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A New Class of Skew-Normal DistributionsReinaldo B. Arellano-Valle a , Héctor W. Gómez b & Fernando A. Quintana a ca Departamento de Estadística , Facultad de Matemática , Pontificia Universidad Católicade Chile , Santiago, Chileb Departamento de Matemática , Facultad de Ingeniería , Universidad de Atacama ,Copiapó, Chilec Departamento de Estadística , Facultad de Matemática , Pontificia Universidad Católicade Chile , Avenida Vicuña Mackenna 4860, 782-0436 MACUL, Santiago, ChilePublished online: 15 Feb 2007.

To cite this article: Reinaldo B. Arellano-Valle , Héctor W. Gómez & Fernando A. Quintana (2004) A New Class of Skew-Normal Distributions, Communications in Statistics - Theory and Methods, 33:7, 1465-1480, DOI: 10.1081/STA-120037254

To link to this article: http://dx.doi.org/10.1081/STA-120037254

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A New Class of Skew-Normal Distributions

Reinaldo B. Arellano-Valle,1 Hector W. Gomez,2

and Fernando A. Quintana1,*

1Departamento de Estadıstica, Facultad de Matematica, PontificiaUniversidad Catolica de Chile, Santiago, Chile

2Departamento de Matematica, Facultad de Ingenierıa, Universidadde Atacama, Copiapo, Chile

ABSTRACT

We introduce a new family of asymmetric normal distributions thatcontains Azzalini’s skew-normal (SN) distribution as a special case.We study the main properties of this new family, showing in parti-cular that it may be generated via mixtures on the SN asymmetryparameter when the mixing distribution is normal. This propertyprovides a Bayesian interpretation of the new family.

Key Words: Asymmetry; Kurtosis; Skew-normal; Skew-curvednormal; Skew-generalized normal distributions.

*Correspondence: Fernando A. Quintana, Departamento de Estadıstica,Facultad de Matematica, Pontificia Universidad Catolica de Chile, AvenidaVicuna Mackenna 4860, 782-0436 MACUL, Santiago, Chile; Fax: (562) 354-7229; E-mail: quintana@mat.puc.cl.

1465

DOI: 10.1081/STA-120037254 0361-0926 (Print); 1532-415X (Online)

Copyright # 2004 by Marcel Dekker, Inc. www.dekker.com

COMMUNICATIONS IN STATISTICS

Theory and Methods

Vol. 33, No. 7, pp. 1465–1480, 2004

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Mathematic Subject Classification: Primary 60E05; Secondary62E10.

1. INTRODUCTION

The class of distributions we study in this article is closely related tothe skew-normal (SN) distribution introduced by Azzalini (1985) anddefined as follows: a random variable X has skew-normal distributionwith asymmetry parameter l 2 R if

f ðx jlÞ ¼ 2fðxÞFðlxÞ x 2 R; ð1:1Þ

where fð�Þ and Fð�Þ are the Nð0; 1Þ density and CDF, respectively.We denote this by X � SNðlÞ. When l ¼ 0 (1.1) becomes the Nð0; 1Þdistribution. Henze (1986) gives a stochastic representation of thisdistribution, obtaining explicitly its odd moments. Azzalini (1986) alsodiscusses stochastic representations in a more general context.Multivariate extensions of the univariate SN distribution have beenstudied by several authors. See, for example, Azzalini and Dalla-Valle (1996), Branco and Dey (2001) and Arellano-Valle and Genton(2003).

A limitation of the SNðlÞ model is that for moderate values of lnearly all the mass accumulates either on the positive or negative num-bers, as determined by the sign of l. In such cases, (1.1) closely resemblesthe half-normal density, with a nearly linear shape in the side with smallermass.

To partially mitigate such limitation we introduce a family ofdistributions that exhibits a better behavior, particularly at the sidewith smaller mass. This class will be referred to as skew-generalizednormal (SGN) because it contains the SN distribution (1.1) as a specialcase.

