Bayesian Estimation for the Centered Parameterization of the Skew-Normal Distribution

Transcription

1 Revista Colombiaa de Estadística Jauary 2017, Volume 40, Issue 1, pp. 123 to 140 DOI: Bayesia Estimatio for the Cetered Parameterizatio of the Skew-Normal Distributio Estimació bayesiaa para la parametrizació cetrada de la distribució ormal-asimétrica Paulio Pérez-Rodríguez a, José A. Villaseñor b, Sergio Pérez c, Javier Suárez d Socioecoomía Estadística e Iformática-Estadística, Colegio de Postgraduados, Texcoco, México Abstract The skew-ormal (SN) distributio is a geeralizatio of the ormal distributio, where a shape parameter is added to adopt skewed forms. The SN distributio has some of the properties of a uivariate ormal distributio, which makes it very attractive from a practical stadpoit; however, it presets some iferece problems. Specifically, the maximum likelihood estimator for the shape parameter teds to ifiity with a positive probability. A ew Bayesia approach is proposed i this paper which allows to draw ifereces o the parameters of this distributio by usig improper prior distributios i the cetered parametrizatio for the locatio ad scale parameter ad a Beta-type for the shape parameter. Samples from posterior distributios are obtaied by usig the Metropolis-Hastigs algorithm. A simulatio study shows that the mode of the posterior distributio appears to be a good estimator i terms of bias ad mea squared error. A comparative study with similar proposals for the SN estimatio problem was udertake. Simulatio results provide evidece that the proposed method is easier to implemet tha previous oes. Some applicatios ad comparisos are also icluded. Key words: Metropolis-Hastigs Algorithm, Poit Estimatio, Prior Distributio. a PhD. perpdgo@gmail.com b PhD. jvillasr@colpos.mx c PhD. sergiop@colpos.mx d PhD. sjavier@colpos.mx 123

2 124 Paulio Pérez-Rodríguez, José A. Villaseñor, Sergio Pérez & Javier Suárez Resume La distribució Normal Asimétrica (SN) es ua geeralizació de la distribució ormal, icluye u parámetro extra que le permite adoptar formas asimétricas. La distribució SN tiee alguas de las propiedades de la distribució ormal uivariada, lo que la hace muy atractiva desde el puto de vista práctico; si embargo preseta alguos problemas de iferecia. Particularmete, el estimador de máxima verosimilitud para el parámetro de forma tiede a ifiito co probabilidad positiva. Se propoe ua solució Bayesiaa que permite hacer iferecia sobre los parámetros de esta distribució asigado distribucioes impropias e la parametrizació cetrada para el parámetro de localidad y el de escala y ua distribució tipo Beta para el parámetro de forma. Las muestras de las distribucioes posteriores se obtiee utilizado el algoritmo de Metropolis-Hastigs. U estudio de simulació muestra que la moda de la distribució posterior parece ser u bue estimador, e térmios de sesgo y error cuadrado medio. Se preseta tambié u estudio de simulació dode se compara el procedimieto propuesto cotra otros procedimietos. Los resultados de simulació provee evidecia de que el método propuesto es más fácil de implemetar que las metodologías previas. Se icluye tambié alguas aplicacioes y comparacioes. Palabras clave: algoritmo de Metropolis-Hastigs, distribucioes a priori, estimació putual. 1. Itroductio The skew-ormal distributio is a three parameter class of distributio with locatio, scale ad shape parameters, ad it cotais the ormal distributio whe the shape parameter equals zero. A cotiuous radom variable Z is said to obey the skew-ormal law with shape parameter λ R ad it is deoted by SN(λ) if its desity fuctio is: f Z (z; λ) = 2φ (z) Φ(λz)I (, ) (z), (1) where φ( ) ad Φ( ) deote the desity ad distributio fuctios of a stadard ormal variable. If Y is a radom variable defied by Y = ξ + ωz, with ξ R, ω R +, the Y is said to have a skew-ormal distributio with locatio-scale (ξ, ω) parameters ad shape parameter λ, ad it is deoted by Y SN D (ξ, ω, λ). The subscript D idicates the use of the direct parametrizatio (Azzalii 1985). The desity fuctio of Y is: f Y (y; ξ, ω, λ) = 2 1 ( ) [ y ξ ω φ Φ λ ω ( y ξ ω )] I (, ) (y). (2) Revista Colombiaa de Estadística 40 (2017)

