# Marginal Distributions of Random Vectors Generated by Affine Transformations of Independent Two-Piece Normal Variables

Authors: Maximiano Pinheiro
Journal: Journal of Probability and Statistics
 (2012)

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## Abstract

Marginal probability density and cumulative distribution functions are presented for multidimensional variables defined by nonsingular affine transformations of vectors of independent two-piece normal variables, the most important subclass of Ferreira and Steel's general multivariate skewed distributions. The marginal functions are obtained by first expressing the joint density as a mixture of Arellano-Valle and Azzalini's unified skew-normal densities and then using the property of closure under marginalization of the latter class.


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## Body

## 1. Introduction

In the literature on probability distributions, there are several approaches for extending the multivariate normal distribution with the introduction of some sort of skewness. Arellano-Valle et al. [1] provide a unified view of this literature. The largest group of contributions was initiated by Azzalini and Dalla Valle [2] and Azzalini and Capitanio [3] and generalizes the univariate skew-normal (SN) distribution studied by Azzalini [4, 5]. These “multivariate skew-normal distributions” are generated from a normal distribution either by conditioning on a truncated variable or by a convolution mechanism.An alternative approach was proposed by Ferreira and Steel [6–8] and is based on nonsingular affine transformations of random vectors with independent components, each having a skewed distribution with probability density function (pdf) constructed from a symmetric distribution using the inverse scaling factor method introduced by Fernández and Steel [9]. (Arellano-Valle et al. [10] consider a general class of asymmetric univariate distributions that includes the distributions generated according to the procedure proposed by Fernández and Steel [9] as a special case.) If the univariate symmetric distribution is the standard normal, then the corresponding univariate skewed distribution becomes (with a different parameterization) the two-piece normal (tpn) analyzed by John [11] (see also Johnson et al. [12]). To overcome an issue of overparameterization, Ferreira and Steel [7, 8] pay particular attention to the subclass associated with transformation matrices that can be factorized as the product of an orthogonal matrix and a diagonal positive definite matrix. Villani and Larsson [13] studied this subclass when the basic univariate skewed distribution is the tpn and named these distributions “multivariate split normal.”Under the acronymSUN (standing for “unified skew-normal”), Arellano-Valle and Azzalini [14] suggested a formulation for the first approach that encompasses the most relevant coexisting variants of multivariate skew-normal distributions. Like the multivariate normal and SN distributions, the class of SUN distributions is closed under affine transformations, marginalization, and conditioning to given values of some components. Besides these important properties, the SUN class is also closed under sums of independent components. However, one limitation of the SUN distributions is that the vector of location parameters does not have a direct interpretation as the mean or the mode of the distribution, which are rather complicated functions of all the parameters. Even in the simplest case of the basic SN, both the mean and the mode (for which there is no closed expression) depend on the parameters regulating dispersion and skewness.Ferreira and Steel’s independent components approach to the construction of multivariate skewed normal distributions (henceforthFS-SN) provides an alternative to the SUN class in applications for which it is important to have some location measure that does not depend on the dispersion and skewness parameters. Indeed, the FS-SN distributions have the convenient feature that the mode is part of the distribution parameters and therefore is invariant to dispersion and skewness. In addition, the FS-SN distributions are closed under nonsingular affine transformations. However, unlike the SUN class, the FS-SN distributions are not closed under marginalization (neither under conditioning) and, to my knowledge, general closed expressions of their marginal pdf and cumulative distribution function (cdf) are not available in the literature.This paper aims at filling the gap and proposing expressions for the marginal density and cumulative distribution functions of anFS-SN distribution. Obviously, the expressions will also apply to the subclass of multivariate split normal distributions studied by Villani and Larsson [13]. The technique used to derive the marginal distributions is simple and consists of expressing the joint FS-SNdistribution as a finite mixture of singular SUN distributions and then making use of their property of closure under marginalization.An area of application of the results presented in this paper is macroeconomic density forecasting. Many institutions that publish macroeconomic forecasts complement their point forecasts with information on the dispersion and skewness of the probability distributions of the forecasting errors. Fan charts are one of the most popular tools to convey the predictive densities, and they gained prominence through their use in inflation reports released by many central banks, with the Bank of England and the Sveriges Riksbank (the Swedish central bank) featuring as pioneers in this respect [15, 16] (see also Wallis [17, 18] and Tay and Wallis [19]). The characterization of the forecast densities is complicated by the fact that typically institutional forecasts are not based on a single model but stem from different competing models combined with judgements by experts (the latter regarding, in particular, the skewness, i.e., the balance of upward and downward risks to the forecasts). Most of the procedures used to generate the fan charts take the point baseline forecasts as given and assume that the sources of uncertainty and asymmetry have univariate tpndistributions. These sources of forecasting error are then aggregated according to a linear mapping, envisaged as an approximation around the baseline to the underlying unknown data generating process. In the absence of closed expressions for the exact distribution of a linear combination of tpn variables, some aggregation procedures resort to informal approximations based on the first moments, while other procedures are based on numerical simulation. Examples of the first approach are Blix and Sellin [16, 20, 21] and Elekdag and Kannan [22], while Pinheiro and Esteves [23] opted to simulate the distribution. The results presented in Section 3 allow to overcome this aggregation difficulty.


