂r}(z_{1}−z_{2})+tanh(z_{1}−z_{2})\frac{∂^{2}}{∂λ∂r}(z_{1}−z_{2})−\frac{∂^{2}}{∂λ∂r}(z_{1}+z_{2}).

The result follows as

(i) | tanh | ≤ 1 and \frac{∂}{∂λ}(z_{1} − z_{2})\frac{∂}{∂r}(z_{1} − z_{2}) = \frac{1}{2}\{h′(λ + r)^{2} − h′(λ − r)^{2}\} ≥ 0, given that h′ is

increasing and positive; (ii) tanh(z_{1} − z_{2}) ≥ 0 with tanh(·) increasing, z_{1} ≥ z_{2} because h is increasing on (0,∞) and even

on (−∞,∞), and \frac{∂^{2}}{∂λ∂r}(z_{1} − z_{2}) = \frac{1}{2}\{h′′(λ + r) + h′′(λ − r)\} ≥ 0 given that h is convex; (iii) and, finally, \frac{∂^{2}}{∂λ∂r}(z_{1} +z_{2}) = \frac{1}{2}\{h′′(λ+r)−h′′(λ−r)\} ≤ 0 given that h is even and h′ is concave

on (0,∞).

3. General dominance results

Here, for an equivariant estimator δ, we provide conditions on (h,m,δ) for which δ dominates δ_{0} on Θ(m), with risk analyzes for θ ∈ [0,m] sufficing given the symmetry of the risk functions involved. As in Marchand and Perron (2001), we proceed by studying improvements with respect to the conditional risks for: (i) |X| > m, (ii) |X| ≤ m, and (iii) |X| = r, r > 0. Notwithstanding the fact that a large portion of the development below does indeed represent an extension of the univariate normal case analyzed by Marchand and Perron (2001) to general models in (1), the technical arguments used match those given by Marchand and Perron (2001), with results of Section 2 called upon for various bounds on monotonicity properties. Hence, we have relegated some proofs to an Appendix. Theorem 1. For a model in (1) with h ∈ H^{∗∗}(∞), and δ a nondecreasing function on (m,∞), the condition

m(2ᾱ_{h}(m) − 1) ≤ δ(x) ≤ m, for all x > m,  (2)

with strict inequalities for some x ∈ (m,∞), is sufficient to have

E_{θ} [|θ − δ_{mle}(X)|^{2} − |θ − δ(X)|^{2}||X| > m] > 0 , for all θ ∈ Θ(m).

Proof. See Appendix. Remark 2. Lemma 2 tells us that the left-hand side of (2) can also be expressed explicitly (or written as we will do frequently hereafter) as m (\frac{1−6F_{0,h}(−2m)}{1+2F_{0,h}(−2m)}). Theorem 2. For a model in (1) with (m,h) such that h ∈ H^{∗}(m) (i.e., m ≤ b_{h}^{∗}) and β̄_{h}(m) ≤ 1, and g a nondecreasing function on (0,m) such that x(x − δ(x)) increases in x for x ∈ (0,m), the condition

(2β̄_{h}(m) − 1)x ≤ δ(x) ≤ x, for all x ∈ (0,m),

with strict inequalities for some x ∈ (0,m), is sufficient to have E_{θ} [|θ − δ_{mle}(X)|^{2} − |θ − δ(X)|^{2}||X| ≤ m] > 0 for all θ ∈ Θ(m).

Proof. See Appendix. Combining the dominance conditions of Theorem 1 and Theorem 2 leads directly to the following. Corollary 1. For a model in (1) with (m,h) such that h ∈ H^{∗}(m) ∩ H^{∗∗}(∞) and β̄_{h}(m) ≤ 1, and δ a nondecreasing function on (0,∞) such that (i) δ(x) < x ∧ m for x > 0, (ii) x(x − δ(x)) increases in x on (0,m), the conditions