The rest of this article is organized as follows. Section 2 gives thedefinition of the SGN class and presents its basic properties. We alsoconsider an important subclass, called the skew-curved normal family,which includes the normal distribution as a special case, but is disjointof Azzalini’s subclass. Section 3 gives important probabilistic propertiesof the new family and Sec. 4 is related to the moments of SGN randomvariables. Section 5 deals with a location-scale extension of the SGNdistribution and establishes an interesting invariance property of suchdistributions under normal sampling. Finally, Sec. 6 gives an applicationbased on real data.

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2. THE SKEW-GENERALIZED NORMAL

DISTRIBUTION

In this section, we introduce a new class of skew normal distribu-tions, which generalizes (1.1) while preserving most of its properties.

Definition 1. We say that a random variable X has the skew-generalizednormal distribution if its density is given by

f ðx j l1; l2Þ ¼ 2fðxÞF l1xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ l2x2

p !

; x 2 R; ð2:1Þ

where l1 2 R and l2 � 0. We denote this by X � SGNðl1; l2Þ. Theresulting distribution for the special case l2 ¼ l21 is called skew-curvednormal (SCN) and is denoted by X � SCNðl1Þ.

We note that (2.1) is indeed a density, because as shown by Ellison(1964), if Z � Nðm; s2Þ then

E½FðZÞ� ¼ Fmffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ s2p� �

: ð2:2Þ

Figure 1 illustrates several of the possible shapes obtained from (2.1)under various choices of ðl1; l2Þ.

Figure 2 compares the SCNðl1Þ and the SNðl1Þ densities for variouschoices of l1.

Some basic properties of the fSGNðl1; l2Þ : l1 2 R; l2 � 0g familyarise directly from (2.1).

Proposition 1.

(1) f ðx jl1 ¼ 0; l2Þ ¼ fðxÞ for all x 2 R and l2 � 0, i.e.,SGNðl1 ¼ 0; l2Þ ¼ Nð0; 1Þ.

(2) f ðx jl1; l2 ¼ 0Þ ¼ 2fðxÞFðl1xÞ for all x 2 R, i.e.,SGNðl1; l2 ¼ 0Þ ¼ SNðl1Þ.

(3) If X � SGNðl1; l2Þ then �X � SGNð�l1; l2Þ.(4) f ðx j � l1; l2Þ þ f ðx jl1; l2Þ ¼ 2fðxÞ for all x 2 R.

(5) liml1!þ1

f ðx j l1; l2Þ ¼ 2fðxÞIfx � 0g and liml1!�1

f ðx j l1; l2Þ¼ 2fðxÞIfx � 0g for all l2.

(6) liml1!�1

f ðx jl1; l21Þ ¼ 2fðxÞFð� signðxÞÞ.

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Remark 1. Item 6 of Proposition 1 gives a hint at the basic motivationfor considering the special case SCNðl1Þ. When l1 grows, the resultingSCN density has heavier tail to the left than the SN distribution. In fact,the asymptotic distribution obtained from the SCNðl1Þ when l1 ! 1 isnot half-normal. Indeed, this corresponds to a distribution concentratingmost of its mass at one side of 0, but still leaving positive probability atthe other side.

From (2.1) the CDF of the SGNðl1; l2Þ distribution can beexpressed as

Fðx j l1; l2Þ ¼Z x

�12fðtÞF l1tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ l2t2p !

dt: ð2:3Þ

Figure 1. Examples of the SGNðl1; l2Þ density for ðl1; l2Þ ¼ ð�5; 20Þ (solidline), ðl1; l2Þ ¼ ð1; 1Þ (dashed line) and ðl1; l2Þ ¼ ð10; 40Þ (dotted line).

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This immediately leads to the following properties:

Proposition 2.

(1) F x jl1 ¼ 0; l2ð Þ ¼ FðxÞ for all x 2 R and l2 � 0.

