Bayes Estimation Under a General Class of

Balanced Loss Functions

Mohammad Jafari Jozani,^{a,}^{1} Eric Marchand,^{a,}^{2} Ahmad Parsian,^{b,}^{3}

a Université de Sherbrooke, Département de mathématiques, Sherbrooke, QC, CANADA, J1K 2R1 b School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, IRAN

ABSTRACT For estimating an unknown parameter θ, we introduce and motivate the use of balanced loss functions of the form L_{ρ,ω,δ_{0}}(θ, δ) = ωρ(δ_{0}, δ) + (1 − ω)ρ(θ, δ), as well as weighted versions q(θ)L_{ρ,ω,δ_{0}}(θ, δ), with q(·) being a positive weight function, where ρ(θ, δ) is an arbitrary loss function, δ_{0} is a chosen a priori “target” estimator of θ, and the weight ω takes values in [0, 1). A general development with regards to Bayesian estimators under L_{ρ,ω,δ_{0}} is given, namely by relating such estimators to Bayesian solutions for the unbalanced case, i.e., L_{ρ,ω,δ_{0}} with ω = 0. Illustrations are given for various choices of ρ, such as for absolute value, entropy, linex, intrinsic (i.e., model based), and a generalization of squared error losses.

Keywords and phrases: Balanced loss function; Bayes estimator; entropy loss; intrinsic loss; Linex loss

MSC: 62C05; 62C10; 62F10; 62F30

1 e-mail: Mohammad.Jafari.Jozani@Usherbrooke.ca

2Research supported by NSERC of Canada (corresponding author; e-mail:

eric.marchand@usherbrooke.ca)

3Research supported by a grant of the Research Council of the University of Tehran. (e-mail: ahmad p@khayam.ut.ac.ir)

1 Introduction

This paper concerns Bayesian point estimation of an unknown parameter θ of a model X = (X_{1}, …, X_{n}) ∼ F_{θ}, under “Balanced” loss functions of the form:

L_{ρ,ω,δ_{0}}(θ, δ) = ωρ(δ_{0}, δ) + (1 − ω)ρ(θ, δ),  (1)

as well as weighted versions q(θ)L_{ρ,ω,δ_{0}}(θ, δ), with q(·) being a positive weight function (see Remark 2). Here ρ(θ, δ) is an arbitrary loss function, while δ_{0} is a chosen a priori “target” estimator of θ, obtained for instance from the criterion of maximum likelihood estimator, least-squares, or unbiasedness among others. Loss L_{ρ,ω,δ_{0}}, which depends on the observed value of δ_{0}(X), reflects a desire of closeness of δ to both: (i) the target estimator δ_{0}, and (ii) the unknown parameter θ; with the relative importance of these criteria governed by the choice of ω ∈ [0, 1).

Loss L_{ρ,ω,δ_{0}} may be viewed as a natural extension to Zellner’s (1994) balanced loss function (in one dimension), the lather being specific to a squared error loss ρ and a least-squares δ_{0}. Subsequent findings, due to Dey, Ghosh and Strawderman (1999), as well as Jafari Jozani, Marchand and Parsian (2006); who also study the use of more general target estimators δ_{0}; involve the issues of Bayesianity, admissibility, dominance, and minimaxity and demonstrate how results for unbalanced (ω = 0) squared error loss serve directly (and necessarily) in obtaining results for the balanced case with ω > 0.

Here we give a general Bayesian estimation connection (Lemma 1) between the cases ω = 0 and ω > 0. This useful connection is quite general, especially with respect to the choices of ρ and δ_{0}. Illustrations are given for various choices of ρ, such as for absolute value, entropy, linex, intrinsic (i.e., model based), and a generalization of squared error losses. We conclude with a discussion of ongoing related work.