\frac{δ(m)}{m} > \frac{1 − 6F_{0,h}(−2m)}{1 + 2F_{0,h}(−2m)}, and \frac{δ(x)}{x} > 2β̄_{h}(m) − 1 for all x ∈ (0,m)  (3)

are jointly sufficient for δ to dominate δ_{mle} on Θ(m). Remark 3. For densities in (1) that are logconcave with logconvex derivatives (i.e., h ∈ H^{∗}(∞)), the conditions on δ of Corollary 1 yield many dominating estimators as long as β̄_{h}(m) ≤ 1, which in turn holds whenever m ≤ G_{h}^{−1}(1) (see Lemma 2). Furthermore, a wide class of solutions consist in selecting δ’s such that \frac{δ(x)}{x} decreases for x ∈ (0,m) (which together with (i) implies (ii)), with \frac{δ(m)}{m} > (2β̄_{h}(m) − 1) ∨ (\frac{1−6F_{0,h}(−2m)}{1+2F_{0,h}(−2m)}).

Example 1. (Truncated Linear and truncated linear minimax estimators) Interesting and simple alternatives to δmle are truncated linear estimators of the form δ_{a}(x) = (ax∧m)sgn(x), for a ∈ (0,1). A particulary appealing choice, available for cases where the variance in (1) exists (say equal to V), is given by the choice a_{0} = \frac{m^{2}}{m^{2}+V} corresponding to the truncation onto [−m,m] of the linear minimax estimator for θ ∈ Θ(m). Although a_{0}X dominates X on Θ(m), there is no guarantee that dominance will carry over to the truncated versions. For the estimator δ_{a} with a ∈ (0,1), it is easy to see that conditions (i) and (ii) of Corollary 1 are satisfied. Hence for models in (1), Corollary 1 (or Remark 3) tells us, for (m,h) satisfying its conditions, that δ_{a} dominates δmle whenever :

(2β̄_{h}(m) − 1) ∨ (\frac{1 − 6F_{0,h}(−2m)}{1 + 2F_{0,h}(−2m)}) ≤ a < 1.  (4)

For the truncated linear minimax estimator δ_{a_{0}}, the evaluation above is more delicate since a_{0} depends on m. The following is specific to the normal case, but similar analyzes can be arrived at for other logconcave densities. Theorem 3. For a normal N(θ,σ^{2}) model with |θ| ≤ m, δ_{a_{0}} dominates δmle whenever m ≤ c_{1} ≈ 0.60936σ, with c_{1} being the unique positive solution in c to the equation ψ(c) = 0, with ψ(c) = \frac{c^{2}}{c^{2}+1} − (\frac{1−6Φ(−2c)}{1+2Φ(−2c)}); Φ being the standard normal cdf.

Proof. Setting σ = 1 without loss of generality and applying (4) with a = \frac{m^{2}}{m^{2}+1}, we seek to determine the values of m such that: (1) \frac{m^{2}}{m^{2}+1} ≥ 2β̄_{h}(m) − 1, and (2) ψ(m) ≥ 0. For (1), make use of part (b) of Lemma 2 with h(y) = (y^{2})/2 to infer that (1) holds as long as \frac{m^{2}}{m^{2}+1} ≥ 2m^{2} − 1; i.e., m ≤ (\frac{1}{2})^{1/4}. With regards to (2), observe that ψ(0) > 0, and that

ψ(m) ≥ 0 ⟺ T (m) ≥ \frac{1}{2}, with T (m) = (4m^{2} + 3)Φ(−2m).

A direct evaluation yields T′(m) = 2Φ(−2m)(4m − (4m^{2} + 3)\frac{ϕ(2m)}{Φ(−2m)}); ϕ being the standard normal pdf. From this and by virtue of the inequality \frac{ϕ(x)}{Φ(−x)} ≥ x (x ≥ 0), we infer that T (m) decreases in m on [0,∞). Since lim_{m→∞} T (m) = 0, we conclude that ψ(m) changes signs once from + to − as m ranges on (0,∞). Finally, a direct evaluation tells us that ψ((\frac{1}{2})^{1/4}) ≤ 0, yielding the result. Theorem 4. (Marchand & Perron, 2005) For a model in (1), the (symmetric) estimator δ dominates δ_{0} on Θ(m) whenever, for all x > 0,

2 sup{θρ_{h}(θ,x) : θ ∈ [0,m]} − (m ∧ x) < δ(x) < (m ∧ x),  (5)

for all x ∈ A_{h,m} = {x > 0 : sup{θρ_{h}(θ,x) : θ ∈ [0,m]} < x}, and δ(x) = x otherwise.