3 Bayesia Estimatio for the Skew-Normal Distributio 125 The mea ad variace of desity i (1) are give by: E(Z) = V ar(z) = 1 2 π 2 λ π, 1 + λ 2 λ λ 2. The coefficiet of skewess of Y is give by: γ 1 = κ 3 = 4 π κ 3/2 2 2 (E(Z)) 3 {1 (E(Z)) 2 } = 4 π 3/2 2 ( 2 π ( 1 2 π ) 3 λ 1+λ 2 ) 3/2, λ 2 1+λ 2 where κ 2, κ 3 are the secod ad third degree cumulats, respectively. The rage of γ 1 is ( c 1, c 1 ) where c 1 = 2(4 π)/(π 2) 3/ The maximum likelihood estimatio of parameters is troublesome. Although the likelihood fuctio ca be calculated without much trouble, several problems arise o maximizig it. For example, if ξ = 0, ω = 1, ad all the observatios are positive (or egative), the the likelihood fuctio is mootoe, ad its maximum occur o the upper (lower) boudary of the parameter space, correspodig to a o fiite estimate of λ. I the case of ukow values for the parameters ξ, ω, λ, the problem ca be eve more difficult because there is always a iflexio poit at λ = 0 for the likelihood fuctio, leadig to the Hessia sigular (Chioga 1998, Azzalii & Geto 2008). The secod problem is solved by usig a cetered parametrizatio of the desity fuctio. The first is a ope problem ad some proposals already exist: Sartori s (2006) stad out, who uses a modificatio of the score fuctio to estimate the λ parameter, combied with maximum likelihood estimators of ξ ad ω. The cetered parametrizatio [Azzalii (1985), Azzalii & Capitaio (1999)] is obtaied as follows give Z SN(λ), the radom variable Y, Y = µ + σ ( Z E(Z) V ar(z) ), is said to be a skew-ormal variable with cetered parameters µ, σ, γ 1, E(Y ) = µ ad V ar(y ) = σ 2. I this case the usual otatio is Y SN C (µ, σ, γ 1 ). Here γ 1 is the skewess parameter of both Y ad Z. Oe ca easily chage from oe parametrizatio to aother by usig the idetities i (3): Revista Colombiaa de Estadística 40 (2017)

4 126 Paulio Pérez-Rodríguez, José A. Villaseñor, Sergio Pérez & Javier Suárez ( 2 λ = sg(γ 1 ) ( 2 ξ = µ σ ω = σ { 1 + ) { 1/3 ( ) 2/3 ( ) } 1/2 γ 1 1/ π π + γ 1 2/3 4 π π 1, ) 1/3, (3) 4 π γ 1 ( 2 4 π γ 1 ) 2/3 } 1/2. Based o the Bayesia approach, a solutio was proposed by Liseo & Loperfido (2006), who focused their iferece o λ, cosiderig ξ, ω as uisace parameters i the direct parameterizatio. Wiper, Giró & Pewsey (2008) also studied the problem i the case of the half-ormal distributio. I this paper we propose a ew Bayesia approach based o the cetered parametrizatio to deal with the estimatio of the shape parameter i the skewormal family. The proposed methodology applies MCMC simulatio techiques by usig the Metropolis Hastigs algorithm (Metropolis, Rosembluth, Teller & Teller 1953). From a classical poit of view, there are at least two reasos to cosider the cetered parametrizatio : i) it provides a more practical iterpretatio of the parameters, ad ii) it solves the well kow problems of the likelihood fuctio uder direct parametrizatio. Arellao-Valle & Azzalii (2008) stated that the stadard likelihood based methods ad also the Bayesia methods are problematic whe they are applied to iferece o the parameters i the direct paramaerizatio ear λ = 0. This is due to the fact that the direct paramerizatio is ot umerically suitable for estimatio. The structure of this paper is the followig: I sectio 2 a Bayesia method is proposed to obtai poit estimates for the cetered parameters, which ca be back trasformed usig the equatios i (3) to obtai poit estimates of the direct parameters. Sectio 3 cotais results from a simulatio study cocerig the proposed methodology. Applicatios are preseted i sectio 4. Some coclusios are icluded i sectio Bayesia Estimatio Let Y = (Y 1,..., Y ) be a radom sample from the skew-ormal distributio with parameters µ, σ, ad γ 1, ad suppose we wish to obtai Bayesia estimators for the parameters. Followig Azzalii (1985), Azzalii & Capitaio (1999), ad Pewsey (2000) we propose to use the cetered parametrizatio to obtai Bayesia estimators for the parameters of iterest. We propose the followig prior distributios: Revista Colombiaa de Estadística 40 (2017)

