ON A CLASS OF SKEWED DISTRIBUTIONS GENERATED BY A MIXING MECHANISM

Transcription

1 Fa East Jounal of Mathematical Sciences (FJMS) 5 Pushpa Publishing House, Allahaba, Inia Publishe Online: Novembe 5 Volume 98, Numbe 7, 5, Pages ISSN: ON A CLASS OF SKEWED DISTRIBUTIONS GENERATED BY A MIXING MECHANISM Hea-Jung Kim a,, Sulki Pak a an Hee Kyung Yoo b a Depatment of Statistics Dongguk Univesity Seoul, Koea kim3hj@ongguk.eu b Depatment of Compute Engineeing Kangwon National Univesity Samcheok-si, Ganwono Koea Abstact This pape consies a mixing mechanism that geneates a skewe istibution to intouce a class of mixe two-piece skew-nomal (MTN) istibutions with a ensity shape that is both asymmetic an uni/bimoal. Some inteesting popeties, along with an efficient estimation of the MTN istibution, ae heeby stuie. The salient featues of the istibution ae its mathematical tactability, statistical applicability an istibutional flexibility. Thus, it is potentially elevant in pactical applications because a few istibutions ae available fo ealing with uni/bimoal ata. An application of the istibution is illustate though the use of one example. Receive: July 3, 5; Accepte: Septembe, 5 Mathematics Subject Classification: 6E, 6E5, 6E5. Keywos an phases: skewe istibution, skewing mechanism, stochastic epesentation, uni/bimoal ata. Coesponing autho Communicate by K. K. Aza

2 858 Hea-Jung Kim, Sulki Pak an Hee Kyung Yoo. Intouction Suppose F is a family of ensities that ae symmetic about an G is an absolutely continuous istibution function such that G F. Using such istibutions, Azzalini [4] popose a skewing mechanism that geneates a skewe ensity of the fom f A ( z θ) = f ( z) G ( θz), z, () ; R whee f F an θ R is a skewing paamete. As a special case of (), Azzalini [4] intouce the skew-nomal ensity by setting f ( z) = φ( z) an G ( θz) = Φ( θz) an efee to the istibution of the anom vaiable Z as a skew-nomal with paamete witten as Z ~ SN( θ). Hee φ ( ) an Φ () enote the N (, ) ensity an the istibution function, espectively. In pactical situations whee a ata exhibits unimoal empiical istibution, but with some skewness as iscusse by Hill an Dixon [7], the SN ( θ) istibution is suitable fo the ata (see Anol et al. [] an Chen et al. [5] among othes). Meanwhile, Kim [9] popose a symmetic ensity f K of the fom f K ( z; θ) = aθ f ( z) G( θ z ), z R () θ that also belongs to F, whee a = f ( z) G ( θz) z. When f () an G () ae equal to those of N (, ), fk euces to the ensity of a two-piece skew-nomal (TN) anom vaiable Z with a istibution law enote by Z ~ TN( θ). Kim [9] has peviously povie a systematic teatment an a pactical application of the TN ( θ) istibution. Anol et al. [] applie Azzalini s skewing mechanism () to f K F in oe to consie a skewe vesion of the TN ( θ) istibution (extene two-piece skew-nomal istibution). See Anol et al. [3] an Jamalizaeh et al. [8] an efeences theein fo plenty of eseach into skewing the TN ( θ) istibution via Azzalini s skewing mechanism.

3 On a Class of Skewe Distibutions Geneate by a Mixing 859 The objective of this pape is to intouce yet anothe skewing mechanism that is applicable to a base ensity f F. This mechanism geneates a skewe istibution by mixing the Azzalini type ensity () to the Kim type ensity (). Then we popose a new class of mixe two-piece skew-nomal (MTN) istibutions by using the mixing mechanism. Futhemoe, this pape consies some inteesting popeties of the class an it also exploes how infeences might be mae by using the MTN istibution. The inteest in stuying the class comes fom both the theoetical an the applie iections. On the theoetical sie, it povies a simple mechanism fo geneating a skewe istibution othe than that of the Azzalini type an the class enjoys a numbe of new fomal popeties that can also be applie to those of the SN ( θ) an TN ( θ) istibutions. Fom the applie point of view, the istibution povies a class of flexible moels that enable us to eal with vaious types of empiically skewe unimoal an bimoal ata... The MTN istibution. The Class of Distibutions The following lemma is useful in oe to achieve the pobabilistic eivation of the MTN istibution an to investigate its istibutional popeties. Lemma. Let f F an G be an absolutely continuous istibution such that G F. Then, fo any eal θ an ε ( < ε < ), f M ( z; ε) = a f ( z) { εg ( θ z ) + ( ε) G ( θz) } (3) is an asymmetic ensity fo z R, whee ε ε a θ, = f( z) { εg( θ z ) + ( ε) G( θz) } z. (4) Poof. The skewing function εg( θ z ) + ( ε) G( θz) multiplying the symmetic ensity f is not symmetic about, because εg ( θ z ) +

