ROBUST BAYESIAN ESTIMATION IN A NORMAL MODEL WITH ASYMMETRIC LOSS FUNCTION

A. B O R A T Y ´N S K A and M. D R O Z D O W I C Z (Warszawa)

ROBUST BAYESIAN ESTIMATION IN A NORMAL MODEL WITH ASYMMETRIC LOSS FUNCTION

Abstract. The problem of robust Bayesian estimation in a normal model with asymmetric loss function (LINEX) is considered. Some uncertainty about the prior is assumed by introducing two classes of priors. The most robust and conditional Γ -minimax estimators are constructed. The situa- tions when those estimators coincide are presented.

1. Introduction and notation. In Bayesian statistical inference the goal of research are optimal decisions under a specified loss function and a prior distribution over the parameter space. However the arbitrariness of a unique prior distribution is a permanent problem. Robust Bayesian inference deals with the problem of expressing uncertainty of the prior information using a class Γ of priors and of measuring the range of a posterior quantity while the prior distribution Π runs over the class Γ . It is interesting not only in calculating the range but also in constructing optimal procedures.

In the problem of estimation of an unknown parameter two concepts of optimality are considered: the idea of conditional Γ -minimax estimators (see DasGupta and Studden [4], Betro and Ruggeri [1]) and the idea of stable estimators developed in M¸ eczarski and Zieli´ nski [6] and Boraty´ nska and M¸ eczarski [3]. The first concept is connected with the problem of ef- ficiency of the estimator with respect to the posterior risk when the priors run over Γ . The second one is connected with the problem of finding an estimator with the smallest oscillation of the posterior risk when the priors run over Γ . Sometimes those two estimators coincide (see M¸ eczarski [5] and Boraty´ nska [2]).

In all papers mentioned above the quadratic loss function was consid- ered. However in many situations a quadratic loss function seems inappro-

1991 Mathematics Subject Classification: Primary 62C10; Secondary 62F15, 62F35.

Key words and phrases: Bayes estimators, classes of priors, robust Bayesian estima- tion, asymmetric loss function.

[85]

priate in that it assigns the same loss to overestimates as to equal under- estimates.

In this paper we estimate an unknown parameter θ and consider the asymmetric loss function (LINEX)

L(θ, d) = exp(a(θ − d)) − a(θ − d) − 1,

where a is a known parameter and a 6= 0. Exhaustive motivations to use LINEX are presented in Varian [7] and Zellner [8]. We find the conditional Γ -minimax estimators and the stable estimators, and present conditions when those estimators coincide, in a normal model with two classes of con- jugate priors given below.

Let X

, . . . , X

be i.i.d. random variables with normal N (θ, b

) distri- bution where θ is unknown and b

is known. Set X = (X

, . . . , X

). Let Π

= N (µ

, σ

) be a fixed prior distribution of θ.

Define X = 1

n

n

X

i=1

X

, v

= n(X − µ

)

b

, λ = λ(σ) =  1 σ

+ n

b



,

m = m(µ) = µ

 1 − n

b

 1 σ

+ n

b



 , w

=  a

2 + nX b

 1 σ

+ n

b



.

If X = x then the posterior distribution is the normal distribution N (µ

+ v

λ

, λ

) = N (m

+ w

− aλ

/2, λ

),

where λ

= λ(σ

) and m

= m(µ

). The posterior risk of an estimator b θ with LINEX loss function is equal to

Ee

− aEθ + ab θ − 1,

where Ey(θ) denotes the expected value of a function y(θ) when θ has the posterior distribution. Thus under the prior Π

,

Ee

= exp(aµ

+ (a

/2 + av

)λ

) = exp(am + aw

) and

Eθ = µ

+ v

λ

= m

+ w

− aλ

/2.

The minimum of the posterior risk as a function of θ is reached for θ = b 1

a ln Ee

.

Thus the Bayes estimator with LINEX loss function is given by the formula θ b

= 1

a ln Ee

= µ

+ (a/2 + v

)λ

= m

+ w

.

Now suppose that the prior distribution is not exactly specified and consider two classes of prior distributions of θ:

Γ

= {Π

: Π

= N (µ

, σ

), σ ∈ (σ

, σ

)}, where σ

< σ

are fixed and σ

∈ (σ

, σ

), and

Γ

= {Π

: Π

= N (µ, σ

), µ ∈ (µ

, µ

)},

where µ

< µ

are fixed and µ

∈ (µ

, µ

). The classes Γ

and Γ

express two types of uncertainty about the elicited prior.

Let R

(µ, σ, b θ ) denote the posterior risk of the estimator b θ when the prior is normal N (µ, σ

). The posterior risk can be expressed by two formulas as a function of λ and m:

R(µ

, σ, b θ) = %

(λ, b θ)

= exp(−ab θ + aµ

+ (a

/2 + av

)λ) − a(µ

+ λv

) + ab θ − 1 and

R(µ, σ

, b θ) = %

(m, b θ)

= exp(−ab θ + am + aw

) − a(m + w

) + a

λ

/2 + ab θ − 1.