2 Bayes estimation under balanced loss

We consider here Bayes estimation under balanced loss function L_{ρ,ω,δ_{0}} as in (1). When ω = 0, we simply use L_{0} instead of L_{ρ,0,δ_{0}} unless we want to emphasize the role of ρ . We show how the Bayes estimator of θ under balanced loss function L_{ρ,ω,δ_{0}} can be derived or expressed in terms of a Bayes estimator of θ under the loss L_{0}. We assume throughout necessary conditions for finite expected posterior loss. In particular, we work with ρ and δ_{0} such that ρ(δ, δ_{0}) < ∞ for at least one δ ≠ δ_{0}.

Lemma 1 For estimating θ under balanced loss function L_{ρ,ω,δ_{0}} in (1) and for a prior π(θ), the Bayes estimator δ_{π,ω}(X) corresponds to the Bayes solution δ^{∗}(X) with respect to (π^{∗}(θ|x), L_{0}), for all x, where

π^{∗}(θ|x) = ω1_{{δ_{0}(x)}}(θ) + (1 − ω)π(θ|x),

i.e., a mixture of a point mass at δ_{0}(x) and the posterior π(θ|x).

Proof: Denoting μ_{X}(·) and ν_{X}(·) as dominating measures of π(θ|x) and π^{∗}(θ|x) respectively, we have by definition of δ_{ω,π}(x), δ^{∗}(x), and L_{0}

δ_{ω,π}(x) = argmin_{δ} ∫_{Θ}{ωρ(δ_{0}, δ) + (1 − ω)ρ(θ, δ)}π(θ|x)dμ_{X}(θ)
= argmin_{δ} ∫_{Θ∪{δ_{0}(x)}} L_{0}(θ, δ)π^{∗}(θ|x)dν_{X}(θ) = δ^{∗}(x).  □

Remark 1 Strictly speaking, δ^{∗}(X) is not a Bayes estimator under loss L_{0} ( because π^{∗}(θ|x) is not the posterior associated with π), while δ_{ω,π}(X) is indeed Bayes under loss L_{ρ,ω,δ_{0}} and prior π.

Remark 2 Lemma 1 also applies for weighted versions of balanced loss function (1), i.e., L_{ρ,ω,δ_{0}}^{q}(θ, δ) = q(θ)L_{ρ,ω,δ_{0}}(θ, δ), with q(θ) being a positive weight function. Indeed, it is easy to see that the Bayes estimator of θ under loss L_{ρ,ω,δ_{0}}^{q} using the prior π_{0} is equivalent to the Bayes estimator of θ under loss L_{ρ,ω,δ_{0}} using prior π ≡ q × π_{0}.

We pursue with various applications of Lemma 1.

Example 1 (A generalization of squared error loss). In (1), the choice ρ(θ, δ) = ρ_{1}(θ, δ) = τ(θ)(δ −θ)^{2}, with τ(·) > 0, leads to loss ωτ(δ_{0})(δ −δ_{0})^{2} +(1−ω)τ(θ)(δ −θ)^{2}. Since under L_{0} and a prior π, the Bayes estimator is given by δ_{0,π}(x) = \frac{E_{π}(θτ(θ)|x)}{E_{π}(τ(θ)|x)} (subject to the finiteness conditions E_{π}(θ^{i}τ(θ)|x) < ∞; i = 0, 1; for all x), Lemma 1 tells us that

δ_{ω,π}(x) = \frac{E_{π^{∗}}(θτ(θ)|x)}{E_{π^{∗}}(τ(θ)|x)} = \frac{ωδ_{0}(x)τ(δ_{0}(x)) + (1 − ω)E_{π}(θτ(θ)|x)}{ωτ(δ_{0}(x)) + (1 − ω)E_{π}(τ(θ)|x)}.