4 86 Hea-Jung Kim, Sulki Pak an Hee Kyung Yoo ( ε) G ( θz) = G( θz) fo z >, while ε ( θ z ) + ( ε) G ( θz) = ε + G ( ε) G( θz) fo z <. The nomalizing constant (4) is immeiate by using (3). ~ It can be easily seen that the ensity (3) is a mixtue of () an () ensities, i.e., π fk ( z θ) + π f A( ; θ) with the mixing popotion = ; z π = εa ε aθ. Futhemoe, (3) efines a ich class of skewe moels. Fo example, when F enotes the family of ensities of π C = F : N(, κ ( λ)), λ ~ π( λ) with κ( λ) >, an π( λ) λ =, (4) leas to skewe an heavy-taile moels. Notice that C enotes a class of scale mixtue of nomal istibutions (see Chen et al. [5] fo etails). Fom now on, we will iscuss in some etail a moel involving nomal components to illustate the potential of the skewe ensity (3). Definition. Let a anom vaiable Y have a ensity function f ( y; ε) given by f ( y; ε) = c, φ( y) { εφ( θ y ) + ( ε) Φ( θy) }, y R. (5) θ ε Then we say that Y is a mixe two-piece skew-nomal (MTN) anom vaiable with paametes θ an ε. We shall wite the istibution law of Y as ε Y ~ MTN( ε), whee ε an c θ, ε = + tan θ. π ε We see that c θ, ε = E[ εφ( θ Y ) + ( ε) Φ( θy )] = + tan θ by Kim π [9]. Figue epicts the shapes of the ensity (5) fo iffeent paamete values, an the figue shows that the ensity inclues both unimoal an bimoal cases that can be expecte to fit a wie vaiety of ata configuations than those of SN ( θ) by Azzalini [4] an TN ( θ) by Kim [9] istibutions.

5 On a Class of Skewe Distibutions Geneate by a Mixing 86 Figue. Vaious shapes of MTN ( ε) istibution; (a) unimoal case whee θ < ; (b) bimoal case whee θ > an ε >.5; (c) unimoal case whee θ > an ε.5; () ensity of TN ( θ) = MTN( ε = ) an SN ( θ) = MTN( ε = ). Let F MTN ( y) be the istibution function of Y ~ MTN( ε), some algebas using the istibution functions of SN ( θ) an TN ( θ) lea to that of MTN ( ε) given by ε c ε tan θ + Φ y T y, θ, if y, π F MTN ( y) = (6) c ( ε ) ( θ) + Φ ε T y, y, if y <. Following stochastic epesentation of the MTN ( ε) anom vaiable is useful fo simulation puposes an fo unestaning the intinsic stuctue of the MTN istibution as well. Theoem. Let U ~ TN( θ) an V ~ SN( θ) be inepenent of S ~ Be ( p ε ), a Benoulli anom vaiable. Then