## 2. TheSUN and the FS-SN Distributions

If theM-dimensional random vector Y~SUNM,N(ξ,γ,ω̅,Ω*), then its pdf and cdf are, respectively, for any point y∈RM(2.1)gY(y∣ξ,γ,ω̅,Ω*)=φM(y-ξ∣Ω)ΦN(γ+Δ′Ω̅-1ω-1(y-ξ)∣Γ-Δ′Ω̅-1Δ)ΦN(γ∣Γ),(2.2)GY(y∣ξ,γ,ω̅,Ω*)=ΦN+M([γω-1(y-ξ)]∣Ω*)ΦN(γ∣Γ),
where φM(y-ξ∣Ω) and ΦM(y-ξ∣Ω) denote, respectively, the pdf and the cdf at point y of a normal distribution NM(ξ,Ω), ξ(M×1) and γ(N×1) are vectors of parameters, Ω(M×M) is a positive definite covariance matrix, ω(M×M) is the diagonal matrix formed by the standard deviations of Ω, Ω̅(M×M) is the correlation matrix associated with Ω (hence Ω=ωΩ̅ω), ω̅=ωιN with ιN=[1⋯1]′(N×1), Γ(N×N) is a positive definite correlation matrix, and Δ(M×N) is such that(2.3)Ω*=[ΓΔ′ΔΩ̅]((N+M)×(N+M))
is also a (semi-definite positive) correlation matrix. (Arellano-Valle and Azzalini [14, Appendix C] consider three cases of singular SUNdistributions: (i) Ω singular; (ii) Γ singular; (iii) Ω* singular with nonsingular Ω and Γ. For our purposes, only the latter case is relevant.) The SUN distribution collapses to the multivariate normal when Δ=0, Δ being the matrix of parameters that regulate skewness. It collapses to the basic multivariate SNdistribution suggested by Azzalini and Dalla Valle [2] when N=1 and γ=0 (implying that Γ=1).Now let the scalar random variableUn be tpndistributed with zero mode. Its pdf may be parameterized as follows:(2.4)fUn(un∣ωn,θn)={2ωn-1(θn+θn-1)-1ϕ(ωn-1θnun)(un≤0),2ωn-1(θn+θn-1)-1ϕ(ωn-1θn-1un)(un>0),
where ϕ() denotes the N(0,1) pdf, ωn(>0)is a scale parameter, and θn(>0) is a shape parameter. When θn=1, the density becomes the normal pdf with zero mean and standard deviation ωn (so that when the latter parameter is 1 the pdf collapses to ϕ(un)). Values of θn above (below) unity correspond to densities skewed to the right (left). Let U be an N-dimensional random vector of independent tpncomponents un with zero mode and unitary scale ωn=1. Its pdf is(2.5)fU(u∣θ)=∏n=1NfUn(un∣1,θn),
where fUn() is as in (2.4) (with ωn=1) and θ=[θ1⋯θN]′. An N-dimensional random vector X is said to be FS-SNN(μ,A,θ) distributed if there is a random vector U with density (2.5) and two vectors μ (the joint mode) and θ (the “shape vector”) and a nonsingular matrix A (the “scale matrix”) such that X=μ+AU. Vector Xhas pdf(2.6)fX(x∣μ,A,θ)=|det⁡(A)|-1fU(A-1(x-μ)∣θ).
It is straightforward to confirm that (i) when θ=0, this density collapses to the pdf of a NN(μ,AA′) distribution, (ii) the FS-SNN(μ,A,θ) distribution is unimodal with mode μ, invariant with respect to A and θ, and (iii) by construction, the FS-SN class is closed under nonsingular affine transformations.