The case (τ(θ) = 1, δ_{0} = least square estimator) leads to balanced squared error loss (and δ_{ω,π}(x) = ωδ_{0}(x) + (1 − ω)E_{π}(θ|x)) as introduced by Zellner (1994), and as further analyzed by Dey, Ghosh and Strawderman (1999). The case L_{ρ_{1},ω,δ_{0}}(θ, δ) with τ(θ) = 1 and arbitrary δ_{0}, as well as its weighted version q(θ)L_{ρ_{1},ω,δ_{0}}(θ, δ), were investigated by Jafari Jozani, Marchand and Parsian (2006) with respect to classical decision theory criteria such as Bayesianity, dominance, admissibility, and minimaxity. Gómez-Déniz (2006) investigates the use of q(θ)L_{ρ_{1},ω,δ_{0}}(θ, δ) with τ(θ) = 1 to credibility premiums.

Example 2 (Entropy balanced loss). The choice ρ(θ, δ) = \frac{θ}{δ} − log \frac{θ}{δ} − 1 in (1) leads to loss:

ω(\frac{δ_{0}}{δ} − log(\frac{δ_{0}}{δ}) − 1) + (1 − ω)(\frac{θ}{δ} − log(\frac{θ}{δ}) − 1).  (2)

Since under L_{0} (i.e., ω = 0 in (2), and a prior π, the Bayes solution is given by δ_{0,π}(x) = E_{π}(θ|x) (subject to the finiteness conditions: E_{π}(θ|x) < ∞ and E_{π}(log(θ)|x) < ∞, for all x). Lemma 1 tells us that δ_{ω,π}(x) = E_{π^{∗}}(θ|x) = ωδ_{0}(x) + (1 − ω)E_{π}(θ|x). As a special example, take the model X ∼ Gamma(α, θ); α known; θ > 0; (density proportional to x^{α−1}e^{−x/θ}); and consider estimating θ under loss (2) with arbitrary δ_{0} (e.g., δ_{0}(x) = δ_{mle}(x) = \frac{x}{α}). With conjugate prior π such that λ = θ^{−1} ∼ Gamma(γ, \frac{1}{h}); γ > −α, h ≥ 0; and with the posterior distribution of θ^{−1} being Gamma(α+γ, \frac{1}{h+x}); the unique Bayes estimator of θ under entropy balanced loss (2) is:

δ_{ω,π}(x) = ωδ_{0}(x) + (1 − ω)\frac{h + x}{α + γ − 1}.

Example 3 (Stein loss with ρ(θ, δ) = (\frac{δ}{θ})^{β} − βlog \frac{δ}{θ} − 1; β ≠ 0). This class of losses, which includes Example 2’s loss (i.e., β = 1), as well as Stein’s loss (i.e., β = −1) illustrates well Lemma 1. Under loss L_{0}, we have the Bayes solution δ_{0,π}(x) = (E_{π}(\frac{1}{θ^{β}} |x))^{− \frac{1}{β}} (subject to risk finiteness conditions). Lemma 1 tells us that δ_{ω,π}(x) = (E_{π^{∗}}(\frac{1}{θ^{β}} |x))^{− \frac{1}{β}} = \{\frac{ω}{(δ_{0}(x))^{β}} + (1 − ω)E_{π}(\frac{1}{θ^{β}} |x)\}^{− \frac{1}{β}}.

Example 4 (Case with ρ(θ, δ) = (\frac{θ}{δ} − 1)^{2}). Under loss L_{0}, for a prior π supported on (0, ∞),we have the Bayes solution δ_{0,π}(x) = \frac{E_{π}(θ^{2}|x)}{E_{π}(θ|x)} (subject to the finiteness condition E_{π}(θ^{2}|x) < ∞, for all x). Lemma 1 tells us that δ_{ω,π}(x) = \frac{E_{π^{∗}}(θ^{2}|x)}{E_{π^{∗}}(θ|x)} = \frac{ωδ_{0}^{2}(x)+(1−ω)E_{π}(θ^{2}|x)}{ωδ_{0}(x)+(1−ω)E_{π}(θ|x)}. Rodrigues and Zellner (1994) introduced and studied Bayesian estimators under such a loss function for a least square δ_{0} and an exponential model.