Observe that λ is an increasing function of σ and therefore if σ ∈ (σ

, σ

) then λ ∈ (λ

, λ

), where λ

= λ(σ

), i = 1, 2. Similarly, m is an increasing function of µ and therefore if µ ∈ (µ

, µ

) then m ∈ (m

, m

), where m

= m(µ

), i = 1, 2. The ranges of the posterior risk of the estimator b θ when the prior runs over Γ

and Γ

are

r

(b θ) = sup

λ∈(λ1,λ2)

%

(λ, b θ) − inf

λ∈(λ1,λ2)

%

(λ, b θ) and

r

(b θ) = sup

m∈(m1,m2)

%

(m, b θ) − inf

m∈(m1,m2)

%

(m, b θ), respectively.

2. The range of the posterior risk for the Bayes estimator.

Consider the prior Π

, note that Π

∈ Γ

and Π

∈ Γ

, and consider the Bayes estimator

θ b

= µ

+ (a/2 + v

)λ

= m

+ w

.

The posterior risk of this estimator under an arbitrary prior Π

∈ Γ

is

%

(λ, b θ

) = exp((a

/2 + av

)(λ − λ

)) − av

(λ − λ

) + a

λ

/2 − 1.

Denote it by f (λ). Now computations lead to the following form of the

oscillation of %

for b θ

while λ runs over (λ

, λ

):

r

(b θ

) =







f (λ

) − f (λ

) if −a/2 ≤ v

< 0 and a > 0, or 0 < v

≤ −a/2 and a < 0, or b λ < λ

, f (λ

) − f (b λ) otherwise,

where

b λ = λ

+ (a

/2 + av

)

ln v

a/2 + v

. Thus

r

(b θ

)

=



 



 



e

[e

− 1] − av

δ if −a/2 < v

< 0 and a > 0, or 0 < v

≤ −a/2 and a < 0, or b λ < λ

,

a

δ/2 if v

= −a/2,

e

+ av

(b λ − λ

− 1/z) otherwise, where z = a

/2 + av

and δ = λ

− λ

.

Consider the class Γ

. The posterior risk of this estimator under an arbitrary prior Π

∈ Γ

is

%

(m, b θ

) = e

+ a(m

− m) + a

λ

/2 − 1 and the oscillation of %

is equal to

r

(b θ

) =

 e

+ a(m

− m

) − 1 for m

≤ m, b e

+ a(m

− m

) − 1 for m

> m, b where

m = m b

+ 1

a ln exp(am

− am

) − 1 a(m

− m

) .

3. Most stable and conditional Γ -minimax estimators. Now the problem is to find most stable estimators b θ

and b θ

, i.e. those satisfying

inf

θb

r

(b θ ) = r

(b θ

) and inf

θb

r

(b θ) = r

(b θ

)

and to find the conditional Γ -minimax estimators e θ

and e θ

, i.e. those satisfying

inf

θb

sup

σ∈[σ1,σ2]

R

(µ

, σ, b θ ) = sup

σ∈[σ1,σ2]

R

(µ

, σ, e θ

) and

inf

θb

sup

µ∈[µ1,µ2]

R

(µ, σ

, b θ ) = sup

µ∈[µ1,µ2]

R

(µ, σ

, e θ

).

We use the following theorem proved by M¸ eczarski [5].

Theorem 1 (M¸ eczarski [5]). Let Γ = {Π

: α ∈ [α

, α

]} be a set of prior distributions, where α is a real parameter. Let %(α, d) be the posterior risk of a decision d based on an observation x when the prior is Π

. Assume that the function %(α, d) satisfies the following conditions:

1. %(α, ·) is a strictly convex function for any α;

2. for any d the minimum point α

(d) of %(·, d) is unique and α

is a strictly monotone function of d;

3. for any α and d such that α

(d ) = α we have

∀d

< d

≤ d %(α, d

) − %(α, d

)

d

− d

< %(α

(d

), d

) − %(α

(d

), d

) d

− d

and

∀d

> d

≥ d %(α, d

) − %(α, d

)

d

− d

> %(α

(d

), d

) − %(α

(d

), d

)

d

− d

;

4. the function %(α

, d) − %(α

, d) is a monotone function of d.

Then

(i) if there exists b d such that sup

α∈[α1,α2]

%(α, b d ) = %(α

, b d ) = %(α

, b d ) then b d is the most stable;

(ii) if b d satisfying (i) belongs to L

= {d : ∀x ∈ X ∃α ∈ [α

, α

] d(x) = d

(x)} then b d is conditional Γ -minimax.

We now prove our results.

Theorem 2. If the class of priors is Γ

then θ b

= b θ

+ 1

a ln exp[a(m

− m

)] − 1 a(m

− m

) and e θ

= b θ

for all values x of the random variable X.

P r o o f. Let us check the conditions of Theorem 1 for

%

(m, b θ) = exp(−ab θ + am + aw

) − a(m + w

) + a

λ

/2 + ab θ − 1.

The function %

(m, ·) is convex and

∂%

(m, b θ)

∂m = a exp(−ab θ + am + aw

) − a,

thus the minimum point m

(b θ) = b θ − w

, and m

is an increasing

function of b θ.