## 3. The MarginalFS-SN Distributions

To establish additional notations, letIN denote the identity matrix of order N, and let  η(z) the number of zero elements in vector z, ψ(z) one if all elements of vector z are nonnegative and zero otherwise, and  k(i)=(k1(i),…,kn(i),…,kN(i)) the generic element of the Nth Cartesian power of {-1;1}(with cardinal2N), K(i)=diag⁡n(kn(i))(N×N), Θ(i)=diag⁡n(θnkn(i))(N×N), Ω(i)=[AΘ(i)K(i)][AΘ(i)K(i)]′=AΘ2(i)A′, ω(i)=[diag⁡(Ω(i))]1/2, ω̅(i)=ω(i)ιN, Δ(i)=ω-1(i)AΘ(i)K(i) and Ω̅(i)=ω-1(i)Ω(i)ω-1(i)=Δ(i)Δ(i)′.Proposition 3.1.
The pdf and the cdf of the N-dimensional random vector(3.1)X∼FS-SNN(μ,A,θ)
with nonsingular scale matrix  A  can be expressed, respectively, as
(3.2)fX(x∣μ,A,θ)=∑i=12N[∏n=1N(1+θn-2kn(i))]-1gX(x∣μ,0,ω̅(i),Ω*(i)),FX(x∣μ,A,θ)=∑i=12N[∏n=1N(1+θn-2kn(i))]-1GX(x∣μ,0,ω̅(i),Ω*(i)),
where  gX() and  GX() are pdfs and cdfs of singular SUNN,N(μ,0,ω̅(i),Ω*(i)) distributions, with Ω*(i)=[INΔ(i)′Δ(i)Ω̅(i)]=[INΔ(i)][INΔ(i)′].   The latter functions may be written as
(3.3)gX(x∣μ,0,ω̅(i),Ω*(i))=2N-η(x-μ)φN(x-μ∣Ω(i))ψ(K(i)Θ-1(i)A-1(x-μ)),GX(x∣μ,0,ω̅(i),Ω*(i))=2NΦ2N([0ω-1(i)(x-μ)]∣Ω*(i))=2N∫{z∣z≤0,AΘ(i)K(i)z≤x-μ}φN(z∣IN)dz.Note that(3.4)∑i=12N[∏n=1N(1+θn-2kn(i))]-1=[∏n=1N(θn+θn-1)]-1∑i=12N[∏n=1Nθnkn(i)]=1.
Hence, the distribution FS-SNN can be envisaged as a finite mixture of singular SUNN,N distributions.As pointed out by Arellano-Valle and Azzalini [14, Appendix C], the rank deficiency of Ω*(i)does not affect the properties of the SUN distributions and its only impact is of a computational nature. In our case, it actually simplifies the computation of the pdf values because the evaluation of a normal cdf is not required anymore, unlike when computing (2.1), the general expression of a SUN pdf.In order to derive the marginal pdfs and cdfs ofX, one needs to consider its partition X=[X1'X2']'with X1 and X2 of dimensions N1 and N2, respectively, and the corresponding partitions(3.5)μ=[μ1μ2],A=[A1A2],Δ(i)=[Δ1(i)Δ2(i)],Ω(i)=[Ω11(i)Ω12(i)Ω12(i)′Ω22(i)],ω(i)=[ω1(i)00ω2(i)],ω̅(i)=[ω̅1(i)ω̅2(i)]
with A1(N1×N), Δ1(i)=ω1-1(i)A1Θ(i)K(i)(N1×N), Ω11(i)=A1Θ2(i)A1′(N1×N1), ω1(i)=[diag⁡(Ω11(i))]1/2, Ω̅11(i)=ω1-1(i)Ω11(i)ω1-1(i)=Δ1(i)Δ1(i)′, and ω̅1(i)=ω1(i)ιN1. Proposition 3.2 follows directly from Proposition 3.1 and from the result of Arellano-Valle and Azzalini [14, Appendix A] on the marginal distributions of members of the SUN class.Proposition 3.2.
LetX=[X1′X2′]′~FS-SNN(μ,A,θ). Then, the marginal pdf and the cdf of the  N1-dimensional subvector X1are, respectively,
(3.6)fX1(x1∣μ1,A1,θ)=∑i=12N[∏n=1N(1+θn-2kn(i))]-1gX1(x1∣μ1,0,ω̅1(i),Ω11*(i)),FX1(x1∣μ1,A1,θ)=∑i=12N[∏n=1N(1+θn-2kn(i))]-1GX1(x1∣μ1,0,ω̅1(i),Ω11*(i)),
where  gX1() and  GX1() are pdfs and cdfs of singular SUNN1,N(μ1,0,ω̅1(i),Ω11*(i)) distributions, with  Ω11*(i)=[INΔ1(i)′Δ1(i)Ω̅11(i)]=[INΔ1(i)][INΔ1(i)′]. The latter functions may be written as
(3.7)gX1(x1∣μ1,0,ω̅1(i),Ω11*(i))=2N-η(x1-μ1)φN1(x1-μ1∣Ω11(i))ψ(K(i)Θ(i)A1′[A1Θ2(i)A1′]-1(x1-μ1)),GX1(x1∣μ1,0,ω̅1(i),Ω11*(i))=2NΦN+N1([0ω1-1(i)(x1-μ1)]∣Ω11*(i))=2N∫{z∣z≤0,A1Θ(i)K(i)z≤x1-μ1}φN(z∣IN)dz.



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*Source: 758975-2012-04-03.xml*