5 Bayesia Estimatio for the Skew-Normal Distributio 127 p(µ) 1, p(σ) 1 σ, ( ) a 1 ( γ1 + c 1 p(γ 1 ) 1 γ ) b c 1 I ( c1,c1)(γ 1 ). 2c 1 2c 1 Note that we have assiged the stadard prior for locatio-scale parameters to µ ad σ (Box & Tiao 1973). I the case of γ 1, we kow that it takes values o ( c 1, c 1 ), so if W Beta(a, b), the the radom variable γ 1 = 2c 1 W c 1 has a desity the kerel of which is show above. The prior desity for γ 1 depeds o two hyperparameters (a ad b) ad could lead to a rich varieties of shapes, just as i the case of the Beta distributio. I this paper, we set a = b = 1, which leads to a uiform distributio o ( c 1, c 1 ), but other values ca be selected, allowig the icorporatio of prior iformatio. The uiform prior is o-iformative, but proper. A ivariat ad o iformative prior for γ 1 could be derived usig a approach similar to that employed by Liseo & Loperfido (2006); however, that approach is complicated from computatioal poit of view because it ivolves umerical itegratio i the -dimesioal space. The priors proposed i this work allow the implemetatio of the well kow Metropolis-Hastigs algorithm i order to sample the posterior desity. As the sample size icreases, the effect of the prior becomes less relevat. The, uder the assumptio of idepedece, the joit prior desity i the cetered parameterizatio is: p(µ, σ, γ 1 ) 1 σ ( γ1 + c 1 2c 1 ) a 1 ( 1 γ ) b c 1 I ( c1,c1)(γ 1 )I (0, )(σ). 2c 1 Applyig Bayes theorem, from (2) ad (3) the posterior joit distributio of µ, σ, γ 1 is: { } p(µ, σ, γ 1 y) f Yi (y i ; µ, σ, γ 1 ) p(µ, σ, γ 1 ). i=1 The implied prior distributio ca be obtaied o the parameters i the origial parameter space by the radom variable trasformatio formulae, for which the iverse trasformatio turs out to be: Revista Colombiaa de Estadística 40 (2017)

6 128 Paulio Pérez-Rodríguez, José A. Villaseñor, Sergio Pérez & Javier Suárez T 1 ) 1/3 ( 2 µ = w 1 (ξ, ω, λ) = ξ + σ 4 π γ 1 [ ( ) ] 2/3 1/2 2 σ = w 2 (ξ, ω, λ) = ω π γ 1 γ 1 = w 3 (ξ, ω, λ) = 4 π 2 ( 2 π ( 1 2 π ) 3 λ 1+λ 2 ) 3/2, λ 2 1+λ 2 ad its Jacobia is [ J = 12 4 π λ 1 + ( λ ) 1 2 ( ) ] 2/3 1/2 2 ( 4 π g (λ) 1 λ 2 ( λ ) ) ( π π 3 2 ( λ ) 1 2 λ + 1 π ( 2 π ) 3 ( λ λ 2 π λ 2 λ λ 2 ) 3/2 1 + λ 2 ) 1 ad, therefore the implied joit prior desity i the origial parametrizatio is give by p (ξ, ω, λ) p µ,σ,γ1 (w 1 (ξ, ω, λ), w 2 (ξ, ω, λ), w 3 (ξ, ω, λ)) J. This implied origial parameterized joit prior desity is quite differet from the oe used by Liseo & Loperfido (2006). To obtai the margial posterior distributios of iterest p(µ y), p(σ y), p(γ 1 y) it is ecessary to use umerical based itegratio as the Markov Chai Mote Carlo techiques. I the case of the Gibbs sampler, it is ecessary to have the complete coditioals p(µ σ, γ 1, y), p(σ µ, γ 1, y), p(γ 1 µ, σ, y) to implemet it; however, their closed forms are ot available, therefore we propose to use the Metropolis-Hastigs algorithm (e.g. Metropolis et al. 1953, Chib & Greeberg 1995). To apply the Metropolis-Hastigs algorithm, the cadidate geeratig distributio has to be selected first. Oe has to be very careful i this step sice a iadequate selectio of such a distributio ca cause the Metropolis-Hastigs algorithm to have a poor performace due to a high rejectio rate. Pewsey (2000) obtaied large sample theory results for the momet s estimators from the cetered parametrizatio. For Y SN C (µ, σ, γ 1 ), let Revista Colombiaa de Estadística 40 (2017)

