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A class of weighted normal distributions and its variants
useful for inequality constrained analysis
Hea-Jung Kim a
aDepartment of Statistics, Dongguk University, Seoul, South Korea
Online Publication Date: 01 October 2007
To cite this Article: Kim, Hea-Jung (2007) 'A class of weighted normal distributions
and its variants useful for inequality constrained analysis', Statistics, 41:5, 421 - 441
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Statistics, Vol. 41, No. 5, October 2007, 421–441
A class of weighted normal distributions and its variants
useful for inequality constrained analysis
HEA-JUNG KIM*
Department of Statistics, Dongguk University, Seoul 100-715, South Korea
(Received 21 June 2006; in final form 5 January 2007)
This article develops a class of the weighted normal distributions for which the probability density
function has the form of a product of a normal density and a weight function. The class constitutes
marginal distributions obtained from various kinds of doubly truncated bivariate normal distributions.
This class of distributions strictly includes the normal, skew–normal and two-piece skew–normal and
is useful for selection modelling and inequality constrained normal mean analysis. Some distribu-
tional properties and Bayesian perspectives of the class are given. Probabilistic representation of the
distributions is also given. The representation is shown to be straightforward to specify distribution
and to implement computation, with output readily adapted for required analysis. Necessary theories
and illustrative examples are provided.
Keywords: Weighted normal distribution; Doubly truncated bivariate normal; Selection model;
Constrained normal means; Probabilistic representation
AMS 2000 Subject Classification: 62E10; 62F30.
1. Introduction
There is growing interest in the literature on family of marginal distributions of a truncated
bivariate normal distribution that occupies a central place in the development of applica-
tions. The marginal model can be taken as a member of the family of weighted distributions
(see [1] and references therein). A weighted distribution arises when the density g(x;θ1)of
a potential observation xgets distorted so that it is multiplied by some non-negative weight
function w(x;θ1,θ2)involving parameter vectors θ1and θ2,an additional parameter vector.
The density of the distribution is given by
f(x;θ1,θ2)=w(x;θ1,θ2)g(x;θ1)
Eθ1[w(X;θ1,θ2)],(1)
where the expectation in the denominator is just a normalizing constant.
In the literature, several weighted distributions have been considered, among others, by
Bayarri and DeGroot [2], Arnold and Beaver [3], Branco and Dey [4], Ma et al. [5] and
*Email: kim3hj@dongguk.edu
Statistics
ISSN 0233-1888 print/ISSN 1029-4910 online © 2007 Taylor & Francis
http://www.tandf.co.uk/journals
DOI: 10.1080/02331880701442726
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422 H.-J. Kim
Kim [6]. Moreover, as elaborated in the articles by Arnold et al. [7], Ma and Genton [8] as
well as in the book edited by Genton [9], the application of weighted distributions extends to
the areas of econometric, astronomy, engineering, medicine as well as psychology in scenarios
where the observed random phenomena can be described by equation (1), i.e., a probability
density function distorted by some multiplicative non-negative weight function.
Although cases of weighted distributions are well addressed in the literature, so far as
we know, there are few results concerning the class of weighted models associated with a
doubly truncated bivariate normal (see [7]). This motivates the investigation in this article. The
interest in the class comes from both theoretical and applied direction. On the theoretical side,
it enjoys a number of formal properties, which resemble those of skew–normal distribution
by Azzalini [10] and the two-piece skew–normal distribution by Kim [6], and plays a role
of intermediate distributions that fill in the gap between the normal and the skew–normal
(or between the normal and two-piece skew–normal) distribution. These are obtained from a
probabilistic representation of the distribution considered by Arnold et al. [7] and its variants.
In addition, the class provides yet other conjugate prior distributions, especially useful for
Bayesian subjective methodology for inequality constrained normal means problems. In the
applied view point, the class provides new models that enable us to analyse variously screened
data sets.
The material in this article is arranged as follows. Section 2 provides some lemmas useful for
studying the class of the weighted normal distributions. Section 3 sets the scene and defines a
class of the weighted normal distributions whose underlying distribution is a doubly truncated
bivariate normal. This is followed by discussing the properties of the distributions based on
their probabilistic representations, which indicate that there exists a sum of a normal and a
doubly truncated normal variate such that it follows a weighted normal distribution useful for
modelling screened and/or skewed data sets. Section 4 proposes the other members of the
class useful for selection modelling and studies their properties. Section 5 contains illustrative
examples, which has been selected for the purpose of validating and motivating the contents
of this article, and section 6 provides discussions and conclusions.
2. Preliminaries
Prior to suggesting a class of weighted normal distributions, we provide lemmas useful for
studying properties of the distributions. For notational convenience, we use U(a,b) to indicate
the distribution of a random variable Uthat is doubly truncated with respective lower and
upper truncation points aand b.
LEMMA 2.1 Let Z∼N(0,1), then for any real cand d,
E[(cZ(a,b) +d)]=L(−d/√1+c2,a,ρ)−L(−d/√1+c, b, ρ)
(a) −(b) ,(2)
where ρ=c/√1+c2,is the standard normal cumulative distribution function (cdf), and
L(α, β, ρ ) =P(V
1>α,V
2>β) is the orthant probability of a bivariate standard normal
variable (V1,V
2).
Proof Let V∼N(0,1)independently of Z. Then
E[(cZ(a,b) +d)]=P(V −cZ(a,b) ≤d) =PV1>−d
√1+c2|a<V
2<b
,
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A class of weighted normal distributions 423
where (V1,V
2)is the standard bivariate normal variable with correlation ρ. Expressing the
conditional probability in terms of the orthant probabilities gives the result.
Computing methods that evaluate L(α, β, ρ) have been given by Donnelly [11] and Joe [12].
Lemma 2.1 immediately gives the following results for Z∼N(0,1)random variable – upon
setting a=−∞and b=∞,we have, for any real cand d,
E[(cZ +d)]=d
√1+c2.(3)
The expectation was derived by a different method (see, for instance, [13], pp. 53–54). When
a=0 and b=∞, equation (2) yields
E[(c|Z|+d)]=d
√1+c2+2Td
√1+c2,c
,(4)
where T (α, β ), α > 0,β > 0,is the function which gives the integral of the standard bivari-
ate density over the right side region bounded by lines x=α, y =0,and y=βx in
the (x, y ) plane (see, [10], for the properties of the function). The cdfof the standard
Cauchy random variable W=V/|Z|is directly obtained from (4); FW(w) =E[(w|Z|)]=
(0)+2T(0,w) =1/2+tan−1w/π.
LEMMA 2.2 Let φ(x) =(2π)−1/2exp(−x2/2)and (x) =x
−∞ φ(t)dt.Then,for any a∈R
and λ∈R,
x2
x1
φ(t)(λt −δa)dt=L(x1,a,ρ)−L(x2,a,ρ),
x2
x1
φ(t)(δa −λt)dt=(x2)−(x1)−L(x1,a,ρ)+L(x2,a,ρ),
where L(α, β, ρ) function is the orthant probability of the standard bivariate normal variable,
λ=ρ/1−ρ2and δ=λ/ρ.
Proof Let (V1,V
2)be a bivariate standard normal variablewith correlation ρ, and let Z1=V1
and Z2=δV2−λV1.Then Z1and Z2are independent standard normal variables. So that the
conditional distribution of V1given V2<ais equivalent to that of Z1given Z2<δa−λZ1.