(2) F �x jl1; l2ð Þ ¼ 1� F x j�l1; l2ð Þ:

(3) liml1!þ1

Fðx jl1; l21Þ ¼2ð1� Fð1ÞÞFðxÞ if x<0

1� 2Fð1Þð1� FðxÞÞ otherwise.

�

(4) liml1!�1

Fðx jl1;l21Þ¼2Fð1ÞFðxÞ if x<0

2½Fð1Þþð1�Fð1ÞÞFðxÞ�1=2� otherwise.

�

Figure 2. The SCNðl1Þ (dotted line) and SNðl1Þ (solid line) densities fordifferent values of l1.

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3. SOME IMPORTANT PROPERTIES

We now derive the main properties of the SGN distribution. The firstresult shows that, just as Azzalini’s SN, the distribution of the absolutevalue of a SGN random variable is half-normal. In particular, this impliesthat all the moments of the SGN distribution are finite, and its evenmoments coincide with those of the standard normal distribution.

Proposition 3. Let X � SGNðl1; l2Þ and Y � Nð0; 1Þ. Then jX j and jY jare identically distributed, i.e., jX j ¼d jY j � HNð0; 1Þ, where HNð0; 1Þdenotes the standard half-normal distribution.

Proof. It is well known that Z ¼jY j has density 2fðzÞIfz > 0g. On theother hand, by (2.1) the density of W ¼ jX j is

fW ðwÞ ¼ fX ðwÞ þ fX ð�wÞ

¼ 2fðwÞF l1wffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ l2w2

p !

þ 2fð�wÞF � l1wffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ l2w2

p !

¼ 2fðwÞ for w > 0;

and since fW ðwÞ ¼ 0 for w � 0, we have fW ðwÞ ¼ fZðwÞ for all w 2 R:&

The following result is a direct consequence of Proposition 3:

Proposition 4. Let X � Nð0; 1Þ, Y � SGNðl1; l2Þ and let h be an evenfunction. Then hðX Þ and hðY Þ are identically distributed. In particular,Y 2 � w2ð1Þ.

An interesting interpretation of the SGN distribution is given next.

Proposition 5. If X jY ¼ y � SNðyÞ and Y � N l1; l2ð Þ then X �SGN l1; l2ð Þ.

Proof.

f ðx jl1; l2Þ ¼ 2ffiffiffiffiffil2

pZ 1

�1fðxÞFðxyÞf y� l1ffiffiffiffiffi

l2p

� �dy

¼Z 1

�12fðxÞFð

ffiffiffiffiffil2

pxzþ l1xÞfðzÞdz

¼ 2fðxÞE Fðffiffiffiffiffil2

pxZ þ l1xÞ

h i:

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The result now follows by applying (2.2) with Z � Nð0; 1Þ, and thedefinition of conditional density. &

Proposition 5 states that the skew-generalized (curved) normaldistribution can be represented as a mixture of the skew-normal on theasymmetry parameter, using a (curved) normal distribution as the mixingdistribution. This stochastic representation is also quite useful fordrawing samples from the SGNðl1; l2Þ distribution via the method ofcomposition (see, e.g., Tanner, 1996) and the SN package of Azzalini,available for downloaded at http:==azzalini.stat.unipd.it. Alternatively,(2.1) is easily shown to be a log-concave density, which can be also usedto draw samples via the Gilks and Wild (1992) algorithm.

4. MOMENTS

Taking hðxÞ ¼ xk for k even in Proposition 4, we conclude thatthe even moments of the skew-generalized and standard normal distribu-tions are identical. This also implies the existence of the odd moments,but there is no explicit expression for them. Nevertheless, an implicitformula can be derived as follows. Let

akðl1; l2Þ ¼Z 1

0

ukffiffiffiffiffiffi2p

p e�u=2Fl1

ffiffiffiu

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ l2u

p� �

du: ð4:1Þ

Note that for all l2 � 0,

akð0; l2Þ ¼ akð0; 0Þ ¼ 2kGðk þ 1Þ=ffiffiffiffiffiffi2p

p: ð4:2Þ

Proposition 6. If X � SGNðl1; l2Þ then for k ¼ 0; 1; 2; . . . . we have

EðX 2kþ1Þ ¼ 2fakðl1; l2Þ � akð0; 0Þg: ð4:3Þ

Proof.