7 Bayesia Estimatio for the Skew-Normal Distributio 129 β 2 = 3 + τ 4 (π 3), β 3 = 10γ 1 + τ 5 (3π 2 40π + 96)/4, β 4 = 15 { 1 + τ 4 (2π 6) } τ 6 (9π 2 80π + 160)/2, where τ = (2/(4 π)γ 1 ) 1/3. By usig the delta method, Pewsey (2000) obtaied: V ar( µ) = σ 2 /, V ar( σ) = σ 2 (β 2 1)/4 + O( 3/2 ), V ar( γ 1 ) = { 9 6β 2 3γ 1 β 3 + β 4 + γ 2 1(35 + 9β 2 )/4 } / + O( 3/2 ), Cov( µ, σ) = σ 2 γ 1 /2 + O( 3/2 ), Cov( µ, γ 1 ) = σ(β 2 3 3γ 2 1/2)/ + O( 3/2 ), Cov( σ, γ 1 ) = σ {2β 3 γ 1 (5 + 3β 2 )} /4 + O( 3/2 ), where µ, σ, γ 1 are the momet estimators of µ, σ ad γ 1. As a cosequece of the defiitio of the momet estimators, Slutsky s Theorem ad the Cetral Limit Theorem, the joit distributio of the estimators is asymptotically trivariate ormal. We ca use these results to select the multivariate ormal distributio as the cadidate geerator i the Metropolis-Hastigs algorithm. To star the Metropolis-Hastigs algorithm, the trivariate ormal distributio N 3 ( θ, Σ) is used as a proposal desity, where: Σ = V ar( µ) Cov( µ, σ) Cov( µ, γ 1 ) Cov( µ, σ) V ar( σ) Cov( σ, γ1 ) Cov( µ, γ1 ) Cov( σ, γ1 ), (4) V ar( γ1 ) ad θ = ( µ, σ, γ 1 ) are the momet s estimators of (µ, σ, γ 1 ) i the cetered parametrizatio. The variaces ad covariaces i (4) are obtaied by pluggig i the parameter estimates ito the variace ad covariace formulae give by Pewsey (2000). It is importat to ote that Σ i (4) ca be adjusted i order to mimic the posterior distributio (Carli & Louis 2000). I this paper, routies were writte i the R software (R Core Team 2015) to estimate the posterior distributios of µ, σ, γ 1. These routies are desiged to update the covariace matrix after havig obtaied s Markov Chai Mote Carlo samples from the parameters of iterest. I this work we set s = 1, 000. The variace covariace matrix is estimated as: Σ = 1 s s ( θj θ ) ( θ j θ ), j=1 where j idexes the Markov Chai Mote Carlo samples, θ = 1 j=1 θ j ad θ j is the j-th Markov Chai Mote Carlo sample. Oce the variace covariace-matrix Revista Colombiaa de Estadística 40 (2017)