Thus, for i=1,2,P(V
1>x
i|V2>a)=L(xi,a,ρ)/{1−(a)}=A(xi), where
A(xi)=P(Z
1>x
i|Z2>δx
i−λZ1)
=∞
xi
φ(z1)(δa −λz1)
1−E[(δa −λZ)]dz1.
Since E[(δa −λZ)]=(a) by equation (3), computing A(x1)−A(x2), we obtain the first
result. The second equation is immediate by using (λt −δa) =1−(δa −λt).
LEMMA 2.3 When Z∼N(0,1), for any a∈R,λ∈R,and t∈R,
E[(δa −λ|Z+t|)]=1−L(t, a +ρt, ρ) −L(−t,a −ρt, ρ),
where L(α, β, ρ) function is the orthant probability of the standard bivariate normal variable,
λ=ρ/1−ρ2,and δ=λ/ρ.
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424 H.-J. Kim
Proof E[(δa −λ|Z+t|)]
=∞
−t
φ(z1)(δa −λ(z1+t))dz1+∞
t
φ(z1)(δa −λ(z1−t))dz1.
The result follows by using Lemma 2.2.
3. The class of weighted normal distributions
According to equation (1), one may define a class of weighted normal distributions – for
X∼N(μ,σ2), if the pdf Xis distorted by a multiplicative non-negative weight function
w(x;θ)such that
fX(x) =σ−1φx−μ
σw(x;θ)
E[w(X;θ)],x∈R,(5)
we say that the distribution is a weighted normal distribution, where φis the standard normal
density function (pdf), and θ({μ,σ}⊂θ) denotes a parameter vector involved in the weight
function.
3.1 Derivation form doubly truncated bivariate normal
A weighted normal distribution of form (5) is obtained from the conditional distribution of a
doubly truncated normal distribution – let (X1,X
2)be a bivariate normal variable with mean
vector (μ1,μ
2), variance vector (σ 2
1,σ2
2)and correlation ρ. If we define the random variable
Xas X=[X1|a<X
2<b],then a direct integration yields the density of Xgiven by
fX(x) =σ−1
1φ(z)(δ u(b) −λz) −(δu(a) −λz)
(u(b)) −(u(a)) ,x∈R(6)
for real constants aand b, where z=(x −μ1)/σ1, u(a) =(a −μ2)/σ2,u(b) =(b −μ2)/σ2,
δ=1/1−ρ2and λ=ρ/1−ρ2.
The distribution function of Xis immediately obtained by using L(α, β, ρ) function, the
orthant probability of the standard bivariate normal variable
FX(x) =1−L(z, u(a ), ρ) −L(z, u(b), ρ)
(u(b)) −(u(a)) ,x∈R.(7)
Since E[(δk −λZ)]=(k) for any real constant kby equation (3), the marginal distri-
bution (6) of a doubly truncated bivariate normal distribution reduces to the weighted normal
distribution (5) with the weight function w(x;θ)=(δu(b) −λz) −(δu(a) −λz), where
θ=(μ1,μ
2,σ
1,σ
2,ρ)
.This indicates that the class weighted normal distributions (5) is asso-
ciated with a bivariate doubly truncated normal distribution, if its weight function has the form
w(x;θ)=(δu(b) −λz) −(δu(a) −λz).
DEFINITION 3.1 If the pdf of Xis equation (6), the random variable Xis said to have the
weighted normal distribution with a parameter vector θ=(μ1,μ
2,σ
1,σ
2,ρ)
and truncation
points (a, b). This is abbreviated by saying Xis WN(a,b) (θ).
It is easily seen that the class of WN(a,b)(θ)distributions strictly includes the normal den-
sity (for b=∞,a =−∞), the skew–normal density by Azzalini [10] (for b=∞,a =μ2).
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A class of weighted normal distributions 425
In fact, the distribution Xis equivalent to that considered by Arnold et al. [7]. Nevertheless,
it is the purpose of this section to give some new properties of the distribution.
3.2 Probabilistic representation and properties
In the following, we represent the WN(a,b) (θ)distribution in terms of independent normal and
doubly truncated normal distributions, and provide some properties for the distribution.
THEOREM 3.1 Suppose U(1)∼N(θ1,τ2
1)and U(2)∼N(θ2,τ2
2)are independent variables.
Then, for any real values c1(c1= 0)and c2satisfying ρ=c2σ2/σ1,
c1U(1)+c2U(2)
(a,b) ∼WN(a,b)(θ), (8)
provided that θ=(μ1,μ
2,σ
1,σ
2,ρ) with μ1=2
i=1ciθi,μ
2=θ2,σ
2
1=2
i=1c2
iτ2
iand
σ2
2=τ2
2.
Proof Note that the conditional distribution of X1given a<X
2<b is equivalent to
the WN(a,b) (θ)distribution, where (X1,X
2)is a bivariate normal variable with mean vec-
tor (μ1,μ
2), variance vector (σ 2
1,σ2
2)and correlation ρ. Let X1=c1U(1)+c2U(2)and
X2=U(2).Then (X1,X
2)is the bivariate normal variable with mean vector (μ1,μ
2), vari-
ance vector (σ 2
1,σ2
2)and correlation ρ=c2σ2/σ1.Since U(1)and U(2)are independent and
the conditional distribution of U(2)given that a<U
(2)<bequals that of U(2)
(a,b) figuring in
the statement of the theorem, we are done.
This representation indicates the kind of departure from the normal law and reveals intrin-
sic structure of WN(a,b)(θ)distributions. Furthermore, equation (8) enables us to implement
a one-for-one method for generating a random variable Xwith the WN(a,b) (θ)distribu-
tion. For generating the doubly truncated normal variable U(2)
(a,b) , the one-for-one method
by Devroye [14] may be used.
Theorem 3.1 provides probabilistic proofs for the following properties of the class of
WN(a,b) (θ)distributions.
PROPERTY 3.1 WN(a,b)(θ)≡WN(u(a),u(b)) (μ1,0,σ
1,1,ρ), where u(a) =(a −μ2)/σ2and
u(b) =(b −μ2)/σ2.
PROPERTY 3.2 Let X∼WN(a,b)(θ). Then,for α∈Rand β∈R,
αX +β∼WN(a,b) (θα,β ), (9)
where θα,β =(αμ1+β, μ2,|α|σ1,σ
2,sign{α}ρ),and sign{α}denotes the sign of α. So that
−X∼WN(a,b) (−μ1,μ
2,σ
1,σ
2,−ρ).
PROPERTY3.3 Suppose X∼WN(a ,b)(θ)and Z=(X −μ1)/σ1.Then Z∼WN(u(a ),u(b)) (θ0),
where θ0=(0,0,1,1,ρ)
.
PROPERTY 3.4 Let X∼WN(a,b)(θ)and X3∼N(μ3,σ2
3)be independent. Then
X+X3∼WN(a,b) (θ+), (10)
where
θ+=μ1+μ3,μ
2,σ2
1+σ2
3,σ
2,σ
1ρσ2
1+σ2
3.
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426 H.-J. Kim
PROPERTY 3.5 Let Z1∼N(0,1)and Z2∼N(0,1)be independent. Then
1
√1+λ2Z1+λ
√1+λ2Z2(a,b) ∼WN(a,b)(θ0), (11)
where λ=ρ/1−ρ2.
Property 3.5gives the probabilistic representation of the distribution V1conditionally on
a<V
2<b,where (V1,V
2)is a bivariate standard normal variable with correlation ρ.This
yields the following property.