The above condition is also used in Marchand and Perron (2007) to determine conditions for which the Bayes estimator with respect to a uniform 2-point prior on {−m,m}, given by δ_{BU}(x) = mρ_{h}(m,x) for x > 0, dominates δ_{0} on Θ(m). Two key observations lead to their Theorem 2 result. First, if (m,h) are such that h ∈ H^{∗∗∗}(m), then we have ρ̄_{h}(m,x) = ρ_{h}(m,x) for all x > 0 (part (a) of Lemma 1). Secondly, if (m,h) are such that h ∈ H^{∗}(m) and m ≤ G_{h}^{−1}(1) (see Table 1), then (5) is satisfied for δ = δ_{BU} as seen as a consequence of part (f) of Lemma 1 (they actually give somewhat weaker conditions along with illustrations). We will not use Theorem 4 in the same way. Instead, we will, as in the next result and the examples that follow, exploit conditions for which the left-hand side of (5) is bounded above, namely by 0, which leaves a large window in (5) for δ to vary and for δ to dominate δ_{0}. Corollary 2. For a model in (1), an equivariant estimator δ of θ, such that δ ≠ δ_{0} and 0 ≤ δ(x) ≤ (m ∧ x) for all x > 0, dominates δ_{0} when either:

(i) 2mρ̄_{h}(m,x) ≤ (m ∧ x) for all x > 0, (ii) or δ is nondecreasing on (m,∞), and (m,h) are such that h ∈ H^{∗}(m) with m ≤ G_{h}^{−1}(\frac{1}{2}) ∧

\frac{1}{2}F_{0,h}^{−1}(\frac{5}{6}).

Proof. The sufficiency of (i) is a direct consequence of (5). For (ii), we verify (5) for all x ∈ (0,m) and (2) for all x > m. For x ∈ (0,m), apply part (f) of Lemma 1 to infer that 2mρ̄_{h}(m,x) ≤ 2mh′(m)x ≤ x as soon as m ≤ G_{h}^{−1}(1/2). For x > m, it is straightforward to see that the mere nonnegativity of g implies that (2) is satisfied whenever ᾱ_{h}(m) ≤ \frac{1}{2}, that is m ≤ \frac{1}{2}F_{0,h}^{−1}(\frac{5}{6}) as seen in Lemma 2. Remark 4. For densities in (1) that are logconcave with logconvex derivatives (i.e., h ∈ H^{∗}(∞)), condition (ii) of Corollary 2 gives, for shrinkage estimators δ which increase on [m,∞), universal dominance with no further conditions on g, but requiring only small enough m. Furthermore, it will be established below (i.e., Lemma 4, Corollary 3) that the conditions on g are shared by all Bayes estimators with respect to symmetric about 0 priors on Θ(m). This universal dominance result, which illustrates vividly the inadequacy of δmle for small parameter spaces (under squared error loss) extends the univariate normal result of Marchand and Perron (2001) (they also give a multivariate normal version). Observe that the estimator δ(X) = 0, which has been further studied by Dou and van Eeden (2006) in the normal case, belongs to the class of dominating estimators. A selection of further examples of models for which h ∈ H^{∗}(∞); which include members of the Exponential power family such as Laplace (or Double-Exponential) distributions, Hyperbolic Secant, and Logistic distributions (see Marchand and Perron, 2007, for more details and examples); along with the corresponding upper bounds on m, are given in Table 1. Example 2. (Student and Cauchy distributions) Consider families of distributions in (1) with densities

f(x| θ) = \frac{1}{B(ν/2,1/2)σ\sqrt{ν}} \{1 + \frac{1}{ν}(\frac{x − θ}{σ})^{2}\}^{− \frac{(ν+1)}{2}} ,  (6)

corresponding to Student distributions with positive scale and shape parameters σ and ν. Even though the associated h functions belong to H^{∗∗}(\frac{σ\sqrt{ν}}{2} ), they do not belong to H^{∗∗}(∞) (see Marchand and Perron, 2007), and part (ii) of Corollary 2 cannot be applied. However, turning to the potential applicability of part (ii) of Corollary 2, ρ̄_{h} admits a useful lower bound as established by Marchand and Perron (2005, Lemma 2 and equation A.4) which tells us that

