The multivariate skew-slash distribution

Transcription

1 Jurnal f Statistical Planning and Inference 136 (6) 9 wwwelseviercm/lcate/jspi The multivariate skew-slash distributin Jing Wang, Marc G Gentn Department f Statistics, Nrth Carlina State University, Bx 83, Raleigh NC , USA Received 31 December 3; accepted 17 June 4 Available nline 14 August 4 Abstract The slash distributin is ften used as a challenging distributin fr a statistical prcedure In this article, we define a skewed versin f the slash distributin in the multivariate setting and derive several f its prperties The multivariate skew-slash distributin is shwn t be easy t simulate frm and can therefre be used in simulatin studies We prvide varius examples fr illustratin 4 Elsevier BV All rights reserved MSC: 6E5; 6H5 Keywrds: Cauchy; Cnfigural plysampling; Kurtsis; Multivariate distributins; Skewness; Skew-nrmal; Slash 1 Intrductin Despite the central rle played by the magic bell-shaped nrmal distributin in statistics, there has been a sustained interest amng statisticians in cnstructing mre challenging distributins fr their prcedures Indeed, if ne can perfrm satisfactrily under tw extreme scenaris, it is cmmnly believed that the perfrmance will be reasnably gd under intermediate situatins This idea is the crnerstne f cnfigural plysampling, a small sample apprach t rbustness described by Mrgenthaler and Tukey (1991) A first family f scenaris can be represented by a finite mixture f nrmal distributins Hwever, this family des nt cntain any real challenge since its members always have a finite variance A secnd mre ppular class in this scenari is the t ν distributin with ν degrees f freedm, Crrespnding authr Tel: ; fax: addresses: jwang6@statncsuedu (J Wang), gentn@statncsuedu (MG Gentn) /$ - see frnt matter 4 Elsevier BV All rights reserved di:1116/jjspi463

2 1 J Wang, MG Gentn / Jurnal f Statistical Planning and Inference 136(6) 9 which cnverges t the nrmal distributin fr ν The ther extreme given by ν = 1is a Cauchy distributin, that is, the distributin f the rati between tw independent standard nrmal randm variables This distributin has heavier tails than the nrmal distributin and des nt pssess finite mments r cumulants Hwever, its sharp central peak has ften been cnsidered as unrealistic in representing real data Anther family f scenaris is described by the standard slash distributin, representing the distributin f the rati X = Z/(U 1/q ) f a standard nrmal randm variable Z t an independent unifrm randm variable U n the interval (, 1) raised t the pwer 1/q, q> When q = 1 we btain the cannical slash, whereas q yields the nrmal distributin The prbability density functin (pdf) f the univariate slash distributin is symmetric abut the rigin and has heavier tails than thse f the nrmal density, with, fr the cannical slash, the same tail heaviness as the Cauchy Hwever, it is less peaked in the center and thus mre realistic in representing data Effectively, straightfrward algebra yields the pdf ψ(x; q) = q 1 u q (xu) du and the cumulative distributin functin (cdf) Ψ(x; q) = q 1 u q 1 Φ(xu) du, (1) where and Φ dente the standard nrmal pdf and cdf, respectively In particular, we have ψ(; q) = (q/(q + 1)) () = q/( π(q + 1)) and Ψ(; q) = 1 Mrever, clsed-frm expressins fr the pdf can be cmputed, fr instance { ( () (x))/x ψ(x; 1) =, x =, ()/, x = and { ((Φ(x) Φ())/x (x))/x ψ(x; ) =, x =, ()/3, x = The expectatin and the variance f the standard slash distributin are given by E(X) = fr q>1and Var(X)=q/(q ) fr q> A general slash distributin is btained by scale multiplicatin and lcatin shift f a standard slash randm variable, see Rgers and Tukey (197), and Msteller and Tukey (1977) fr further prperties Kafadar (198) discussed the maximum likelihd estimatin f the lcatin and scale parameters f this family The slash distributin has been mainly used in simulatin studies because it represents an extreme situatin, see fr example Andrews et al (197), Grss (1973), and Mrgenthaler and Tukey (1991) In this paper, we intrduce an additinal challenge by defining a skewed versin f the slash distributin Anther apprach t intrduce challenges fr statistical prcedures is t define skewed distributins A simple departure frm the nrmal distributin has been prpsed byazzalini (1985) wh defined the skew-nrmal distributin with pdf (x; μ, σ )Φ(α(x μ)), ()