PROPERTY 3.6 Let Z1∼N(0,1)and Z2∼N(0,1)be independent. Then the distribution of
Z1conditionally on λZ1−b/δ < Z2<λZ
1−a/δ is WN(a,b)(θ0).
PROPERTY 3.7 Suppose X∼WN(a,b)(θ0)and Z3∼N(0,1)are independent,and suppose
Y=(X +Z3)/√2.Then
Y=1
√1+λ∗2Z1+λ∗
√1+λ∗2Z2(a,b) ∼WN(a,b)(θ∗
0), (12)
where θ∗
0=(0,0,1,1,ρ∗),Z
1∼N(0,1)and Z2∼N(0,1)are independent,and λ∗=
ρ∗/1−ρ∗2with ρ∗=ρ/√2.
PROPERTY 3.8 The WN(0,∞)(θ0)distribution is equivalent to SN(λ) distribution,the skew–
normal distribution introduced by Azzalini [10].
PROPERTY 3.9 The WN(μ2,∞)(θ)distribution is equivalent to the distribution of X=μ1+
σ1Y, where Y∼SN(λ).
PROPERTY 3.10 If ρ=0,then WN(a,b)(θ)≡N(μ1,σ2
1). If ρ=1,then WN(a,b) (θ)≡
σ1N(u(a),u(b)) (0,1)+μ1,where N(u(a),u(b))(0,1)denotes a doubly truncated standard normal
with respective upper and lower truncation points u(a) and u(b).
In addition to the above properties, the following corollary is useful for practical applications
of the weighted normal distribution.
COROLLARY 3.1 Let X∼Np(μ,), ap-variate normal distribution,where X=
(X1,...,X
p),μ=(μ1,...,μ
p),and ={σij }.Then the distribution of Xkcondition-
ally on a<ηX<bfollows WN(a,b) (μk,η
μ, σkk,ηη,ρ
k)distribution for k=1,...,p,
where η=(η1,η
2,...,η
p)and
ρk=p
j=1ηjσkj
√σkk(ηη).
So that it is equivalent to the distribution of
Y=c1U(1)+c2U(2)
(a,b) ,(13)
where U(1)∼N(θ1,τ2
1)and U(2)∼N(θ2,τ2
2)are independent variables with μk=
2
i=1ciθi,θ
2=ημ,σ
kk =2
i=1c2
iτ2
i,τ
2
2=ηηand ρk=c2τ2/√σkk.
Proof Xk∼N(μk,σ
kk)and ηX∼N(ημ,ηη), and their correlation is ρk.Applying
these distributions to Theorem 3.1 gives the result.
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A class of weighted normal distributions 427
3.3 Moments
The moment generating function (mgf) and the moments of the WN
(a,b) (θ)distribution have
been given by Arnold et al. [7]. When X∼WN
(a,b) (θ), the mgf of Xis
MX(t) =eμ1t+σ2
1t2/2(u(b) −ρσ1t) −(u(a) −ρσ1t)
(u(b)) −(u(a)) for t∈R.(14)
and the mean and the variance are
E[X]=μ1−ρσ1
φ(u(b)) −φ (u(a))
(u(b)) − (u(a)) ,
V(X) =σ2
11−ρ2u(b)φ(u(b)) −u(a)φ(u(a))
(u(b)) −(u(a)) −ρ2φ (u(b)) −φ (u(a ))
(u(b)) − (u(a)) 2,
respectively. Higher moments could be obtained from the general formula by Arnold et al. [17].
We recall the functions α3(k1,k
2)and α4(k1,k
2)studied by Sugiura and Gombi [15]. For
real values of k1and k2(k1<k
2), α3(k1,k
2)and α4(k1,k
2)give respective skewness and
kurtosis of a doubly truncated standard normal distribution. Here, k1and k2denote the left and
right truncation point of the distribution. Using these functions, we have the following result.
THEOREM 3.2 For X∼WN
(a,b) (θ), θ=(μ1,μ
2,σ
1,σ
2,ρ)
,the skewness (α3)and kurtosis
(α4)of the distribution are
α3=ρ3β1
β23/2
α3(u(a), u(b)),
α4=3(1−ρ2)2+ρ4β2
1α4(u(a), u(b)) +6ρ2(1−ρ2)β1
β2
2,
where
β1=1+u(b)φ(u(b)) −u(a)φ(u(a))
(u(b)) −(u(a)) −φ (u(b)) −φ (u(a ))
(u(b)) − (u(a)) 2,
β2=1−ρ2u(b)φ(u(b)) −u(a)φ(u(a))
(u(b)) −(u(a)) +φ (u(b)) −φ (u(a ))
(u(b)) − (u(a)) 2.
Proof Since α3and α4for Xand Z=(X −μ1)/σ1distributions are the same, it is enough
to obtain α3and α4for the distribution of Zrandom variables. From the Property 3.3
and Property 3.5, we see that Z=1−ρ2Z1+ρZ2(u(a),u(b)),where Z1and Z2are inde-
pendent standard normal variables. The variance of Xyields V(Z) =β2.Therefore, some
algebra using the fact that E[Z2(u(a),u(b)) ]={φ(u(b)) −φ (u(a))}/{(u(b)) −(u(a ))}and
V(Z
2(u(a),u(b)) )=β1gives the result. See, for example, [16, pp. 156–158] for the moments of
the doubly truncated standard normal variable Z2(u(a),u(b)).
COROLLARY 3.2 Fo r X∼WN(a,b )(μ1,μ
2,σ
1,σ
2,ρ) with a<b,the skewness of the distri-
bution is as follows: (i) If |u(a)|< u(b), the distribution of Xvariable is skewed to the right
(left)when ρ>0(ρ < 0).(ii) If |u(a)|> u(b), the distribution of Xvariable is skewed
to the right (left)when ρ<0(ρ > 0), respectively. (iii)If |u(a)|=u(b) for u(a ) = 0,the
distribution of Xvariable is symmetric,where u(a) =(a −μ2)/σ2and u(b) =(b −μ2)/σ2.
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428 H.-J. Kim
Figure 1. Shapes of WN(a,b) (θ0)density with θ0=(0,0,1,1,0.5):(1)N(0,1)density; (2) WN(−0.5,3.5)(θ0)
density; (3) SN(λ) density with λ=0.577;(4) WN(3,4.5)(θ0)density.
Proof It is obvious that α3(u(a), u(b)) > 0 for |u(a)|< u(b), α3(u(a), u(b)) < 0 for
|u(a)|> u(b), and α3(u(a), u(b)) =0 for |u(a)|=u(b), for u(a) = 0.This fact and the
formula of α3in Theorem 3.3 immediately give the result.
It is noted that α3=0 for ρ=0,because WN
(a,b) (μ1,μ
2,σ
1,σ
2,0)≡N(μ1,σ2
1).
Also note that α3=(4/π −1)√2/πρ3(1−2ρ2/π)−3/2for a=μ1and b=∞,because
α3(0,∞)=√2(4−π)/(π −2)3/2(see, for instance, [15]). The latter case defines the skew-
ness of the skew–normal distribution by Property 3.9 and the value agrees with that given
in [7]. Figure 1 depicts the shapes of equation (6) (the pdf of WN
(a,b) (θ)distribution).
4. Variants of the distribution
We postulate that our distributions should be weighted versions of normal densities based on
a truncated bivariate normal density other than equation (6).