mρ̄_{h}(m,x) ≤ \frac{m^{2}x}{m^{2} + x^{2} + σ^{2}ν} (1 + ν), for all x > 0,

with equality for all x > 0 if and only if ν = 1 (i.e., Cauchy case). Simple analysis with the upper bound tells us that: (a) 2mρ̄_{h}(m,x) ≤ x for x ∈ (0,m) whenever m ≤ σ\sqrt{\frac{ν}{1+2ν}}, and (b) 2mρ̄_{h}(m,x) ≤ m for x > m whenever m ≤ σ\sqrt{\frac{1}{ν+2}}. Therefore, unifying both conditions for ν ≥ 1, we infer that any shrinkage estimator δ (towards 0) with respect to δ_{0} is a dominator of δ_{mle} under (6) as soon as m ≤ σ \frac{1}{\sqrt{ν+2}}. In the Cauchy case, the bound becomes m ≤ σ\sqrt{\frac{1}{3}}. We conclude by mentioning that, for the specific case of the Bayes estimator δ_{BU} with respect to the boundary uniform prior on {−m,m}, Marchand and Perron (2005) establish the stronger condition m ≤ σ for dominance. But the new condition above is universal in the sense that it applies to all shrinkage estimators. Remark 5. Although this paper does not address multivariate extensions with a constraint of the form ∥θ∥ ≤ m, as in Marchand and Perron (2001,2005), one wonders about such extensions. Several difficulties persist, namely with regards to Bayesian estimators, but we point out that the above Student case admits the following relatively straightforward extension, which we establish in the Appendix.

Theorem 5. For X − θ distributed p-variate student with d degrees of freedom and ||θ|| ≤ m, any equivariant shrinkage estimator δ with respect to δmle (i.e., δ(x) = g(∥x∥)\frac{x}{∥x∥}, with 0 ≤ g(r) ≤ m∧r for all r > 0) dominates δmle as soon as m ≤ \sqrt{\frac{p^{2}}{d+2p} ∧ \frac{d}{3}}.

4. Bayesian dominance results

We now turn our attention to Bayesian estimators δ_{π}, associated with symmetric about 0 priors π, and work towards conditions on π which will justify the applicability of the dominance results of Section 3. The next lemma summarizes the key relationships between π and δ_{π} which we will make use of. Lemma 4. Let π be a symmetric about 0 prior density on Θ(m) (with respect to a symmetric σfinite measure ν), and δ_{π} the corresponding Bayes estimator for squared error loss under model (1). Then,

(a) if h ∈ H^{∗∗}(∞), then the family of posterior densities π(·|x) possess increasing monotone likelihood ratio;

(b) δ_{π} is nondecreasing whenever h ∈ H^{∗∗}(∞);

(c) δ_{π} admits the representation: δ_{π}(x) = E_{x}[Uρ_{h}(U,x)] x ≥ 0, where U has density (with respect to ν) proportional to {(e^{−h(x−u)} + e^{−h(x+u)}) I_{( 0 , m]}(u) + e^{−h(x)} I_{{0}}(u) }π(u);

(d) if (m,h) are such that h ∈ H^{∗}(m), then δ_{π}(x) ≤ x ∧ m, for all x > 0, whenever (i) E_{x}[Uh′(U)] ≤ 1, or (ii) m ≤ G_{h}^{−1}(1);

(e) if h is an increasing function on (0,∞), and if π is a continuous, logconcave, and nondecreasing density on (0,m), then both x − δ_{π}(x) and x(x − δ_{π}(x)) increase in x, x ≥ 0; (f) If π^{∗} is another prior (also with respect to ν) such that \frac{π^{∗}(θ)}{π(θ)} increases in θ on [0,m], and if

(m,h) are such that h ∈ H^{∗∗∗}(m) ∩ H^{∗∗}(∞), then δ_{π^{∗}}(x) ≥ δ_{π}(x) for all x ≥ 0;

(g) For h ∈ H^{∗}(∞) any any prior π, the family of posterior densities of |θ| given x, x ≥ 0, have increasing monotone likelihood ratio in |θ|, with x taken as the parameter.