Example 5 (Linex loss with ρ(θ, δ) = e^{a(δ−θ)} −a(δ −θ)−1; a ≠ 0; (e.g., Zellner, 1986; Parsian and Kirmani, 2002)). Under loss L_{0}, we have the Bayes solution δ_{0,π}(x) = −\frac{1}{a}logE_{π}(e^{−aθ}|x) (subject to risk finiteness conditions). Hence, using Lemma 1, we have δ_{ω,π}(x) = −\frac{1}{a}logE_{π^{∗}}(e^{−aθ}|x) = −\frac{1}{a}log(ωe^{−aδ_{0}(x)} + (1 − ω)E_{π}(e^{−aθ}|x)) = −\frac{1}{a}log(ωe^{−aδ_{0}(x)} + (1 − ω)e^{−aδ_{0,π}(x)}).

The following development concerns intrinsic balanced loss functions, where ρ in (1) is derived in some automated manner from the model. For a general reference on intrinsic losses and additional details we refer to Robert (1996).

Example 6 (Intrinsic balanced loss functions). For a model X|θ ∼ f(x|θ), the choice ρ(θ, δ) = d(f(·|θ), f(·|δ)) in (1); i.e., the distance between f(·|θ) and f(·|δ); where d(·, ·) is a suitable distance function, leads to intrinsic balanced loss functions of the form,

ωd (f(·|δ_{0}), f(·|δ)) + (1 − ω)d (f(·|θ), f(·|δ)) .  (3)

Two candidates of interest for d in (3) are Hellinger and Kullback-Leibler distance leading to:

• Kullback-Leibler Balanced Loss,

L_{ω,δ_{0}}^{KL}(θ, δ) = ωE_{δ_{0}}[Ln\frac{f(X|δ_{0})}{f(X|δ)}] + (1 − ω)E_{θ}[Ln\frac{f(X|θ)}{f(X|δ)}];  (4)

• Hellinger Balanced Loss,

L_{ω,δ_{0}}^{H}(θ, δ) = ω \frac{1}{2}E_{δ_{0}}[(\sqrt{\frac{f(X|δ)}{f(X|δ_{0})}} − 1)^{2}] + (1 − ω)\frac{1}{2}E_{θ}[(\sqrt{\frac{f(X|δ)}{f(X|θ)}} − 1)^{2}]; (5)

or equivalently,

L_{ω,δ_{0}}^{H}(θ, δ) = 1 − ωE_{δ_{0}}[\sqrt{\frac{f(X|δ)}{f(X|δ_{0})}}] − (1 − ω)E_{θ}[\sqrt{\frac{f(X|δ)}{f(X|θ)}}].  (6)

Hence, with known Bayesian representations in the unbalanced case ; i.e., ω = 0; (e.g., Brown, 1986; Robert, 1996) and with Lemma 1, we can build a catalog of Bayesian representations for losses of the type L^{KL} and L^{H}. For instance, natural parameter exponential family of distributions with densities f(x|θ) = e^{θT (x)−ψ(θ)}h(x) (with respect to a σ-finite measure ν on X), and unknown natural parameter θ, lead to Kullback-Leibler balanced loss functions:

ω [(δ_{0} − δ)ψ′(δ_{0}) + ψ(δ) − ψ(δ_{0})] + (1 − ω) [(θ − δ)ψ′(θ) + ψ(δ) − ψ(θ)] .  (7)

Furthermore, under L_{0} (i.e., ω = 0 in (7)), and for prior π, the Bayes estimator of θ is given as a solution of ψ′(δ_{0,π}(x)) = E_{π}[ψ′(θ)|x] in δ_{0,π}(x). Hence, together with Lemma 1, we have that the Bayes estimator δ_{ω,π}(x) admits the following representation

ψ′(δ_{ω,π}(x)) = ωψ′(δ_{0}(x)) + (1 − ω)ψ′(δ_{0,π}(x)),

with the special case of estimating E_{θ}[T (X)] = ψ′(θ) (as for X ∼ N(θ, 1), T (X) = X), yielding δ_{ω,π}(x) = ωδ_{0}(x)+(1−ω)δ_{0,π}(x); i.e., a convex linear combination of δ_{0} and the Bayes estimator δ_{0,π}(x). Also, it is easy to see that for natural exponential family of distributions as above, Hellinger balanced loss in (6) reduces to:

1 − ωe^{\{ψ( \frac{δ_{0}+δ}{2} )− \frac{ψ(δ_{0})+ψ(δ)}{2} \}} − (1 − ω)e^{\{ψ( \frac{θ+δ}{2} )− \frac{ψ(θ)+ψ(δ)}{2}\}}.  (8)

For instance, a N(θ, 1) model leads to Hellinger balanced loss in (8) given by

1 − ωe^{− \frac{(δ_{0}−δ)^{2}}{8}} − (1 − ω)e^{− \frac{(θ−δ)^{2}}{8}} ,

and which we can refer to as a Balanced bounded reflected normal loss (e.g., Spiring (1993) for the case ω = 0).

Remark 3 In Example 2, loss (2) is equivalent to Kullback-Leibler balanced loss (7) for a Gamma(α, θ) model with known α > 0, and unknown scale parameter θ (see also Parsian and Nematollahi, 1996; Robert, 1996).

Example 7 (Balanced absolute loss or L_{1} loss). The choice ρ(θ, δ) = |δ − θ| in (1) leads to balanced L_{1} loss:

ω|δ − δ_{0}| + (1 − ω)|δ − θ|.  (9)

Since Bayesian solutions associated with L_{1} losses are posterior medians, Lemma 1 tells us that the Bayes estimate of θ, δ_{ω,π}(X), with respect to loss (9) and priors π is a median of π^{∗}(θ|x), for all x. Working now with priors π and models f(·|θ) leading to unique posterior medians, setting α(x) = (1 − ω)P_{π}(θ ≤ δ_{0}(x)|x) and F_{θ|x}^{−1} as the inverse posterior cdf, it is easy to see that for ω ≥ \frac{1}{2}, δ_{ω,π}(x) = δ_{0}(x), (i.e., the point mass of δ_{0}(x) has large enough probability to force the median to be δ_{0}(x)), while for ω < \frac{1}{2} a straightforward evaluation leads to:

δ_{ω,π}(x) = F_{θ|x}^{−1}(\frac{1 − 2ω}{2(1 − ω)})1_{[0, \frac{1}{2} −ω)}(α(x)) + δ_{0}(x)1_{[ \frac{1}{2} −ω, \frac{1}{2} )}(α(x)) + F_{θ|x}^{−1}(\frac{1}{2(1 − ω)})1_{[ \frac{1}{2} ,1)}(α(x)),

(10)

for all x. Now, we give some applications of (10),

(A) (Case of a positive normal mean). Let X|θ ∼ N(θ, σ^{2}) with θ ≥ 0, σ^{2} known, and cdf F_{0}. Decision theoretic elements concerning the estimation of θ have a long history (e.g., see Marchand and Strawderman, 2004, for a review). Here, take δ_{0}(x) = δ_{mle}(x) = max(0, x) and ω < \frac{1}{2} in (9), and consider the estimator δ_{ω,π}(x) associated with the flat prior π(θ) = 1_{(0,∞)}(θ). In evaluating (10), we have: (i) P_{π}(θ ≥ y|x) = \frac{F_{0}(x−y)}{F_{0}(x)} for y ≥ 0; (ii) F_{θ|x}^{−1}(z) = x − F_{0}^{−1}((1 − z)F_{0}(x)), and (iii) α(x) = (1 − ω)P_{π}(θ ≤ δ_{0}(x)|x) = (1 − ω)(1 − \frac{1}{2F_{0}(x)})1_{(0,∞)}(x). Now, since α(x) < \frac{1}{2}, and x ≤ F_{0}^{−1}(1 − ω), we obtain

δ_{ω,π}(x) = F_{θ|x}^{−1}(1 − \frac{1}{2(1 − ω)})1_{(−∞,F_{0}^{−1}(1−ω))}(x) + δ_{0}(x)1_{[F_{0}^{−1}(1−ω),∞)}(x)
= \{x − F_{0}^{−1}(\frac{F_{0}(x)}{2(1 − ω)})\}1_{(−∞,F_{0}^{−1}(1−ω))}(x) + x1_{[F_{0}^{−1}(1−ω),∞)}(x)
= x − \{F_{0}^{−1}(\frac{F_{0}(x)}{2(1 − ω)})1_{(−∞,F_{0}^{−1}(1−ω))}(x)\}.