EðX 2kþ1Þ ¼ 2

Z 1

0

2x2kþ1fðxÞF l1xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ l2x2

p !

dx� 2

Z 1

0

x2kþ1fðxÞdx

¼ 2akðl1; l2Þ � 2kþ1Gð1þ kÞ=ffiffiffiffiffiffi2p

p: &

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In deriving further properties of moments of the SGN distribution,the next result turns out to be useful:

Lemma 1. For any l1 > 0, l2 > 0 and k ¼ 0; 1; 2; . . . ,

akð0; 0Þ � akðl1; l2Þ � 2akð0; 0ÞF l1ffiffiffiffiffil2

p� �

ð4:4Þ

and

akð0; 0Þ � akðl1; l2Þ � akðl1; 0Þ � 2akð0; 0Þ: ð4:5Þ

Proof. Since F is an increasing function, we have for any l1 > 0 andu > 0 that

1

2� F

l1ffiffiffiu

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ l2u

p� �

¼ Fl1ffiffiffiffiffil2

pffiffiffiffiffiffiffil2u

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ l2u

p� �

� Fl1ffiffiffiffiffil2

p� �

and the proof of (4.4) follows by combining this last set of inequalitieswith (4.1) and using the fact that

Z 1

0

uðkþ1Þ�1

2kþ1Gðk þ 1Þ e�u=2 du ¼ 1:

The argument to show (4.5) is very similar, but now using

1

2� F

l1ffiffiffiu

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ l2u

p� �

� Fðl1ffiffiffiu

p Þ: &

It can be easily seen that the inequalities in (4.4) and (4.5) arereversed when l1 < 0. Therefore, (4.4) and (4.3) imply that EðXÞ > 0if and only if l1 > 0. This property is generalized next.

Proposition 7. If X � SGNðl1; l2Þ and Y � SNðl1Þ then

jEðXrÞ j � jEðYrÞ j for all l1 2 R; l2 > 0 and r ¼ 1; 2; . . .

Proof. The equality is trivially true for all r when l1 ¼ 0. When r is evenit is easily seen from Proposition 4 that EðXrÞ ¼ EðYrÞ. When r is odd,let us first consider the case where l1 > 0. By (4.3)

EðXrÞ ¼ 2fa2ðr�1Þðl1; l2Þ � a2ðr�1Þð0; 0Þg and

EðYrÞ ¼ 2fa2ðr�1Þðl1; 0Þ � a2ðr�1Þð0; 0Þg

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and the result follows from (4.5). Finally, the case l1 < 0 is handled in asimilar way. &

Remark 2. As an immediate consequence of Proposition 7 we have thatif X � SGNðl1; l2Þ and Y � SNðl1Þ then VarðY Þ � VarðX Þ � 1 for alll1 2 R and l2 > 0. For the special case X � SCNðl1Þ the skewnessand kurtosis coefficients

ffiffiffiffiffib1

pand b2 verify

ffiffiffiffiffib1

p¼ 0:5ð4� pÞ E2ðXÞ

VarðXÞ� �3=2

; b2 ¼ 2ðp� 3Þ E2ðXÞVarðXÞ� �2

: ð4:6Þ

Considering all the possible values for l1 we get

�0:376 �ffiffiffiffiffib1

p� 0:376 and 3:0000 � b2 � 4:332: ð4:7Þ

If X � SNðlÞ the corresponding intervals are shown to be

�0:9953 �ffiffiffiffiffib1

p� 0:9953 and 3:0000 � b2 � 3:8692: ð4:8Þ

Thus, the skew-curved normal model provides a wider range of kurtosisbut less skewness than the skew-normal model. On the other hand, wefind that the SGN family provides identical skewness range as the SNcase, while allowing for kurtosis ranging over a slightly wider intervalthan the SCN distribution (data not shown). This is attained when l2takes on values somewhat larger than l21. However, for large values ofl1 the skewness and kurtosis of the SGNðl1; l2Þ and SNðl1Þ distributionsexhibit nearly identical behavior.