To check condition 3 it is enough to show the inequalities

∀θ

< θ

≤ b θ e

e

− e

θ

− θ

< −a and

∀θ

> θ

≥ b θ e

e

− e

θ

− θ

> −a.

These hold by the Lagrange formula. The last condition of Theorem 1 is also true, thus b θ

is a solution of the equation

%

(m

, b θ) = %

(m

, b θ).

To obtain the conditional Γ -minimax estimator note that for all values x of the random variable X we have b θ

(x) ∈ [b θ

(x), b θ

(x)].

Theorem 3. Let the class of priors be Γ

. Then the most stable esti- mator b θ

of θ in the class of all estimators of θ exists only for the values of X satisfying

v

(v

+ a/2) > 0 or v

= −a/2.

For v

(v

+ a/2) > 0,

b θ

= b θ

+ 1

a ln e

− 1 av

(λ

− λ

) .

For v

= −a/2 the range of the posterior risk does not depend on the value of b θ.

The conditional Γ -minimax estimator is

θ e

=







θ b

if v

(v

+ a/2) > 0 and

exp[(λ

− λ

)(a

/2 + av

)] + av

(λ

− λ

)) ≥ 1, θ b

otherwise.

The most stable estimator in the class

L = {b θ : ∀x ∃σ ∈ [σ

, σ

] b θ(x) = b θ

(x)}

is equal to the conditional Γ -minimax estimator in the class of all estima- tors.

P r o o f. Let us check the conditions of Theorem 1 for

%

(λ, b θ) = exp(−ab θ + aµ

+ (a

/2 + av

)λ) − a(µ

+ λv

) + ab θ − 1.

The function %

(λ, ·) is convex and

∂%

(λ, b θ)

∂λ = (a

/2 + av

) exp(−ab θ + aµ

+ λ(a

/2 + av

)) − av

.

Thus the minimum point is

λ

(b θ) = ab θ − aµ

+ ln

n

a

/2 + av

and λ

exists iff v

(v

+ a/2) > 0.

For v

satisfying v

(v

+ a/2) ≤ 0 the function %

(·, b θ) is an increasing function of λ and the oscillation of the posterior risk

r

(b θ) = − av

(λ

− λ

) + exp(−ab θ + aµ

+ (a

/2 + av

)λ

)

× [exp((a

/2 + av

)(λ

− λ

)) − 1]

is a monotone function of b θ (decreasing for a > 0 and −a/2 < v

≤ 0, constant for v

= −a/2 and increasing for a < 0 and 0 ≤ v

< −a/2).

Thus the most stable estimator does not exist for v

(v

+ a/2) ≤ 0 and v

6= −a/2. For v

= −a/2 the oscillation r

(b θ) = a

(λ

− λ

)/2 does not depend on the value of b θ. The conditional Γ -minimax estimator e θ

is equal to b θ

.

Let us consider the situation when v

(v

+ a/2) > 0. The minimum point λ

and the function %

(λ

, ·) − %

(λ

, ·) are monotone functions of θ. Condition 3 of Theorem 1 is similar to that in Theorem 2 so we obtain b the most stable estimator as a solution of the equation

%

(λ

, b θ

) = %

(λ

, b θ

).

To find the conditional Γ -minimax estimator we check when b θ

∈ L.

For v

+ a/2 > 0 we have b θ

< b θ

. Solving the inequalities θ b

≤ b θ

≤ b θ

we obtain the condition

(∗) exp[(λ

− λ

)(a

/2 + av

)] + av

(λ

− λ

) ≥ 1.

For v

+ a/2 < 0 we have b θ

> b θ

. Solving the inequalities θ b

≥ b θ

≥ b θ

we also obtain (∗). Thus if v

(v

+ a/2) > 0 and (∗) is true then e θ

= b θ

. If v

+ a/2 > 0 and v

> 0 and (∗) is not true then

θ b

< b θ

< b θ

and

sup

λ∈[λ1,λ2]

%

(λ, b θ) =  %

(λ

, b θ) if b θ ≤ b θ

,

%

(λ

, b θ) if b θ ≥ b θ

,

and the oscillation r

(b θ) is a decreasing function for b θ < b θ

.

If v

+ a/2 < 0 and v

< 0 and (∗) is not true then θ b

> b θ

> b θ

and

sup

λ∈[λ1,λ2]

%

(λ, b θ) =  %

(λ

, b θ) if b θ ≤ b θ

,

%

(λ

, b θ) if b θ ≥ b θ

, and the oscillation r

(b θ) is an increasing function for b θ > b θ

.

Thus if v

(v

+ a/2) > 0 and (∗) is not true then e θ

= b θ

and b θ

is the most stable estimator in the class L.

The monotonicity of the function r

shows that b θ

is also the most stable estimator in the class L for v

(v

+ a/2) ≤ 0.

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Agata Boraty´nska

Institute of Applied Mathematics University of Warsaw

Banacha 2

02-097 Warszawa, Poland E-mail: agatab@mimuw.edu.pl

Monika Drozdowicz Wojciechowskiego 22 02-495 Warszawa, Poland

Received on 2.9.1998;

revised version on 3.12.1998