4.1 A symmetric weighted normal distribution
Let (X1,X
2)be a bivariate normal variablewith mean vector (μ1,μ
2), variance vector (σ 2
1,σ2
2)
and correlation ρ. When the conditional distribution of X1given a<sign{X1}X2<b is
considered, we see that it is equivalent to that of X1given a<X
2<bfor the region x1>0
and that of X1given −b<X
2<−afor x1<0.Applying this fact to the distribution (6), the
conditional density in xof X1given a<sign{X1}X2<bis given by
gX(x) =σ−1
1φ(z) (δu(b) −λ|z|)−(δu(a) −λ|z|)
(u(b)) −(u(a)) +2T (u(a ), λ) −2T (u(b ), λ) ,x∈R,(15)
where z=(x −μ1)/σ1and the normalizing constant E[(δu(b) −λ|Z|)]−E[(δu(a ) −
λ|Z|)]=(u(b)) −(u(a)) +2T (u(a ), λ) −2T (u(b), λ) is obtained by using equation (4).
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A class of weighted normal distributions 429
Thus, the pdf of Xis that of a weighted normal distribution (5) with the weight function
w(z;θ)=(δu(b) −λ|z|)−(δu(a) −λ|z|). This gives the following definition.
DEFINITION 4.1 If the pdf of Xis equation (15), the random variable Xis said to have a sym-
metric weighted normal distribution with parameters θ=(μ1,μ
2,σ
1,σ
2,ρ)
and truncation
points (a, b). This is abbreviated by saying Xis SWN(a,b) (θ).
The SWN(a,b) (θ)distribution with the pdf (15) has the following properties.
PROPERTY 4.1 Let X∼SWN(a,b)(θ)and Z=(X −μ1)/σ1,then
Z∼SWN(u(a),u(b)) (θ0), where θ0=(0,0,1,1,ρ).
PROPERTY 4.2 If Zis a SWN(u(a),u(b))(θ0)random variable,−Zis a SWN(u(a),u(b ))(θ0)
random variable,and hence the SWN(a,b) (θ)is symmetric about μ1.
PROPERTY 4.3 If (V1,V
2)is a bivariate standard normal variable with correlation ρ, then
the conditional distribution of V1given V1V2>0is SWN(0,∞)(θ0)≡TN(λ), where TN(λ)
is the two-piece skew–normal distribution introduced by Kim [6].
PROPERTY 4.4 The SWN(−∞,∞)(θ0)distribution is N(0,1)distribution.
Because of Property 4.1,it is sufficient to deal only with the distribution function of Z∼
SWN(u(a),u(b)) (θ0). Denote by GZ(z) the distribution function of Z, i.e.,
GZ(z) =cz
−∞
φ(t){(δu(b) −λ|t|)−(δu(a) −λ|t|)}dt, (16)
where c−1=(u(b)) −(u(a)) +2T (u(a), λ) −2T (u(b), λ). For positive values of z,
G(z) =1
2+cz
0
φ(t){(δu(b) −λt) −(δu(a) −λt )}dt
=1
2+c{L(z, u(b), ρ) +L(0, u(a ), ρ) −L(z, u(a ), ρ) −L(0, u(b), ρ )}
by direct application of Lemma 2.2. Since the pdf of the distribution is symmetric, i.e.,
GZ(−z) =1−GZ(z), the general expression of the distribution function is, therefore,
GZ(z) =⎧
⎪
⎨
⎪
⎩
1
2+cL1if z≥0,
1
2−cL2if z<0,
where L1=L(z, u(b), ρ) +L(0, u(a), ρ) −L(z, u(a ), ρ) −L(0, u(b), ρ) and L2=L(−z,
u(b), ρ) +L(0, u(a ), ρ) −L(−z, u(a ), ρ) −L(0, u(b), ρ ).
Property 3.5 and the argument for deriving the density (15)give the following theorem.
THEOREM 4.1 Let Z∼SWN(u(a),u(b)) (θ0), θ0=(0,0,1,1,ρ)
,then the distribution is
equivalent to
Z=⎧
⎪
⎪
⎨
⎪
⎪
⎩
Tw.p. 1
2,
−Tw.p. 1
2,
(17)
where U=Z1/√1+λ2+λZ2(u(a),u(b))/√1+λ2,T =U(0,∞),λ =ρ/1−ρ2and Z1and
Z2are independent standard normal variables.
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430 H.-J. Kim
Proof Suppose (V1,V
2)is a bivariate standard normal variable with correlation ρ. Then
the distribution of Zis the same as that of V1conditionally on u(a) < sign{V1}V2< u(b).
For the support z≥0ofZ, the conditional distribution reduces to V1given u(a) < V2< u(b),
whereas for z<0,the distribution reduces to V1given −u(b) < V2<−u(a). Thus, for z≥0,
the distribution of Zis equivalent to U(0,∞)by (11).We also see that the conditional distribution
V1given −u(b) < V2<−u(a) is equivalent to Z1/√1+λ2+λZ2(−u(b),−u(a))/√1+λ2≡
−Z1/√1+λ2−λZ2(u(a),u(b)) /√1+λ2for Z2(−u(b),−u(a)) =−Z2(u(a ),u(b)).Thus, for z<0,
the distribution of Zis the same as that of −U(0,∞).Since P(Z ≥0)=P(Z < 0)=1/2by
equation (16), we are done.
This representation (17) of SWN(u(a),u(b)) (θ0)distribution is useful for simulating obser-
vations with density (15). For generating the doubly truncated standard normal variable
Z2(u(a),u(b)) the one-for-one method by Devroye [14] may be used, while the rejection method
(discarding ineligible observations sampled from the Udistribution) may be used to draw an
observation from the U(0,∞)distribution. Once we have drawn observations of Zvariable,
then generating a random variable with distribution SWN(a,b)(θ)can be followed by using
Property 4.1.
COROLLARY 4.1 Let Z1∼N(0,1)and Z2∼N(0,1)be independent. Then the distribution
of Z1conditionally on λ|Z1|−δu(b) < Z2<λ|Z1|−δu(a) is SWN(u(a),u(b)) (θ0).
Proof Let (V1,V
2)be a bivariate standard normal variable with correlation ρ.Then one can
express Z1and Z2in terms of V1and V2so that Z1=V1and Z2=−δV2+λV1.Using the dis-
tributions, we see that, for Z1>0,the conditional distribution Z1|(Z1>0,λZ
1−δu(b) <
Z2<λZ
1−δu(a )) is equivalent to that of V1|(V1>0, u(a) < V2< u(b)). Alternatively, one
can express Z1and Z2in terms of V1and V2so that Z1=V1and Z2=δV2−λV1.Thus,
for Z1<0,the conditional distribution Z1|(Z1<0,−λZ1−δu(b) < Z2<−λZ1−δu(a))
is equivalent to V1|(V1<0,−u(b) < V2<−u(a)). Combining these two conditional distri-
butions, we see that the distribution of Z1conditionally on λ|Z1|−u(b)/δ < Z2<λ|Z1|−
u(a)/δ is equivalent to that of V1,givenu(a ) < sign{V1}V2< u(b).
PROPERTY 4.5 If Z1∼N(0,1)and Z2∼N(0,1)are independent,then the distribution of
Z1conditionally on Z2<λ|Z1|is SWN(0,∞)(θ0).
4.2 Moments
Note from Property 4.2 that the odd moments of Z∼SWN(u(a),u(b)) (θ0)are equal to zero. For
computing the even moments, we needs the moment generating function of SWN(u(a),u(b))(θ0)
distribution.