The moment generating function (MGF) of X � SGNðl1; l2Þ can beeasily obtained as

MX ðtÞ ¼ 2et2=2E F

l1ðY þ tÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ l2ðY þ tÞ2

q0B@

1CA

264

375; ð4:9Þ

where Y � Nð0; 1Þ. An alternative representation is derived next.

Proposition 8. Let W � Nða; b2Þ be independent of Z � Nðc; d2Þ. Then

E FcWffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ d2W 2p� �� �

¼ E½FðWZÞ�:

In particular, if U and V are i.i.d. Nð0; 1Þ then E F UVð Þ½ � ¼ 1=2.

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Proof.

E½FðWZÞ� ¼ 1

bd

Z 1

�1

Z 1

�1

Z w

�1zfðzuÞf z� c

d

� �f

w� a

b

� �du dz dw

¼ 1

b

Z 1

�1

Z cwffiffiffiffiffiffiffiffiffi1þd2w2

p

�1fðxÞf w� a

b

� �dx dw

¼ E FcWffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ d2W 2p� �� �

: &

Proposition 9. Let X � SGNðl1; l2Þ. Then

MX ðtÞ ¼ 2et2=2E½FðWZÞ�;

where W � Nðt; 1Þ is independent of Z � N l1; l2ð Þ.

Proof. Let Y � Nð0; 1Þ and W ¼ Y þ t. Using (4.9) we obtain

MX ðtÞ ¼ 2et2=2E F

l1Wffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ l2W 2

p !" #

;

and the result follows from Proposition 8. &

5. LOCATION-SCALE EXTENSION

The same basic principle used for the definition of (2.1) can beextended to a location-scale family.

Definition 2. The location-scale skew-generalized normal distribution isdefined as that of Z ¼ mþ sX , where X � SGNðl1; l2Þ, m 2 R ands > 0. Its density is given by

f ðz j hÞ ¼ 2

sf

z� ms

� �F

l1ðz� mÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þ l2ðz� mÞ2

q0@

1A; ð5:1Þ

where h ¼ ðm; s; l1; l2Þ. We denote this by Z � SGNðm; s; l1; l2Þ. In thespecial case l2 ¼ l21 we say that X follows a location-scale skew-curvednormal distribution and denote this by X � SCNðm; s; l1Þ.

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When l1 ¼ 0, the family of densities in (5.1) becomes the Nðm; s2Þdistribution. Additionally, when l2 ¼ 0, (5.1) coincides with theSNðm; s2; l1Þ density, i.e., the distribution of mþ sA with A � SNðl1Þ.A straightforward stochastic representation of the location-scale SGNdistribution is obtained as an extension of Proposition 5.

Proposition 10. If Z jY ¼ y � SNðm; s; yÞ and Y � N l1; l2ð Þ thenZ � SGNðm; s; l1; l2Þ, where SNðm; s; yÞ is the location-scale skew-normaldistribution.

The stochastic representation stated by Proposition 10 results in aninteresting invariance property of the skew-normal distribution undernormal sampling models. The same type of invariance property alsoholds for the skew-curved normal distribution, as we show next.

Proposition 11. Let Z jY ¼ y � SCNðm; s; yÞ and Y � N 0; t2

. Then

Z � Nðm; s2Þ and Y jZ ¼ z � SCN0; t2; t=sðz� mÞð Þ2.