8 130 Paulio Pérez-Rodríguez, José A. Villaseñor, Sergio Pérez & Javier Suárez is adjusted, the N 3 ( θ, Σ) is used as the proposal distributio. Due to the fact that σ is positive ad γ 1 is restricted to the iterval ( c 1, c 1 ), iadmissible values ca be avoided simply by rejectig samples that do ot meet these coditios. I order to verify the covergece of the Metropolis-Hastigs algorithm, the Gelma & Rubi (1992) covergece test is used. 3. Simulatio Study I this sectio, we preset results from a simulatio study cocerig the proposed Bayesia procedure described i the previous sectio. Samples of size = 20, 50 ad 100 were simulated from SN C (µ, σ, γ 1 ) for differet values of µ, σ ad γ 1. A compariso with the method of momets (Pewsey 2000) ad modified maximum likelihood estimators (Sartori 2006) i terms of the bias ad the mea squared error is also icluded. For this compariso, the estimates i the direct parameterizatio were trasformed to the cetered parameterizatio. Next, the algorithm used to estimate the bias ad mea squared error usig the Bayesia approach is briefly described. 1. Set i = Set (µ, σ, γ 1 ). 3. Geerate a radom sample of size from SN C (µ, σ, γ 1 ). 4. Estimate (µ, σ, γ 1 ) usig the method of momets. 5. Obtai the iitial variace-covariace estimator usig the Pewsey s (2000) result. a) Geerate a trivariate ormal sample usig the results of steps 4) ad 5). b) Reject the samples that do ot meet the coditios: σ > 0 ad γ 1 ( c 1, c 1 ), or keep the samples that meet the coditios to select oe with the Metropolis algorithm. c) Repeat steps a) ad b), B = 30, 000 times; update the variace-covariace matrix of the proposed distributio after 1,000 iteratios. d) Obtai the margial posterior desities of (µ, σ, γ 1 ); discard the first 25,000 iteratios (bur-i). e) Obtai the mode of the margial posterior desity of (µ, σ, γ 1 ) usig the results i step d), say (ˆµ i, ˆσ i, ˆγ 1,i ). 6. Set i = i Repeat steps 3 to 6, 5,000 times. 8. Obtai the mea of ˆµ 1,..., ˆµ 5,000, ˆσ 1,..., ˆσ 5,000, ˆγ 1,1,..., ˆγ 1,5000. Revista Colombiaa de Estadística 40 (2017)

9 Bayesia Estimatio for the Skew-Normal Distributio Compute the bias ad mea squared error of ˆµ, ˆσ ad ˆγ 1. The algorithm to estimate the bias ad mea squared error for the estimators obtaied usig the method of momets is as follows: 1. Set i = Set (µ, σ, γ 1 ). 3. Geerate a radom sample of size from SN C (µ, σ, γ 1 ). 4. Obtai the momet estimators of (µ, σ, γ 1 ) usig Pewsey s (2000) results, say (ˆµ i, ˆσ i, ˆγ 1,i ). 5. Set i = i Repeat steps 3 to 6, 5,000 times. 7. Obtai the mea of µ 1,..., µ 5,000, σ 1,..., σ 5,000, γ 1,1,..., γ 1, Compute the bias ad mea squared error of µ, σ ad γ 1. The algorithm to estimate the bias ad mea squared error for the estimators obtaied usig the modified maximum likelihood (Sartori 2006) is aalogous to that used i the momet s estimator case. However, the estimates i the direct parameterizatio were trasformed to the cetered parameterizatio before computig the bias ad the mea squared error. Tables 1-3 preset the bias ad mea squared error obtaied whe usig the Bayesia approach, the method of momets, ad the modified maximum likelihood method respectively. I all the cases cosidered the bias ad the mea squared error decrease as the sample size icreases for the three methods. The bias for the locatio ad scale parameters are small i geeral. Sartori (2006) poited out that the bias for the shape parameter i the skew-ormal distributio has a lower asymptotic bias tha the maximum likelihood estimator. Revista Colombiaa de Estadística 40 (2017)

10 132 Paulio Pérez-Rodríguez, José A. Villaseñor, Sergio Pérez & Javier Suárez Table 1: Bias ad mea squared error for Bayesia poit estimators. The estimates for µ, σ, ad γ 1 were obtaied from 5,000 simulated samples of size from SN C(µ, σ, γ 1). The mode of the margial posterior desities was used as a poit estimate of the true parameters. µ = , σ = , γ 1 = E(ˆµ µ) E(ˆσ σ) E(ˆγ 1 γ 1 ) E(ˆµ µ) 2 E(ˆσ σ) 2 E(ˆγ 1 γ 1 ) µ = , σ = , γ 1 = E(ˆµ µ) E(ˆσ σ) E(ˆγ 1 γ 1 ) E(ˆµ µ) 2 E(ˆσ σ) 2 E(ˆγ 1 γ 1 ) µ = , σ = , γ 1 = E(ˆµ µ) E(ˆσ σ) E(ˆγ 1 γ 1 ) E(ˆµ µ) 2 E(ˆσ σ) 2 E(ˆγ 1 γ 1 ) µ = , σ = , γ 1 = E(ˆµ µ) E(ˆσ σ) E(ˆγ 1 γ 1 ) E(ˆµ µ) 2 E(ˆσ σ) 2 E(ˆγ 1 γ 1 ) µ = , σ = , γ 1 = E(ˆµ µ) E(ˆσ σ) E(ˆγ 1 γ 1 ) E(ˆµ µ) 2 E(ˆσ σ) 2 E(ˆγ 1 γ 1 ) µ = , σ = , γ 1 = E(ˆµ µ) E(ˆσ σ) E(ˆγ 1 γ 1 ) E(ˆµ µ) 2 E(ˆσ σ) 2 E(ˆγ 1 γ 1 ) Revista Colombiaa de Estadística 40 (2017)