THEOREM 4.2 The moment generating function of Z∼SWN(u(a),u(b)) (θ0)is
MZ(t) =et2/2Lt
(u(b)) −(u(a)) +2T (u(a ), λ) −2T (u(b ), λ) ,t∈R,(18)
where Lt=L(t, (u(a) +ρt ), ρ) +L(−t, (u(a ) −ρt),ρ) −L(t, (u(b) +ρt ), ρ) −L(−t,
(u(b) −ρt),ρ).
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A class of weighted normal distributions 431
Proof From the pdf (19), MZ(t) is written as
∞
−∞
et2/2((δu(b) −λ|z|)−(δu(a ) −λ|z|))
√2π((u(b)) −(u(a)) +2T (u(a ), λ) −2T (u(b), λ))e−(z−t)2/2dz.
Using the transformation W=Z−t, one finds that MZ(t) is
et2/2(E[(δu(b) −λ|W+t|)]−E[(δu(a ) −λ|W+t|))
(u(b)) −(u(a)) +2T (u(a ), λ) −2T (u(b), λ) ,(19)
where W∼N(0,1). Applying Lemma 2.3 to equation (19), we have the result.
Property 4.1 notes that the moment generating function of X∼SWN(a,b) (θ)is immedi-
ate from equation (18). Naturally, the moments of Zcan be obtained by using the moment
generating function differentiation. Unfortunately, for higher moments, this rapidly becomes
tedious. An alternative and simple procedure for calculating the moments is available – under
the pdf of Z∼SWN(u(a),u(b)) (θ0), integrating by parts gives
E[(k +1)Zk−Zk+2]
=ρk+2φ(u(b)) ∞
−λu(b)
(w +λu(b))k+1φ(w)dw−λu(b)
−∞
(w −λu(b))k+1φ(w)dw
−φ(u(a)) ∞
−λu(a)
(w +λu(a))k+1φ(w)dw−λu(a)
−∞
(w −λu(a))k+1φ(w)dw
×λk+1{(u(b)) −(u(a)) +2T (u(a ), λ) −2T (u(b), λ)}−1,
for k=−1,0,1,2,3,....
By setting k=−1,0,we obtain that EZ =0 and Var(Z) is
1−2ρ2{φ(u(b))[φ (λu(b)) +λu(b)(λu(b))]−φ (u(a))[φ(λu(a)) +λu(a)(λu(a))]}
λ{(u(b)) −(u(a)) +2T (u(a ), λ) −2T (u(b), λ)}.
Thus, for X∼SWN(a,b) (θ), E[X]=μ1and Var(X) =σ2
1Va r (Z) by Property 4.1.
4.3 An extension
As the class of SWN distributions does not contain asymmetric distributions, this section
introduces an extension of the class by introducing an additional shape parameter ξ.
THEOREM 4.3 Assume Z1and Z2are independent N(0,1)random variables and let λ∈R
and ξ∈R.Then,the density function in zof Z1conditionally on λ|Z1+ξ|−δb < Z2<
λ|Z1+ξ|−δa is
hZ(z) =c(ρ,ξ ) φ(z){(δb −λ|z+ξ|)−(δa −λ|z+ξ|)},z∈R,(20)
where c−1
(ρ,ξ ) =L(−ξ, (a −ρξ ), ρ) +L(ξ, (a +ρξ),ρ) −L(−ξ, (b −ρξ), ρ) −L(ξ, (b +
ρξ),ρ), λ =ρ/1−ρ2and δ=λ/ρ.
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432 H.-J. Kim
Figure 2. Shapes of EWN(a,b) (θ0,ζ) density with θ0=(0,0,1,1,0.5):(1) EWN(−1,−2)(θ0,1)density; (2)
EWN(1,2)(θ0,2)density; (3) SWN(θ0)density; (4) EWN(1,2)(θ0,1)density.
Proof The assertion follows immediately from P(Z
1≤z|λ|Z1+ξ|−δb < Z2<λ|Z1+
ξ|−δa)
=P(Z
1≤z, λ|Z1+ξ|−δb < Z2<λ|Z1+ξ|−δa)
×[P(λ|Z1+ξ|−δb < Z2<λ|Z1+ξ|−δa)]−1
=z
−∞
φ(z1){(δb −λ|z1+ξ|)−(δa −λ|z1+ξ|)}dz1
×{E[(δb −λ(Z1−ξ))]−E[(δa −λ(Z1−ξ))]}−1
and by differentiation and use of Lemma 2.3.
Note that the density (20) is another member of the family of weighted normal densities
(5), because E[(δb −λ|Z+ξ|)]−E[(δa +λ|Z+ξ|)]=c−1
(ρ,ξ ) .This broader class of
distributions, which will be termed EWN(a,b) (θ0,ξ) for brevity is defined by the density
(20) with the weight function w(z;θ0,ξ) =(δb −λ|z+ξ|)−(δa −λ|z+ξ|), where
θ0=(0,0,1,1,ρ)
.Observe that SWN(a,b) (θ0)=EWN(a,b) (θ0,0).
Figure 2 depicts various shapes of EWN(a,b) (θ0,ξ)densities.
PROPERTY 4.6 Let Z∼EWN(a,b)(θ0,ξ),then −Z∼EWN(a,b) (θ0,−ξ).
PROPERTY 4.7 Let Z∼EWN(a,b)(θ0,ξ)and X=μ1+σ1Z, then
X∼EWN(a,b) (θ∗),
where θ∗=(μ1,0,σ
1,1,ρ).
An acceptance–rejection technique, which generates a random variable Zwith density (20)
is the following one. Sample Z1and Z2from independent N(0,1)distributions. If λ|Z1+
ξ|−δb < Z2<λ|Z1+ξ|−δa, then put Z=Z1, otherwise restart sampling a new pair of
variables Z1and Z2, until the inequality λ|Z1+ξ|−δb < Z2<λ|Z1+ξ|−δa is satisfied.
Figure 2 shows shapes of density (20) for different pairs of values (λ,ξ,a,b).
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A class of weighted normal distributions 433
Similar proof of Theorem 4.2 immediately leads to the moment generating function of
Z∼EWN(a,b) (θ0,ξ)),
MZ(t) =c(ρ,ξ ) L∗
tet2/2,t∈R,(21)
where
L∗
t=L(t∗,a+ρt∗,ρ)+L(−t∗,a−ρt∗,ρ)−L(t∗,b+ρt∗,ρ)−L(−t∗,b −ρt∗,ρ)
and t∗=t+ξ.
The moments of Zcan be obtained by tiresome differentiation. As done before, following
simple alternative procedure could be adopted to calculate the moments – Under the pdf (20),
integrating by parts yields
E[(k +1)Zk−Zk+2]=c(ρ ,ξ )ρk+2λ−(k +1)
×φ(b1)∞
−β2
B1(w)k+1φ(w)dw−φ(b2)β1
−∞
B2(w)k+1φ(w)dw
−φ(a1)∞
−α2
A1(w)k+1φ(w)dw−φ(a2)α1
−∞
A2(w)k+1φ(w)dw
,
for k=−1,0,1,2,3,...,where
a1=a−ρξ, a2=a+ρξ, b1=b−ρξ, b2=b+ρξ,
α1=λa −ρξ/λ, α2=λa +ρξ/λ, β1=λb −ρξ/λ, β2=λb +ρξ /λ,
A1(w) =w+λa1,A
2(w) =w−λa2,B
1(w) =w+λb1,B
2(w) =w−λb2.