6 86 Hea-Jung Kim, Sulki Pak an Hee Kyung Yoo Y S S = U V ~ MTN( ε), whee p ε = εc ε cθ an c θ = + tan θ. π Poof. As given by Kim [9] an Azzalini [4], let f TN ( u) an f SN ( v) be the ensity functions of U an V, espectively. Suppose h ( y S = s) is the conitional ensity of Y given S = s, =,, U S V S is s= s the ensity function of h ( y S = s) p( S = s) = f ( y) p + f ( y)( p ) TN ε SN ε = c ε φ ( y) { εφ( θ y ) + ( ε) Φ( θy) } Y = fo y R, because it is staightfowa to see that p = ( ε) c ε ε an p [, ]. This is equivalent to (5). ~ ε The stochastic epesentation inicates that the MTN ( ε) istibution is a mixtue of the TN ( θ) an SN ( θ) istibutions with the Be mixing istibution. Theoem also povies the following coollay: p ε Coollay. If Y ~ MTN( ε), then the stochastic epesentation of the istibution of W S S = Y is W = U ( V ) with its ensity given by f ( w; ε) = c, φ( w) { εφ( θ w ) + ( ε) Φ( θw) }, w R. θ ε Poof. The statement is staightfowa to note that V ~ SN( θ), an U ~ TN( θ) which is a symmetic istibution about. ~ Futhemoe, Theoem povies a metho to geneate a anom vaiable with istibution MTN ( ε). We then, espectively, efe to Kim [9] an Henze [6] fo the etaile methos on geneating U ~ TN( θ) an V ~ SN( θ) anom vaiables. Fo applications, it will be convenient to

7 On a Class of Skewe Distibutions Geneate by a Mixing 863 a the location an scale paametes to the istibution Y ~ MTN( ε). If X = λ + λ, whee λ R an λ, then we will wite X ~ Y MTN ( λ, λ, ε) o at times X ~ MTN( λ), whee λ enotes the vecto of the paametes ( λ, λ, ε). Theefoe, the stochastic epesentation of the anom vaiable X is by Theoem. > S ( S X = λ + λ ) U V ~ MTN( λ, λ, ε) When we moify the theoem by assuming that U ~ TN( θ) an V ~ SN ( β) ae inepenent of S Be, then the anom vaiable ~ p θ, ε Y = U S V S yiels an extene MTN istibution, witten Y ~ EMTN( θ, β, ε) with an aitional skewing paamete β R, an its ensity is given by.. Moments f E ( y; β, ε) = c, φ( y) { εφ( θ y ) + ( ε) Φ( βy) }, y R. (7) θ ε The moments of a tuncate SN ( θ) anom vaiable Z, tuncate by the lowe tuncation point, is useful to calculate those of the MTN anom vaiables. The pobability ensity function of the tuncate anom vaiable Z is f TSN ( z) = cθφ( z) Φ( θz), z >, whee c θ = + tan θ. Using a special integation technique, Anol et π al. [] eive a ecusive fomula fo the moments of Z:, c θ E[ Z ] = ( θ) = θ +, π + θ θc ( ) ( θ) + π( + θ ) θ Γ[ ], =, =,. (8)

8 864 Hea-Jung Kim, Sulki Pak an Hee Kyung Yoo The stochastic epesentation in Theoem an the moments (8) can be utilize to povie a ecusive fomula which calculates the moments of a anom vaiable Y ~ MTN( ε). Coollay. Let Y ~ MTN( ε). Then the moments of Y ae as follows. If is an o numbe, then c ε ε E[ Y ] = ( + ( ) ) ( θ) c θ ( ) + ( ε) θ! ( ) ( ) π + θ ν = ν ν! ( θ). ( ν + )!(( ) ν)! If is an even numbe, then c ε ε E[ Y ] = ( ) + θ + c θ ( ε) Γ[ ( + ) ], π whee ( θ) is the ecusive fomula given by (8). Poof. Let U ~ TN( θ) an V ~ SN( θ), the stochastic epesentation in Theoem gives S S ε ε E[ Y ] = EE[ U V S = s] = p E[ U ] + ( p ) E[ V ], whee p = εc c, p = ( ε) c, an ε ε θ θ ε ε E[ U ] = c u φ( u) Φ( θ u ) u ( ) = + c θu φ( u) Φ( θu) u = ( + ( ) ) ( θ) by (9). Now the moments of V ~ SN( θ) ae as follows. If is even, then E[ V ] = Γ[ ( + ) ] π, because V χ. If is o, then Henze [6] showe that ~