Proof. Note first that

fZðzÞ ¼ 2

st

Z 1

�1f

z� ms

� �F

yðz� mÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þ y2ðz� mÞ2

q0B@

1CAf

y

t

� �dy

¼ 1

sf

z� ms

� �Z 1

�12fðtÞF tðz� mÞtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s2 þ t2ðz� mÞ2t2q

0B@

1CAdt

¼ 1

sf

z� ms

� �:

To prove the other part observe that

fY jZðyÞ ¼2st f

z�ms

F yðz�mÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s2þy2ðz�mÞ2p� �

f yt

1s f

z�ms

¼ 2

tf

y

t

� �F

ts ½ðz� mÞ�yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

t2 þ t2s2 ½ðz� mÞ�2y2

q0B@

1CA: &

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The result of Proposition 11 can be particularly useful in the contextof Bayesian inference. Unlike the usual conjugate models, we have shownthat for a skew-curved normal likelihood, a normal prior yields a normalmarginal and a skew-curved normal posterior. Finally, the next resultshows that a similar invariance property is also true for the skew-generalized normal distribution, but no learning from the data is possiblefor the second parameter under this particular choice of prior.

Proposition 12. Let Z j ðY1 ¼ y1;Y2 ¼ y2Þ � SGNðm; s; y1; y2Þ, where thejoint distribution of ðY1;Y2Þ is defined as Y1 jY2 ¼ y2 � N 0; t2ðy2Þ

and

Y2 � fY2. Then Z and Y2 are independent, Z � Nðm; s2Þ, and

fY1jZðy1Þ ¼Z 1

0

2

tðy2Þfy1

tðy2Þ� �

Ftðy2Þðz� mÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þ y2ðz� mÞ2

q y1tðy2Þ

0B@

1CA fY2

ðy2Þdy2:

Proof. Note first that

fZjY2ðzÞ ¼

Z 1

�1

2

sf

z� ms

� �F

y1ðz� mÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þ y2ðz� mÞ2

q0B@

1CA 1

tðy2Þfy1

tðy2Þ� �

dy1

¼ 1

sf

z� ms

� �Z 1

�12fðtÞF tðy2Þðz� mÞtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s2 þ y2ðz� mÞ2q

0B@

1CAdt

¼ 1

sf

z� ms

� �:

Since this conditional density does not depend on y2, we have that Z andY2 are independent, with Z � Nðm; s2Þ. It is now easy to see that

fY1jZ;Y2ðy1Þ ¼ 2

tðy2Þfy1

tðy2Þ� �

Ftðy2Þðz� mÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þ y2ðz� mÞ2

q y1tðy2Þ

0B@

1CA;

that is,

Y1 j ðY2 ¼ y2;Z ¼ zÞ � SN 0; t2ðy2Þ; tðy2Þðz� mÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þ y2ðz� mÞ2

q0B@

1CA;

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and the rest of the proof follows by noting that the independence of Zand Y2 implies

fY1jZðy1Þ ¼Z

fY1jY2;Zðy1ÞfY2ðy2Þdy2: &

The above proof makes it clear that the inability of learning about Y2

from Z is due precisely to their independence. In fact, the above calcula-tions imply that

fY1;Y2jZðy1; y2Þ ¼ fY1jY2;Zðy1ÞfY2ðy2Þ:

Of course, this is only true under the zero–mean normality assumptionfor the conditional prior fY1jY2

ðy1Þ and the fact that this depends on they2 values only through the scale parameter.

6. DATA ILLUSTRATION

We consider now data concerning the heights (in centimeters) of 100Australian athletes. The data have been previously analyzed in Cook andWeisberg (1994) and are available for download at http:==azzalini.stat.unipd.it=SN=index.html. Table 1 shows summary statistics for thesedata. Note that the sample skewness is not included in the range sup-ported by the SCNðl1Þ family and the same holds for the sample kurtosiswith respect to the SNðl1Þ distribution.