By setting k=−1,0,we obtain two expressions, which may be solved to yield the first
two moments of Z.
E[Z]=c(ρ,ξ ) ρ{[φ(a1)(α2)−φ(a2)(α1)]−[φ(b1)(β2)−φ(b2)(β1)]},
E[Z2]=1+c(ρ,ξ ) ρ2λ−1{φ(a1)[(λa1)(α2)+φ(α2)]+φ(a2)[(λa2)(α1)+φ(α1)]
−φ(b1)[(λb1)(β2)+φ(β2)]−φ(b2)[(λb2)(β1)+φ(β1)]}.
Higher moments could be found similarly.
COROLLARY 4.2 If (V1,V
2)is a bivariate standard normal variable with correlation
ρ, then the conditional distribution of V1given a<sign{V1+ξ}(V2+ρξ)<b is the
EWN(a,b) (θ0,ξ)distribution.
Proof Let Z1=V1and Z2=λV1−δV2,so that Z1and Z2are independent N(0,1)vari-
ables. Then, the conditional distribution of Z1given λ|Z1+ξ|−δb < Z2<λ|Z1+ξ|−δa
is equivalent to that of V1given λ|V1+ξ|−δb < λV1−δV2<λ|V1+ξ|−δa. This is
the EWN(a,b) (θ0,ξ) distribution by Theorem 4.3. For the case V1>−ξ, the conditional
distribution can be written as
V1|(V1>−ξ,a < V2+ρξ < b).
Similar argument using another transformation Z1=V1and Z2=δV2−λV1gives, for
V1<−ξ,
V1|(V1<−ξ,−b<V
2+ρξ < −a).
Combining these two conditional distributions, we have the result.
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434 H.-J. Kim
5. Applications
5.1 Conjugate priors for the normal mean
A well-known property of the normal distribution is that, if Yis N(X,τ2)where the a priori X
is a normal random variable, then the a posteriori distribution Xis still normal. An analogous
fact is true for the WN(a,b)(θ)prior. If the a priori Xhas the probability density function in
equation (6). Some simple algebra shows that the a posteriori density function of Xgiven that
Y=yis still of type (6) with (u(a), u(b), μ1,σ
1,λ)replaced by
1+λ2u(a) −(y −μ1)λσ1
τ2+σ2
11+λ2τ2
τ2+σ2
1−1/2
,
1+λ2u(b) −(y −μ1)λσ1
τ2+σ2
11+λ2τ2
τ2+σ2
1−1/2
,
y/τ2+μ1/σ 2
1
1/τ 2+1/σ 2
1
,1
τ2+1
σ2
1−1/2
,λ
1+σ2
1
τ2−1/2
.
The well-known conjugated property of the normal distribution also applies to the EWN
distribution. That is, if Yis N(X,τ2), where a priori Xis an EWN variable, then the a
posteriori distribution Xis still an EWN distribution. Let the a priori Xhave the probability
density function
hX(x) =c(λ ,ξ)σ−1
1φ(z){(δb −λ|z+ξ|)−(δa −λ|z+ξ|)},x∈R,(22)
where X=μ1+σ1Zwith Z∼EWN(a,b) (θ0,ξ). This distribution be denoted by X∼
EWN(a,b) (μ1,0,σ
1,1,ρ,ξ). Then some simple algebra with the relation δ=√1+λ2
shows that the posterior density function of Xgiven that Y=yis still of type (22) with
(a,b,μ
1,σ
1,λ,ξ)replaced by
(1/σ 2
1+1/τ 2)1/2a
(1/σ 2
1+1/{(1+λ2)τ 2})1/2,(1/σ 2
1+1/τ 2)1/2b
(1/σ 2
1+1/{(1+λ2)τ 2})1/2,y/τ2+μ1/σ 2
1
1/τ 2+1/σ 2
1
,
1
τ2+1
σ2
1−1/2
,λ
1+σ2
1
τ2−1/2
,and ξ+(y −μ1)σ1
τ2+σ2
11+σ2
1
τ21/2
.
Note that, for both WN(a,b) (θ)and EWN(a,b )(μ1,0,σ
1,1,ρ,ξ) priors, λ(or ρ) shrinks
toward 0, independently of y, and that the updating formulae of the parameters μ1and σ1are
the same of the normal case. As given in the next examples, this conjugate property of the
WN and EWN priors may be useful for subjective Bayesian constrained estimation of normal
means.
5.2 A constrained estimation of normal mean vector
Let X1,X
2,...,X
n,n>p, be an independent sample of size nfrom Np(μ,), μ=
(μ1,...,μ
p).Now suppose that it is known that μsatisfies an inequality constraint of the form
a≤dμ≤b(23)
for any vector d= 0,and suppose that one wants to estimate μunder the constraint by using a
Bayesian approach. In fact, this inequality constrained estimation problem has not been seen
in the literature.
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A class of weighted normal distributions 435
As a joint prior density in the analysis of the inequality constrained multivariate nor-
mal model, we assume that our prior information about μi’s and are priori independent.
Accordingly, assume
π(μ,|a≤dμ≤b) =p
i=1
π(μi|a≤dμ≤b)π(). (24)
Suppose that the prior information is readily assessable by the distributions about parameters;
μ∼Np(μ0,) with constraint (23) and π() ∝||−(p+1)/2,a diffuse prior for , where
μ0=(μ01,...,μ
0p)and D=diag{ψ11,...,ψ
pp}are arbitrary parameters to be predeter-
mined by an analyst. By enriching the prior distribution of μand using (24) and Corollary 3.1,
the prior may be written as
π(μ,|a≤dμ≤b) ∝||−(p+1)/2
p
i=1
φ(μ∗
i)(δiu(b) −λiμ∗
i)−(δiu(a) −λiμ∗
i)
(u(b)) −(u(a)) ,
where μ∗
i=(μi−μi0)/√ψii, u(a) =a/√dDd, u(b) =b/√dDd,λ
i=ρi/1−ρ2
i,
ρi=p
j=1djψij/√ψii dDd,and δi=λi/ρi.
The joint posterior density for μand is found by multiplying the likelihood function of
X1,X
2,...,X
n,
p(μ,|a≤dμ≤b, Data)
∝exp −1
2n(μ−¯
X)−1(μ−¯
X) +(μ−μ0)D−1
(μ−μ0)
×||−(n+p+1)/2exp −1
2tr −1V×
p
i=1
(δiu(b) −λiμ∗
i)−(δiu(a) −λiμ∗
i)
(u(b)) −(u(a)) ,
where Data ={Xk;k=1,...,n},¯
X=n
k=1Xk,and V=n
k=1(Xk−¯
X)(Xk−¯
X).
Marginal posterior densities of μi,i=1,...,k, and are complicated. Instead, simple
algebra gives the Gibbs sampler that can be used for inference of the constrained normal
model. To apply the Gibbs sampler, we need full conditional posterior distributions of μi,
i=1,...,k, and . The full conditionals are as follows.