11 Bayesia Estimatio for the Skew-Normal Distributio 133 Table 2: Bias ad mea squared error for modified maximum likelihood estimators (Sartori, 2006). The estimates for µ, σ, ad γ 1 were obtaied from 5,000 simulated samples of size from SN C(µ, σ, γ 1). µ = , σ = , γ 1 = E( µ µ) E( σ σ) E( γ 1 γ 1 ) E( µ µ) 2 E( σ σ) 2 E( γ 1 γ 1 ) µ = , σ = , γ 1 = E( µ µ) E( σ σ) E( γ 1 γ 1 ) E( µ µ) 2 E( σ σ) 2 E( γ 1 γ 1 ) µ = , σ = , γ 1 = E( µ µ) E( σ σ) E( γ 1 γ 1 ) E( µ µ) 2 E( σ σ) 2 E( γ 1 γ 1 ) µ = , σ = , γ 1 = E( µ µ) E( σ σ) E( γ 1 γ 1 ) E( µ µ) 2 E( σ σ) 2 E( γ 1 γ 1 ) µ = , σ = , γ 1 = E( µ µ) E( σ σ) E( γ 1 γ 1 ) E( µ µ) 2 E( σ σ) 2 E( γ 1 γ 1 ) µ = , σ = , γ 1 = E( µ µ) E( σ σ) E( γ 1 γ 1 ) E( µ µ) 2 E( σ σ) 2 E( γ 1 γ 1 ) Revista Colombiaa de Estadística 40 (2017)

12 134 Paulio Pérez-Rodríguez, José A. Villaseñor, Sergio Pérez & Javier Suárez Table 3: Bias ad mea squared error for momet estimates (Pewsey, 2000). The estimates for µ, σ, ad γ 1 were obtaied from 5,000 simulated samples of size from SN C(µ, σ, γ 1). µ = , σ = , γ 1 = E( µ µ) E( σ σ) E( γ 1 γ 1 ) E( µ µ) 2 E( σ σ) 2 E( γ 1 γ 1 ) µ = , σ = , γ 1 = E( µ µ) E( σ σ) E( γ 1 γ 1 ) E( µ µ) 2 E( σ σ) 2 E( γ 1 γ 1 ) µ = , σ = , γ 1 = E( µ µ) E( σ σ) E( γ 1 γ 1 ) E( µ µ) 2 E( σ σ) 2 E( γ 1 γ 1 ) µ = , σ = , γ 1 = E( µ µ) E( σ σ) E( γ 1 γ 1 ) E( µ µ) 2 E( σ σ) 2 E( γ 1 γ 1 ) µ = , σ = , γ 1 = E( µ µ) E( σ σ) E( γ 1 γ 1 ) E( µ µ) 2 E( σ σ) 2 E( γ 1 γ 1 ) µ = , σ = , γ 1 = E( µ µ) E( σ σ) E( γ 1 γ 1 ) E( µ µ) 2 E( σ σ) 2 E( γ 1 γ 1 ) Revista Colombiaa de Estadística 40 (2017)