9 On a Class of Skewe Distibutions Geneate by a Mixing 865 θ! ( ) [ ] ν! θ E V =. ~ π + θ ν= ( ν + )!(( ) ν)! In fact, the moment geneating function of Y ~ MTN( ε) exists fo t R. It is t M Y ε () t = c e εφ( z) Φ( θ z + t ) + ( ε) φ( z) Φ( θz + θt) z ν θt θ ε I( θ ) T, + ( ε) Φ + θ θ + θ t t = c εe by using the esult by Kim [9]. One can easily check that both Coollay an the moment geneating function M Y () t yiel the same moments of Y ~ MTN( ε). The fist two moments ae ( ε) θ E [ Y ] = c θ, ε an E [ Y ] π( + θ ) = + c θ, ε εθ π( + θ ). The moments of X ~ MTN( λ, λ, ε) can be obtaine by using the elationship X = λ + λ. Fo =,,,..., we have Y.3. Distibutional popeties [ ] λ j = λ j j E X E[ Y ] j j=. It is clea that the MTN ( ε) ensity in (5) is continuous at fo evey θ an ε an the ensity can be eithe unimoal o bimoal. Howeve, it is not iffeentiable at unless ε is. Futhe, the MTN ( ε) istibution is flexible enough to stictly inclue seveal well known istibutions. Popety. The istibutional law MTN( ) = SN( θ). Popety. The istibutional law MTN( ) = TN( θ).

10 866 Hea-Jung Kim, Sulki Pak an Hee Kyung Yoo Popety 3. The istibutional law MTN(, ) = MTN(, ) = N(, ). Popety 4. Let Y ~ MTN( ) an Y ~ MTN(, ). Then ~ χ, a chi-squae istibution with egee of feeom. Y = Y Popety 5. As θ, f ( y; θ, ε) φ( y). In contast, as θ, f ( y; ε) φ( y) I( y < ), a ensity of the tuncate stana nomal with the uppe tuncation point at y =. Popety 6. Fo θ > an ε.5, the ensity (5) is bimoal, i.e., in each egion of y (, ] an y [, ε), log f ( y; ε) is a concave function of y. Popety 7. Fo θ > an ε >.5, two moes of the ensity (5) ae locate at y = y ( y < ) an y = y ( y > ), whee y an y ae solutions of ( ε) θφ( θy ) θφ( θy ) y = an y ε + ( ε) Φ( θy ) = Φ( θy ), espectively. Popety 8. Fo θ <, the single moe of the ensity (5) is locate at y = y, because f ( y; ε) < fo y > y y an f ( y; ε) > fo y y < y, whee { y = Min, y } an y is the solution of = y ( ε) θφ( θy ) ε + ( ε) Φ( θy ). Fo ε, we see that the istibution function (6) calculates that F ( ) = fo θ =, F ( ) < fo θ > an F ( ) > fo MTN MTN MTN θ <. This inicates the following popety of the MTN ( ε) istibution. Popety 9. The MTN ( ε) istibution is left-skewe fo θ >, while it is ight-skewe fo θ <.

11 On a Class of Skewe Distibutions Geneate by a Mixing Infeence In pactice, one typically woks with the family of istibutions geneate by intoucing location an scale paametes to (5). Let X ~ MTN ( λ, λ, ε). Then the esulting fou paamete family of the ensity will be of the fom: c ε x λ λ θ λ = φ x εφ θ x f x; λ + ε Φ λ λ λ λ fo x R, whee λ = ( λ, λ, λ3, λ4 ) T with λ 3 = θ an λ 4 = ε. Let { X i}, i =,..., n, be a anom sample awn fom the MTN ( λ, λ, ε) istibution. Then the log-likelihoo fo the paametes is n i = i on a single obsevation x. That is, ( λ; x ), whee ( λ; x) is the log-likelihoo fo the paametes base c ε x λ λ; x log + φ log λ λ + log εφ θ n i x λ λ The scoe function is = ( λ; Xi ), whee 3 λ λ λ λ 4 ( ) θ x λ + ε Φ. λ ( λ; X ) = (,,, ) = ( λ; X ) λ. The maximum likelihoo estimato (MLE) λˆ is the solution of λ fo which n the scoe function = ( λ; X ) i i vanishes. Accoing to the asymptotic efficiency of the MLEs, we see that as n, ( ˆ n λ λ) N ( I( λ) ) (9) 4, in istibution, whee I ( λ) is the infomation matix of one sample X. One can use the bbmle package fo the R pogam to calculate λˆ an the