Let μ(i) =(μi,μ(i )
\i)∼Np(¯
X(i) ,
(i) /n) for i=1,...,p, where μ(i)
\idenotes the sub-
vector of μobtained from excluding the element μifrom μ,and let the arrays of ¯
X(i) and (i )
correspond to that of μ(i) .Then, the conditional distribution of μi,given μ(i)
\i,is N(Ai,ζ
ii),
where
Ai=¯
Xi+(i)
12 (i)−1
22 (μ(i)
\i−¯
X(i)
\i), ζii =(i)
1.2
n,
(i)
1.2=σ2
i−(i)
12 (i)−1
22 (i)
21 ,
(i)
12 /n =Cov(μi,μ(i)
\i), and (i)
22 /n =Va r (μ(i)
\i).
Under these notations, one finds the full conditional distributions of μiand ,
μi|(μ(i)
\i,,a ≤dμ≤b, Data)∼WN(u∗(a ),u∗(b))(θ∗
i)for i=1,...,p (25)
and
|(μ,a ≤dμ≤b, Data)∼W−1
p(n +p+1,V +n(μ−¯
X)(μ−¯
X)), (26)
a inverted Wishart distribution with scale matrix V+n(μ−¯
X)(μ−¯
X)and n+p+1
degrees of freedom. Some algebra using the conjugate property of the WN distribution shows
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436 H.-J. Kim
that θ∗
i=(μ∗
i,0,σ∗
i,1,ρ∗
i),u
∗(a) and u∗(b) are defined by
μ∗
i=Ai
ζii +μ0i
ii 1
ζii +1
ψii −1
,σ
∗
i=1
ζii +1
ψii −1/2
,ρ
∗
i=λ∗
i
1+λ∗
i
,
u∗(a) =1+λ2
iu(a) −λi
(Ai−μ0i)√ψii
ψii +ζii 1+λ2
iζii
ψii +ζii −1/2
,
u∗(b) =1+λ2
iu(b) −λi
(Ai−μ0i)√ψii
ψii +ζii 1+λ2
iζii
ψii +ζii −1/2
,
where λ∗
i=λi(1+ψii/ζii )−1/2.The Gibbs sampler proceeds by alternatively sampling from
the p+1 distributions. For sampling from WN distributions, we may use the one-for-one
method described in subsection 3.2. Odell and Feiveson [17] provide an easy and efficient
algorithm for the sampling from the invertedWishart distribution. If we assume an information
prior for , an inverted Wishart prior may be used for π().
5.3 A constrained regression
Consider a constrained linear regression setting
yk=β0+β1x1k+β2x2k+εk,k=1,...,n,
subject to (β1,β
2)∈C,(27)
where C={(β1,β
2);0<sign{βj}βi<b
i,b
i>0,i = j, i, j =1,2}and εks are i.i.d.
N(0,τ2)with known τ2.The constraint Cof equal signs with bounds in regression coef-
ficients frequently arises in econometric work. For example, an economic theory says that, as
explanatory variables, the disposable income and the rate of price change have the same effect
(positive or negative) on an economic dependent variable (see, for example, [18]). When the
bounds of coefficients are priori unknown, we may put bi=∞.In this case, Creduces to
{(β1,β
2);β1β2>0}.
Our information about regression coefficients is informative and it is represented by taking
β0,β
1and β2as independent. Accordingly, assume
π(β0,β
1,β
2|(β1,β
2)∈C)=π(β0)π(β1|(β1,β
2)∈C)π(β2|(β1,β
2)∈C).
Suppose that the prior information is readily assessable by the distribution about the coeffi-
cients; β0∼N(μ0,σ2
0)and (β1,β
2)∼N2(0,) with the constraint C,where μ0σ2
0and
={σij},σij =σiσjρfor i= j, are arbitrary parameters available to be predetermined
by analyst, then we obtain the constrained prior distributions of β1/σ1and β2/σ2using
Property 4.1.
β1
σ1
C∼SWN(0,b2/σ2)(θ0)I (−b1<β
1<b
1)(28)
and
β2
σ2
C∼SWN(0,b1/σ1)(θ0)I (−b2<β
2<b
2), (29)
for βi/σi|C≡βi/σi|(0<sign{βi}βj<b
j,−bi<β
i<b
i)for i, j =1,2.Here, θ0=
(0,0,1,1,ρ)
and I(·)denotes an indicator function. Thus equations (30) and (31) are
truncated SWN distributions with truncation regions specified by the indicator functions.
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A class of weighted normal distributions 437
The joint posterior density for β0,β
1and β2is found by
p(β0,β
1,β
2|C,Data)∝e−{n
k=1(yk−β0−β1xik−β2x2k)2/(2τ2)+(β0−μ0)2/(2σ2
0)}
×e−2
i=1β2
i/(2σ2
i)
2
i=1,i=jδbj
σj−λ
βi
σi
−−λ
βi
σiI(−bi<β
i<b
i), (30)
where λ=ρ/1−ρ2and δ=λ/ρ. Although the form of marginal posterior densities of
β0,β
1and β2are not easily calculable, by applying the Gibbs sampler to the conditionals in
equation (30) any characteristics of the marginal densities can be obtained. The Gibbs sampler
consists of the following full conditional posterior distributions. The full conditional of β0is
β0|(β1,β
2,C,;Data)∼N(μ(0), σ 2(0)), (31)
where
μ(0)=μ0τ2/σ 2
0+n
k=1(yk−β1x1k−β2x2k)
n+τ2/σ 2
0
and σ2
0=n
τ2+1
σ2
0−1/2
.
The full conditional of β1and β2are obtained from the conjugate property of the EWN(a,b)
(θ0,ξ)distribution with density (20). They are
β1|(β1,β
2,C,;Data)∼EWN(0,b∗
2)(θ1,ξ
1)I (−b1<β
1<b
1)(32)
and
β2|(β1,β
2,C,;Data)∼EWN(0,b∗
1)(θ2,ξ
2)I (−b2<β
2<b
2), (33)
where
b∗
1=(1/σ 2
2+1/τ 2)1/2(b1/σ1)
(1/σ 2
2+1/{(1+λ2)τ 2})1/2,b
∗
2=(1/σ 2
1+1/τ 2)1/2(b2/σ2)
(1/σ 2
1+1/{(1+λ2)τ 2})1/2.
The other parameters θi=(μ(i), 0, σ (i), 1, ρ(i))and ξi,i =1,2,are defined by
μ(1)=n
k=1x1k(yk−β0−β2x2k)
n
k=1x2
1k+τ2/σ 2
1
,μ(2)=n
k=1x2k(yk−β0−β2x1k)
n
k=1x2
2k+τ2/σ 2
1
,
σ(1)=n
k=1
x2
1k
τ2+1
σ2
1−1/2
,σ(2)=n
k=1
x2
2k
τ2+1
σ2
1−1/2
,
ρ(1)=λ(1)
1+λ(1)2,ρ(2)=λ(2)
1+λ(2)2,ξ
1=μ(1)
σ(1),ξ
2=μ(2)
σ(2),
where λ(1)=λσ (1)/σ1and λ(2)=λσ (2)/σ2.
In fact, all of the preceding distributions are available in closed form. Generation of the
required Gibbs samples may be done directly. Note that a sample from the truncated EWN dis-
tributions, EWN(0,b∗
j)(θi,ξ
i)I (−bi<β
2<b
i), i = j;i, j =1,2,is available immediately, if
we use the acceptance–rejection sampling scheme, described in section 4.3.
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438 H.-J. Kim
5.4 Screening in a normal model
The problem of screening has received considerable attention in the literature (see, [19] and
references therein). Suppose that a certain measurement on an individual is described by a
normal random variable Y. We assume that we are given a specification region CYsuch that
if Y∈CYthe individual is considered success. Then γ=P(Y ∈CY)is the probability of
success or the proportion of successful individuals in the population. Typically, CYis of the
form [L, U ].