13 Bayesia Estimatio for the Skew-Normal Distributio Some Examples I this sectio, we preset three examples of the proposed Bayesia method. For each example, we estimate the margial posterior desities of µ, σ, ad γ 1 usig three parallel Metropolis samplig chais; each are ru for 50,000 iteratios ad a bur-i period of 25,000 iteratios. I all the cases the proposal acceptace rate was at least 30%. Covergece was checked by ispectig trace plots ad applyig the Gelma & Rubi (1992) test Example 1: The Frotier Data The data i this example correspods to = 50 radom umbers geerated from SN D (0, 1, 5) or equivaletly SN C (0.7824, , ). The data are available withi the R s s package (Azzalii 2016). The maximum of the likelihood fuctio occurs o the upper boudary of the parameter space, correspodig to a ifiite estimate of λ. So, oe could expect to obtai a estimate ear for γ 1. Figure 1 shows histograms of simulated draws from the posterior distributio for µ, σ, ad γ 1. Table 4 shows the posterior summaries for µ, σ, ad γ 1. If the posterior mode of the margial posterior distributio is used for estimatio purposes, the ˆµ = , ˆσ = , ˆγ 1 = Note that those poit estimates are similar to those obtaied with the method proposed by Sartori (2006), that is µ = ( ), σ = (0.0777), γ 1 = (0.0570), where the values i paretheses are the stadard errors. Table 5 shows the result for the Gelma & Rubi s test. We used three parallel Metropolis samplig chais with differet iitial values. Approximated covergece is diagosed whe the upper C.I. limit is close to 1. Table 4: Posterior summary for the frotier data. Quatile 2.5% Media Mea Mode Sd Quatile 97.5% µ σ γ Table 5: Results of the covergece test (Gelma & Rubi 1992). Potetial Scale Reductio Factors Poit est. Upper C.I. µ σ γ Multivariate psrf 1.03 Revista Colombiaa de Estadística 40 (2017)

14 136 Paulio Pérez-Rodríguez, José A. Villaseñor, Sergio Pérez & Javier Suárez a) µ b) σ c) γ1 Figure 1: Histogram of simulated draws from the posterior distributios for a)µ, b)σ ad c)γ Example 2: Sartori s Data The data i Table 6 are 20 radom umbers from SN D (0, 1, 10) or equivaletly SN C (0.7939, , ) published i Sartori s article (2006). The usual maximum likelihood estimator for λ ad that obtaied by usig Sartori s method are both fiite. Table 6: Sartori s data Figure 2 shows the histograms of simulated draws from the posterior distributio for µ, σ ad γ 1. Table 7 shows the posterior summaries for the mea, stadard deviatio, ad the coefficiet of skewess. Whe the posterior mode of the margial posterior distributio is used as poit estimate, the ˆµ = , ˆσ = , ˆγ 1 = Note that those poit estimates are similar to those obtaied with the method proposed by Sartori (2006): that is µ = (0.2376), σ = (0.1171), γ 1 = (0.3884). Revista Colombiaa de Estadística 40 (2017)

15 Bayesia Estimatio for the Skew-Normal Distributio 137 a) µ b) σ c) γ1 Figure 2: Histogram of simulated draws from the posterior distributios for a)µ, b)σ ad c)γ 1. Table 7: Posterior summary Sartori s data. Quatile 2.5% Media Mea Mode Sd Quatile 97.5% µ σ γ Example 3: Half-Normal Case Perhaps, oe of the hardest cases to estimate the parameters of the SN D (0, 1, λ) is whe λ is large. Because it is kow that as λ teds to ifiity, ad the SN D (0, 1, λ) teds to the half-ormal desity, we test the proposed estimatio procedure for the half-ormal case ad compared it with the results of Wiper et al. (2008). I this case, we expect to obtai a estimate ear for γ 1. I Table 8, we preset the simulatio results for the proposed estimatio method whe a sample comes from a half-ormal radom variable with parameters ξ = 0, η = 1; HN(0, 1). This is approximately a SN C (0.7979, , ). The poit estimate for γ 1 is very close to the expected value, , whe = 50, 100; this result is a good oe because if oe graphed together the halfor- Revista Colombiaa de Estadística 40 (2017)