Proof. (a,b) Part (b) follows immediately from (a) since δ_{π}(x) is a posterior expectation. For (a), observe that, for any (x_{1},x_{2}) such that x_{1} < x_{2},

\frac{π(θ|x_{2})}{π(θ|x_{1})} ∝ e^{h(θ−x_{1})−h(θ−x_{2})} ,

which is indeed increasing in θ, θ ∈ Θ(m), whenever h is convex; i.e, h ∈ H^{∗∗}(∞).

(c) Start with the expression

δ_{π}(x) = \frac{∫_{[−m,m]} te^{−h(x−t)} π(t) dν(t)}{∫_{[−m,m]} e^{−h(x−t)} π(t) dν(t)} ,

set u = −t for t ∈ [−m, 0 ], and exploit the symmetry of both h and π to obtain

δ_{π}(x) = \frac{∫_{(0,m]} t\{e^{−h(x−t)} − e^{−h(x+t)} \}π(t) dν(t)}{∫_{(0,m]} \{e^{−h(x−t)} + e^{−h(x+t)} \} π(t) dν(t) + e^{−h(x)} π(0)ν(\{0\})} .

Finally, the representation holds with the definition of ρ_{h}, and since \frac{e^{−A}−e^{−B}}{e^{−A}+e^{−B}} = tanh(\frac{A−B}{2} ), for all A,B.

(d) Clearly δ_{π}(x) ≤ m for all x ≥ m. Now, to have the property δ_{π}(x) ≤ x for x ∈ [0,m], condition (i) suffices as seen by making use of part (f) of Lemma 1. Condition (ii), which says that mh′(m) ≤ 1, implies (i) given that h is convex on (0,m) by assumption that h ∈ H^{∗}(m).

(e) The increasing property of x − δ_{π}(x) is established within Proposition 3.1 of Kubokawa (2005), and implies as well that x(x − δ_{π}(x)) increases for x ≥ 0 (since δ_{π}(0) = 0 given the symmetry of π and h).

(f) In part (c) above, set g_{π, x} and g_{π^{∗},x} as the densities of U associated with observation x and with priors π and π^{∗}, respectively. Since the increasing property of the ratio \frac{π^{∗}(u)}{π(u)} transfers over to the ratio \frac{g_{π^{∗},x}(u)}{g_{π,x}(u)}; u ∈ [0,m]; we infer that δ_{π^{∗}}(x) ≥ δ_{π}(x); x ≥ 0; given that uρ_{h}(u,x) is, by part (a) of Lemma 1 and since h ∈ H^{∗∗∗}(m), increasing in u; u ∈ [0,m]. From the above, we now infer the following as a consequence of Corollary 2.

(g) The result and proof are perfectly analogous to those of Lemma 3. Corollary 3. (a) For models in (1), with (m,h) such that h ∈ H^{∗}(m) ∩ H^{∗∗}(∞), any Bayes

estimator δ_{π} with respect to a symmetric about 0 prior dominates δmle on Θ(m) as soon as m ≤ G_{h}^{−1}(\frac{1}{2}) ∧ \frac{1}{2}F_{0,h}^{−1}(\frac{5}{6}).

(b) For models in (1), and an arbitrary Bayes estimator δ_{π} with respect to a symmetric about 0 prior, condition (i) of Corollary 2 suffices for δ_{π} to dominate δ_{0}.

Proof. By virtue of Corollary 2, we only need to show that δ_{π}(x) is a nondecreasing shrinkage estimator in (a); and a shrinkage estimator in (b). With the given assumption on (m,h), the nondecreasing and shrinkage properties required in (a) follow from parts (a) and (d, ii) (respectively) of Lemma 4. Now, suppose condition (i) of Corollary 2 holds. Then, the shrinkage property required in (b) follows from part (c) of Lemma 4 as Uρ_{h}(U,x) ≤ mρ̄_{h}(m,x) (with probability one, for all x > 0), which implies δ_{π}(x) ≤ mρ̄_{h}(m,x) ≤ 2mρ̄_{h}(m,x) ≤ m ∧ x, for all x > 0. Example 2 (continued) For Student densities in (6), ν ≥ 1, it is immediate from the above that dominance holds universally for all symmetric Bayes estimators as soon as m ≤ σ \frac{1}{\sqrt{ν+2}}. Now, concerning the applicability of Corollary 1 to Bayesian estimators, the results of Lemma 4 lead to the following.