12 868 Hea-Jung Kim, Sulki Pak an Hee Kyung Yoo infomation matix. Fo moel compaison, one may compute the value of Akaike s Infomation Citeion (AIC) fo moels with a iffeent numbe of paametes, an then we select the best fitte moel that yiels the minimum AIC estimate. 4. An Illustative Example In oe to illustate the applicability of the class of the MTN moels that ae intouce in this pape, we consiee a ata set that consists of the heights of Austalian female athletes. This ata set has been use extensively in the liteatue (see, e.g., Jamalizaeh et al. [8]), an is available in the sn package fo R pogam (o html). Table shows the summay statistics fo this ata set. Base on the ata, we estimate the paametes accoing to the Maximum Likelihoo (ML) estimation metho by using the bbmle package fo the R pogam. Table. Desciptive statistics Data set n Mean s.. Skewness Kutosis Female athletes ata In this example, we have fitte five moels to the ata. The five moels consist of the SN, TN, MTN, EMTN efine by the ensity (7), an GSTPSN (genealize skew two-piece skew-nomal) consiee by Jamalizaeh et al. [8]. The GSTPSN ensity is anothe skewe istibution with two skewing paametes whose ensity is given by f G ( x; λ, λ, β, ρ) = c ( β, ρ) x λ θ λ β( λ ) φ x x Φ, ; ρ λ λ λ λ fo x R, whee Φ (, ; ρ) enotes the f of N (,,, ; ρ) an c ( β, ρ) is the nomalizing constant (see Jamalizaeh et al. [8] fo the exact fomula).

13 On a Class of Skewe Distibutions Geneate by a Mixing 869 Table. ML estimation esults Moels λ λ θ β ε ρ ( λ) SN ( λ, λ, θ) TN ( λ, λ, θ) MTN ( λ, λ, ε) EMTN ( λ, λ, β, ε) GSTPSN ( λ, λ, β, ρ) Figue. The cuves epesent ensities fitte by ML metho: MTN ( λˆ, λˆ, θˆ, εˆ ) (black line), EMTN ( λˆ, λˆ, θˆ, βˆ, εˆ ) (black ashe line), SN ( λˆ, λˆ, θˆ ) (gey line), TN ( λˆ, λˆ, θˆ ) (gey ashe line). The ML estimation esults fo the five moels ae summaize in Table. As seen in the table, the AIC citeion avocates the MTN moel as the best fitting moel among the five moels, i.e., the MTN moel achieves the minimum AIC value (AIC value = 7.4) fo all moels une consieation. Thus, we can conclue that MTN ( ε, λ, λ ) moel is the best fitting moel fo the ata set among the five moels une consieation.

14 87 Hea-Jung Kim, Sulki Pak an Hee Kyung Yoo Figue also confims this conclusion. Note that we eliminate the estimate ensity of the GSTPSN istibution fom the figue because it nealy ovelaps with that of the MTN. See Jamalizaeh et al. [8] fo the figue of the estimate GSTPSN ensity fitte to the same ata. 5. Concluing Remaks In this pape, we suggeste a new class of skewe istibutions which is that of the mixe two-piece skew-nomals (MTN) an is useful fo analyzing asymmetic an uni/bimoal ata. In oe to eive the MTN istibution, instea of using the usual Azzalini s metho, we use the skewing mechanism that geneates a mixtue of symmetic TN an asymmetic SN istibutions. This mechanism enables us to have the class of MTN istibutions at han with the following chaacteistics: (i) istibutional flexibility that is sufficient fo the stict inclusion of the SN, TN, an nomal istibutions, (ii) mathematical tactability in stuying the etaile popeties of the MTN istibution, (iii) statistical applicability with a wie ange of shapes of the ensity as escibe in Figue. Fom the point of view that was applie, as illustate in Section 4, a key featue of the MTN moel has a simple fom in contast to othe extene two-piece skew-nomal moels such as the GSTPSN. Futhemoe, it is seen fom Figue that the moel has flexibility an aapts to a wie ange of coefficients of skewness an kutosis. Acknowlegement This eseach was suppote by Basic Science Reseach Pogam though the National Reseach Founation of Koea (NRF) fune by the Ministy of Science, ICT, an Futue Planning (3RAAA479). Refeences [] B. C. Anol, R. J. Beave, R. A. Goenevel an W. Q. Meeke, The nontuncate maginal of a tuncate bivaiate nomal istibution, Psychometica 58 (993),

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