16 138 Paulio Pérez-Rodríguez, José A. Villaseñor, Sergio Pérez & Javier Suárez Table 8: Performace of the proposed estimatio method whe a sample comes from a HN(0, 1). The mea square error (mse) ad bias are obtaied whe the mode of the margial posterior desities is used as a poit estimate of µ = , σ = , γ 1 = Results are based o 5,000 samples of size. E(ˆµ µ) E(ˆσ σ) E(ˆγ 1 γ 1 ) E(ˆµ µ) 2 E(ˆσ σ) 2 E(ˆγ 1 γ 1 ) mal ad the SN C (0.7979, , ) desities, the we would ot be able to distiguish betwee the two desities. Results show that the bias ad mea squared error of the ˆγ 1 teds to zero as icreases, which provides evidece that the proposed estimator is cosistet (Table 8). Although the followig is ot a fair compariso because i the Wiper s case oly two parameters are estimated while i the SN case three parameters are estimated, at least we have a idea how well the proposed estimator is workig i this case. For the locatio parameter, the proposed estimator seems to be better sice its mea squared error is smaller tha the oe proposed by Wiper et al. s (2008). For the scale parameter, the Wiper et al. s (2008) estimator seems to be better tha the oe proposed i this paper (Table 9). Table 9: Estimated bias ad mea square error of the Bayesia posterior mea estimators whe a sample comes from HN(0, 1). Take from Wiper s et al. (2008) results. E(ˆξ ξ) E(ˆη η) E(ˆξ ξ) 2 E(ˆη η) Cocludig Remarks The simulatio results i Sectio 3 provide evidece o the advatages of usig the Metropolis Hastigs algorithm to estimate the parameters of the skew ormal family. The results from a simulatio study show that the bias ad the mea squared error decreases as the sample size icreases. Also, it seems that the mode appears to be a precise sytetic idex of the posterior distributio. It turs out that the estimates provided by the ew methodology for the shape parameter are fiite i compariso with the estimates obtaied by usig the direct parameterizatio ad the maximum likelihood method, which ca lead to covergece problems. Sice we are usig the Bayesia approach, it is possible to obtai HPD for the parameters of iterest, similarly to Wiper et al. (2008). Revista Colombiaa de Estadística 40 (2017)

17 Bayesia Estimatio for the Skew-Normal Distributio 139 [ Received: Jauary 2016 Accepted: November 2016 ] Refereces Arellao-Valle, R. B. & Azzalii, A. (2008), The cetred parametrizatio for the multivariate skew-ormal distributio, Joural of Multivariate Aalysis 99, Azzalii, A. (1985), A class of distributios which icludes the ormal oes, Scadiavia Joural of Statistics 12, Azzalii, A. (2016), The R package s: The Skew-Normal ad Skew-t distributios (versio 1.4-0), Uiversità di Padova, Italia. * Azzalii, A. & Capitaio, A. (1999), Statistical applicatios of the multivariate skew ormal distributio, Joural of the Royal Statistical Society 61, Azzalii, A. & Geto, M. G. (2008), Robust likelihood methods based o the skew-t ad related distributios, Iteratioal Statistical Review 76, Box, G. & Tiao, G. (1973), Bayesia iferece i statistical aalysis, Addiso- Wesley series i behavioral sciece: quatitative methods, Addiso-Wesley Pub. Co. Carli, B. P. & Louis, T. A. (2000), Bayes ad Empirical Bayes Methods for Data Aalysis, secod ed, Chapma - Hall/CRC, New York. Chib, S. & Greeberg, E. (1995), Uderstadig the Metropolis-Hastigs Algorithm, The America Statisticia 49(4), Chioga, M. (1998), Some results o the scalar skew-ormal distributio, Joural of the Italia Statistical Society 7, Gelma, A. & Rubi, D. B. (1992), Iferece from iterative simulatio usig multiple sequeces, Statistical Sciece 7, Liseo, B. & Loperfido, N. (2006), A ote o referece priors for the scalar skewormal distributio, Joural of Statistical Plaig ad Iferece 136, Metropolis, N., Rosembluth, A. W., Teller, M. & Teller, E. (1953), Equatios of state calculatios by fast computig machies, Joural of Chememical Physics 21, Pewsey, A. (2000), Problems of iferece for Azzalii s skew-ormal distributio, Joural of Applied Statistics 27, Revista Colombiaa de Estadística 40 (2017)

18 140 Paulio Pérez-Rodríguez, José A. Villaseñor, Sergio Pérez & Javier Suárez R Core Team (2015), R: A Laguage ad Eviromet for Statistical Computig, R Foudatio for Statistical Computig, Viea, Austria. * Sartori, N. (2006), Bias prevetio of maximum likelihood estimates for scalar skew ormal ad skew t distributios, Joural of Statistical Plaig ad Iferece 136, Wiper, M., Giró, F. J. & Pewsey, A. (2008), Objective Bayesia iferece for the half-ormal ad half-t distributios, Commuicatios i Statistics-Theory ad Methods 37(20), Revista Colombiaa de Estadística 40 (2017)