Corollary 4. For models in (1), with (m,h) such that h ∈ H^{∗}(m) ∩ H^{∗∗}(∞) and m ≤ G_{h}^{−1}(1), and for prior densities π which are symmetric, continuous, logconcave, and nondecreasing on (0,m), condition (3) is sufficient for δ_{π} to dominate δmle.

Proof. It suffices to check that conditions (i) and (ii) of Corollary 1 are satisfied. On one hand,

(i) holds given part (d, ii) of Lemma 4 and part (b) of Lemma 2. On the other hand, part (e) of Lemma 4 tells us that (ii) is satisfied with the given assumptions on the prior π. Example 3. (Uniform and other symmetric, logconcave priors) Corollary 4 applies to the important and benchmark choice of a fully uniform prior on the parameter space Θ(m) and the corresponding Bayes estimator δ_{U}. The universal dominance result of Corollary 3 already applies, but the conditions here of Corollary 1 (or 4) give us weaker conditions (on m) for δ_{U} to dominate δmle. Hence, we proceeded to evaluate, for a selection of admissible models with h ∈ H^{∗}(∞), the set:

C_{U} = {m : m ≤ G_{h}^{−1}(1) and (3) is satisfied for δ_{U}}.

In Table 1, these sets C_{U} are reported on and contrasted with the universal cutoff points, as well as the dominance cutoffs points (i.e., m < G_{h}^{−1}(1)) for the boundary uniform prior. For other continuous, symmetric, logconcave priors π, which are also nondecreasing on (0,m), it follows from part (f) of Lemma 4 and Corollary 4 that the dominance set

C_{π} = {m : m ≤ G_{h}^{−1}(1) and (3) is satisfied for δ_{π}}

will contain the set C_{U}. Furthermore, again by virtue of part (f) of Corollary 4), the steeper π is on (0,m), the larger C_{π} is, with the steepest π being boundary uniform in the limit with corresponding maximal C_{π} given by the interval [0,G_{h}^{−1}(1)]. Finally, the above elements are partially illustrated in Table 2 for various models and the logconcave priors π(θ) ∝ |θ|^{k} I_{(−m,m)}(θ) ; k = 0,1,2,4.

TABLE 1

Sufficient conditions for dominance of δmle by different Bayesian estimators

Type of Prior

Density (f(x|θ) ∝) Symmetric Boundary Uniform Uniform
(m ≤ G_{h}^{−1}(\frac{1}{2}) ∧ \frac{1}{2}F_{h}^{−1}(\frac{5}{6})) (m ≤ G_{h}^{−1}(1))  (\{m : m ≤ G_{h}^{−1}(1) and

(3) is satisfied for δ_{U}})

Normal (e^{− \frac{1}{2} ( \frac{|x−θ|}{σ})^{2}})  m ≤ 0.4837σ  m ≤ σ  m ≤ 0.5230σ
Laplace (e^{−| \frac{x−θ}{σ} |})  m ≤ 0.5σ  m ≤ σ  m ≤ 0.59720σ
Logistic (\frac{e^{( \frac{|x−θ}{σ} |)}}{(1+e^{(| \frac{x−θ}{σ} |)})^{2}} )  m ≤ 0.8047σ  m ≤ 1.5434σ  m ≤ 0.9004σ
Hyp. Sech. (sech(|\frac{x−θ}{σ} |)) m ≤ 0.6585σ  m ≤ 1.1997σ  m ≤ 0.7640σ

TABLE 2

Dominance set C_{π} for different models and priors

Type of Prior π(θ) ∝

Density (f(x|θ) ∝)  I_{(−m,m)}(θ)  |θ|I_{(−m,m)}(θ)  θ^{2}I_{(−m,m)}(θ) θ^{4}I_{(−m,m)}(θ)
Normal (e^{\frac{1}{2} ( \frac{|x−θ|}{σ})^{2}})  (0,0.5230]  (0,0.6213]  (0,0.6624]  (0,0.7189]
Laplace (e^{−| \frac{x−θ}{σ} |})  (0,0.5972]  (0,0.7270]  (0,0.7805]  (0,0.8441]
Logistic (\frac{e^{( \frac{|x−θ}{σ} |)}}{(1+e^{(| \frac{x−θ}{σ} |)})^{2}} )  (0,0.9004]  (0,0.9783]  (0,1.0222]  (0,1.0760]
Hyperbolic secant (sech(|\frac{x−θ}{σ} |)) (0,0.7640]  (0,0.8938]  (0,0.9534]  (0,1.0237]

Acknowledgements

The research support of NSERC of Canada for both Marchand and Perron is gratefully acknowledged

by both authors.