The motivation behind screening is to try to increase the proportion of successes by
eliminating some individuals. This is done by recording a second measurement Xon each
individual, where Xand Yare correlated and a normal variable Xis easier to measure than Y.
Our first problem of screening is to determine the shortest length of the specification region
CYfor a given CX.C
Xdenotes the specification region for the Xvalues so that the individual
will be retained or screened out if X∈CXor not, respectively. Thus the aim is to specify the
shortest length CYso that P(Y ∈CY|X∈CX)=α, for some specified α>γ.
A simple method may be followed if we use the df(7) of the WN distribution– although at
the screening stage only the dichotomy on Yis important, we assume throughout that there is
an underlying variable, which can be measured, and that we have available data in the form
of a random sample (y1,x
1), (y2,x
2),...,(y
n,x
n)from the unscreened data.
The shortest region of the form CY=[L, U]might require a two-sided region such that
P(L ≤Y≤U|a≤X≤b) =α. (34)
This is equivalent to
L(zL, u(a), ρ ) −L(zL, u(b), ρ)
(u(b)) −(u(a)) −L(zU, u(a), ρ) −L(zU, u(b), ρ)
(u(b)) −(u(a)) =α(35)
by the df(7) of the WN(a,b) (θ)distribution, where θ=(μ1,μ
2,σ
1,σ
2,ρ)
,a bivariate normal
parameter vector, zL=(L −μ1)/σ1,z
U=(U −μ1)/σ1, u(a) =(a −μ2)/σ2and u(b) =
(b −μ2)/σ2.When θis unknown this can be estimated by using the sample.
Thus the problem is to find values of Land Uthat minimizes the length U−Lunder the
restriction
FY(U) −FY(L) =U
L
fY(y)dy=α,
where FY(y) and fY(y ) are the dfand pdf of the WN(a ,b)(θ)distribution defined by equa-
tions (7) and (6), respectively. By taking derivatives of the restricting equation with respect
to U, we see that dL/dU=f (U )/f (L). By setting d(U −L)/dU=0,the constrained
minimization yields a criterion that Uand Lmust satisfy fY(U) =fY(L), i.e.,
φ(zU)
φ(zL)=(δu(b) −λzL)−(δu(a) −λzL)
(δu(b) −λzU)−(δu(a ) −λzU)(36)
by equation (7). This condition along with equation (35) will be used to calculate the values
Uand L.
Let our second problem of screening be to determine a specification region CXfor a given
specification region CY.Then, the aim is to specify CXso that P(Y ∈CY|X∈CX)=α, for
some specified α>γ.In this case, there are many pairs of (a, b) that satisfy equation (35), so
some additional criterion becomes necessary. One such criterion is obtained by considering
the probability attached to the possible mistake that an individual screened out by the process
would have been a success. It seems sensible, therefore, to choose that pair (a, b), which
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A class of weighted normal distributions 439
maximizes β=P(X ∈CX)=(u(b)) −(u(a)) A Lagrangian constrained maximization
analysis (maximizing βsubject to equation (35)) yields the additional criterion
Q(u(a)) =Q(u(b)), (37)
where Q(c) =zU
zLf(z
1|z2=c)dz1and f(z
1,z
2)is the pdf of the bivariate standard normal
random variables (Z1,Z
2)with correlation ρ. Thus
Q(c) =zU−ρc
1−ρ2−zL−ρc
1−ρ2.
The solution of equation (35) satisfying equation (37) gives an optimal (a, b) provided that
a<band α>γ.
5.5 Sum of values from a normal and a truncated normal
Suppose an item, which one makes has two parts that are assembled additively with regard to
length. The length of both parts are normally distributed but, before assembly, one of parts is
subject to an inspection which removes all individuals blow a specific length.
As an example, suppose that U(1)comes from a normal distribution with a mean of 100
and standard deviation of 6, and U(2)comes from a normal distribution with mean 50 with
standard deviation 3, but with the restriction that U(2)≥44.How can we find the chance that
U(1)+U(2)is equal to or less than a given value, say 138?This problem is originally answered
by Weinstein [20].
A simple alternative answer presented here uses the df(7) of the WN(a,b)(θ)
distribution. From Theorem 3.1, we see that U(1)+U(2)
(44,∞)∼WN(a,b) (θ), where θ=
(μ1,μ
2,σ
1,σ
2,ρ)
=(150,50,√45,3,3/√45).Thus using equation (7), we see that
FX(138)=0.03276,and this probability is the same as the Weinstein’s answer. When the
parameters of U(1)and U(2)distributions are unknown, the probability FX(138)can be approx-
imately calculated by using the ML estimates of the parameters. See [16] for the ML estimation
of the truncated normal parameters.
6. Concluding remarks
The primary focus of this paper is to study a class of weighted normal distributions and its
variants associated with doubly truncated bivariate normal. From applied viewpoint, they are
useful for the inequality constrained analysis and selection modelling. Procedures of solving
those problems by using the distributions are illustrated in section 5. On the theoretical side,
they enable us to have at hand a family of densities with the following properties: (i) Strict
inclusion of the normal, skew–normal, and two-piece skew–normal distributions; (ii) mathe-
matical tractability; (iii) wide range of the indices of skewness and kurtosis as described in
figures 1 and 2; (iv) an important Bayesian property (providing a family of conjugate priors
to normal mean) and (v) availability for the multivariate extension.
For the p-dimensional extension of the density (6), we may define a multivariate variable
X=μ+1/2Zsuch that each component of Zis WN
(u(a),u(b)) (θ0i), i =1,...,p, where
θ0i=(0,0,1,1,ρ
i).It is then natural to define the joint distribution of Zas being mul-
tivariate WN distribution. To this end, consider a p-dimensional normal random variable
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440 H.-J. Kim
Y=(Y1,...,Y
p)∼Np(0,) with standardized marginals independent of Y0∼N(0,1),
where =Corr(Y). Then
Zi=1−ρ2
iYi+ρiY0(u(a),u(b)) ,i=1,...,p,
by Property 3.5. In this regard, a multivariate extension of the WN distribution can be defined.
Computation of the distribution of Z=(Z1,...,Z
p)is trivial but lengthy; the basic steps
are similar to those for the computation of the multivariate skew–normal density function by
Azzalini and Dalla Valle [21]. The final expression of its density is
fZ(z;0,)=φp(z;) (Qu(b) −αz)−(Qu(a) −αz)
(u(b)) −(u(a)) ,z∈Rp,
where
Q=(1+λ−1λ)−1/2,α=Qλ−1,=diag{δ1/2
1,...,δ
1/2
p},
λ=(λ1,...,λ
p),=−1( +λλ)−1,
0=(θ
01,...,θ
0p).
Here, λi=ρi/1−ρ2
i,δ
i=λi/ρi,and φp(z;) is the density function of the p-dimensional
multivariate normal distribution with standardized marginals and correlation matrix . Thus,
the joint density of Xis obtained from the transformation so that
fX(x;, ) =||−1/2fZ(z;0,), x∈Rp
where z=−1/2(x−μ), ={μ,,
0}.The p-dimensional extension of the density (20)
can be obtained from similar method.
Acknowledgements
This research was supported by the Korean Research Foundation Grant funded by the Korean
Government (MOEHRD) KRF-2006-311-C00240.
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