Note that the expression holds for the unbalanced case as well (i.e., ω = 0), while a similar development holds for models X|θ ∼ f_{0}(x − θ) with θ ≥ 0, and known and symmetric f_{0}.

(B) Let X|θ ∼ N(θ, σ^{2}); θ ∈ ℜ, σ^{2} known; with cdf Φ(\frac{x−θ}{σ}). Using conjugate prior θ ∼ π(θ) = N(μ, τ^{2}); known μ and τ^{2}; and balanced L_{1} loss in (9) with δ_{0}(X) = X, ω < \frac{1}{2}, and since the posterior θ|x ∼ N(\frac{τ^{2}}{τ^{2}+σ^{2}} x + \frac{σ^{2}}{τ^{2}+σ^{2}} μ, \frac{τ^{2}σ^{2}}{τ^{2}+σ^{2}} ), we have α(x) = (1 − ω)P_{π}(θ ≤ x|x) = (1 − ω)Φ(d(x − μ)); with d = \frac{σ}{τ\sqrt{τ^{2}+σ^{2}}} . With F_{θ|x}^{−1}(α) = \frac{τ^{2}}{τ^{2}+σ^{2}} x + \frac{σ^{2}}{τ^{2}+σ^{2}} μ + \sqrt{\frac{τ^{2}σ^{2}}{τ^{2}+σ^{2}}} Φ^{−1}(α), an evaluation of (10) leads to the piecewise linear estimators

δ_{ω,π}(x) = \{ \begin{matrix}\frac{xτ^{2}+μσ^{2}}{τ^{2}+σ^{2}} + \sqrt{\frac{τ^{2}σ^{2}}{τ^{2}+σ^{2}}} Φ^{−1}(\frac{1−2ω}{2(1−ω)}) & if x ≤ μ + \frac{1}{d}Φ^{−1}(\frac{1−2ω}{2(1−ω)}) \\ x & if x ∈ (μ + \frac{1}{d}Φ^{−1}(\frac{1−2ω}{2(1−ω)}), μ + \frac{1}{d}Φ^{−1}(\frac{1}{2(1−ω)})) \\ \frac{xτ^{2}+μσ^{2}}{τ^{2}+σ^{2}} + \sqrt{\frac{τ^{2}σ^{2}}{τ^{2}+σ^{2}}} Φ^{−1}(\frac{1}{2(1−ω)}) & if x ≥ μ + \frac{1}{d}Φ^{−1}(\frac{1}{2(1−ω)}). \end{matrix}

3 Concluding Remarks

The findings here do permit us in many cases (unique Bayes, proper priors and finite Bayes risk) to draw inferences on the admissibility of resulting Bayesian estimators δ_{ω,π}, as given by Lemma 1 under balanced loss L_{ρ,ω,δ_{0}}. However, a general admissibility and dominance connection between losses L_{0} and L_{ρ,ω,δ_{0}}, such as the one known for squared error ρ, is lacking and worth further investigation. In this regard, we do wish to point out (following a suggestion by Alexandre Leblanc) that most of the general results in Jafari Jozani, Marchand and Parsian (2006) (i.e, Lemma 1, 2; Corollary 1,2; Theorem 1) hold under the following modification of squared error loss : ωq(δ_{0})(δ − δ_{0})^{2} + (1 − ω)q(δ_{0})(δ − θ)^{2}.

Acknowledgements

We thank Alexandre Leblanc for his suggestions as referred to in Section 3.

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