References

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5. Appendix

The property that the covariance Cov(f_{1}(X),f_{2}(X)) is nonnegative for increasing f_{1} and f_{2} is used below. Proof of Theorem 1 Decomposing the difference in conditional risks for θ ∈ [0,m], we obtain

E_{θ} [|θ − δ_{mle}(X)|^{2} − |θ − δ(X)|^{2}||X| > m]
= E_{θ} [{ (m − δ(|X|))sgn(X) }{ ( (m + δ(|X|) sgn(X) ) − 2θ }||X| > m]
= E_{θ} [(m − δ(|X|))(δ(|X|) − {2θρ_{h}(θ,|X|) − m})||X| > m]
≥ E_{θ} [(m − δ(|X|))(δ(|X|) − {2θα_{h}(m,θ) − m})||X| > m]
≥ E_{θ} [(m − δ(|X|))(δ(|X|) − {2θᾱ_{h}(m) − m})||X| > m]
> 0,

where (i) both the second equality and inequality exploit property (c) of Lemma 1 as well as the definitions of ρ_{h} and ᾱ_{h}, (ii) the first inequality holds with the increasing properties of ρ_{h}(θ,·) (Lemma 1) and δ(·), and (iii) the strict inequality follows with the given bounds on δ.

Proof of Theorem 2 Decomposing the difference in conditional risks for θ ∈ [0,m], we obtain

E_{θ} [|θ − δ_{mle}(X)|^{2} − |θ − δ(X)|^{2}||X| ≤ m]
= E_{θ} [{ (|X| − δ(|X|))sgn(X) }{ ( (|X| + δ(|X|) sgn(X) ) − 2θ }||X| ≤ m]
= E_{θ} [(|X| − δ(|X|)) (δ(|X|) − \{2θ\frac{ρ_{h}(θ,|X|)}{|X|} − 1\} |X|) ||X| ≤ m]
≥ E_{θ} [(|X| − δ(|X|)) (δ(|X|) − {2β_{h}(m,θ) − 1}|X|)||X| ≤ m]
≥ E_{θ} [(|X| − δ(|X|)) (δ(|X|) − {2β̄_{h}(m) − 1}|X|)||X| ≤ m]
> 0

where (i) both the second equality and inequality exploit property (c) of Lemma 1 as well as the definitions of ρ_{h} and β̄_{h}, (ii) the first inequality holds given that δ(x), x(x − δ(x)) increase in x; x ∈ (0,m); and \frac{ρ_{h}(θ,x)}{x} decreases in x; x > 0; (part e of Lemma 1), and (iii) the strict inequality follows with the given assumptions on δ. Proof of Theorem 5. It follows from Marchand and Perron (2005) (Lemma 2, Theorem 2) that a sufficient condition for a shrinkage δ to dominate δ_{mle} is:

\frac{2m^{2}rγ}{m^{2} + r^{2} + d} ≤ m ∧ r, for all r > 0, with γ = 1 + (1 ∨ \frac{d}{p}).

For r ≤ m, the above requires: \frac{2m^{2}γ}{m^{2}+d} ≤ 1 ⇔ m^{2} ≤ \frac{d}{2γ−1}.

For r > m, we require:

\frac{2m^{2}rγ}{m^{2} + r^{2} + d} ≤ m for all r > m ⇔ \frac{2m^{2} γ \sqrt{m^{2} + d}}{m^{2} + (m^{2} + d) + d} ≤ m
⇔ m^{2} ≤ \frac{d}{γ^{2} − 1}.

Finally, the result follows since γ^{2}−1 ≥ 2γ−1 given that γ ≥ 2, and by verifying that \frac{d}{γ^{2}−1} = \frac{p^{2}}{d+2p}∧\frac{d}{3}.