We estimate parameters by numerically maximizing the likelihoodfunction

log LðhÞ ¼ � n

2log

ps2

2� 1

2s2Xni¼1

zi � mð Þ2

þXni¼1

log Fl1ðzi � mÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s2 þ l2ðzi � mÞ2q24

35

with respect to the components of h ¼ ðm; s; l1; l2Þ. The results aresummarized in Table 2, considering the full SGNðl1; l2Þ model and threespecial cases: normal, skew-normal and skew-curved normal.

Table 1. Summary statistics for the heights of 100 Australian athletes.

n Z S2ffiffiffiffiffib1

pb2

100 174.594 67.934 �0.568 4.321

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Table 2. Maximum likelihood parameter estimates for the heights of Australianathletes data under the SGN model and three particular sub-models: Normalðl1 ¼ l2 ¼ 0Þ, skew-normal ðl2 ¼ 0Þ and skew-curved normal ðl2 ¼ l21Þ.Parameter estimates N SN SCN SGN

m 174.594 174.5834 180.751 170.320s2 67.255 67.255 105.160 85.518l1 – 0.0016 �4.628 4.381l2 – – – 24.184Log-likelihood �352.318 �352.318 �351.066 �347.239

Figure 3. Histogram of heights of 100 Australian athletes. The lines representdistributions fitted using maximum likelihood estimation: SGNðmm; ss; l1l1; l2l2Þ (solidline); SCNðmm; ss; l1l1; ll21Þ (dotted line); and SNðmm; ss; l1l1Þ (dashed line), which in thiscase is almost indistinguishable from the Nðmm; ssÞ density.

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These summaries illustrate a key point of the SGN model. Theflexibility and increased skewness and kurtosis ranges were able tocapture features of the data that the other models missed. For instance,it is clear that fitting a SNðl1Þ model to these data would be inadequatedue to the sample kurtosis value. And this is reflected in that the fittedskew-normal is almost indistinguishable from a Normal model. Thesepoints are further illustrated in Fig. 3, where a histogram of the data isplotted together with the fitted densities.

Finally, it is possible to carry out asymptotic likelihood ratio tests tocompare the SGN model against the sub-models mentioned above.Doing so we conclude that the SGN model is the most appropriate forthe particular example analyzed here (data not shown).

ACKNOWLEDGMENTS

The first author is partially supported by grant FONDECYT1040865. The second author is partially supported by grant MECESUPPUC-0103. The third author is partially supported by grant FONDE-CYT 1020712.

REFERENCES

Arellano-Valle, R. B., Genton, M. G. (2003). On Fundamental SkewDistributions, Technical Report; Institute of Statistics, NorthCarolina State University.

Azzalini, A. (1985). A class of distributions which includes the normalones. Scand. J. Statist. 12:171–178.

Azzalini, A. (1986). Further results on a class of distributions whichincludes the normal ones. Statistica 46:199–208.

Azzalini, A., Dalla Valle, A. (1996). The multivariate skew-normaldistribution. Biometrika 83:715–726.

Branco, M., Dey, D. (2001). A general class of multivariate ellipticaldistributions. J. Multivariate Anal. 79:99–113.

Cook, R. D., Weisberg, S. (1994). An Introduction to Regression Graphics.New York: Wiley.

Ellison, B. (1964). Two theorems for inferences about the normaldistribution with applications in acceptance sampling. J. Amer.Statist. Assoc. 59:89–95.

Gilks, W. R., Wild, P. (1992). Adaptive rejection sampling for Gibbssampling. J. Appl. Statist. 41:337–348.

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Henze, N. (1986). A probabilistic representation of the skew-normaldistribution. Scand. J. Statist. 13:271–275.

Tanner, M. A. (1996). Tools for statistical inference. In: Methods for theExploration of Posterior Distributions and Likelihood Functions.3rd ed. New York: Springer-Verlag.

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