3 J Wang, MG Gentn / Jurnal f Statistical Planning and Inference 136(6) 9 11 where (x; μ, σ ) is the pdf f a nrmal randm variable with mean μ and variance σ, and α is a shape parameter cntrlling skewness Fr α =, the pdf () reduces t the nrmal ne, whereas fr α > rα < the pdf is skewed t the right r the left, respectively An extensin f () t the multivariate setting was prpsed by Azzalini and Dalla Valle (1996), defining the pdf p (x; μ, Σ)Φ(α T (x μ)), x R p, (3) where p (x; μ, Σ) is the p-dimensinal nrmal pdf with mean μ and crrelatin matrix Σ, Φ( ) is the standard nrmal cdf N(, 1), and α is a p-dimensinal shape parameter A p-dimensinal randm vectr X with a multivariate skew-nrmal distributin is dented by X SN p (μ, Σ, α) Its expectatin and variance are given by E(X)=μ + π δ, Var(X)=Σ π δδt, (4) (5) where δ = Σα/ 1 + α T Σα, see eg Gentn et al (1) It is nw natural t cnstruct univariate and multivariate distributins that cmbine skewness with heavy tails Fr instance, ne can define skew-t distributins (Branc and Dey, 1; Jnes and Faddy, 3; Sahu et al, 3), skew-cauchy distributins (Arnld and Beaver, ), skew-elliptical distributins (Azzalini and Capitani, 1999; Branc and Dey, 1; Sahu et al, 3; Gentn and Lperfid, 5), r ther skew-symmetric distributins (Wang et al, 4) In this article, we define a multivariate skew-slash distributin and study its prperties and applicatins This article is rganized as fllws In Sectin, we define the multivariate slash distributin and derive varius f its prperties Fr example, we shw that the slash distributin is invariant under linear transfrmatins and that its mments are analytically tractable In Sectin 3, we define the multivariate skew-slash distributin and prvide several examples revealing its skewness and tail behavir The results f a small simulatin study are reprted in Sectin 4 alng with tw illustrative applicatins We cnclude in Sectin 5 The multivariate slash distributin In this sectin, we define a multivariate slash distributin and derive its pdf We shw that the multivariate slash distributin is invariant under linear transfrmatins and derive its mments In the sequel, we dente the p-dimensinal multivariate nrmal distributin with mean vectr μ and cvariance matrix Σ by N p (μ, Σ), its pdf by p (x; μ, Σ), and the standard unifrm distributin n the interval (, 1) by U(, 1) Definitin 1 A randm vectr X R p has a p-dimensinal slash distributin with lcatin parameter μ, psitive definite scale matrix parameter Σ, and tail parameter q>, dented

4 1 J Wang, MG Gentn / Jurnal f Statistical Planning and Inference 136(6) 9 by X SL p (μ, Σ,q),if X = Σ 1/ Y + μ, U 1/q (6) where Y N p (,I p ) is independent f U U(, 1) When μ = and Σ = I p, X in (6) has a standard multivariate slash distributin The pdf f the randm vectr X in (6) is easily shwn t be ψ p (x; μ, Σ,q)=q = 1 u q+p 1 p (ux; uμ, Σ) du q (q+p)/ 1 γ((q+p)/; Σ 1/ (x μ) /) (π) p/ Σ 1/ (x μ) q+p, x =, q q+p ( 1 π )p/, x =, where γ(a; z)= z ta 1 e t dt= ( 1) k z a+k k= k!(a+k), and Σ 1/ (x μ) = (x μ) T Σ 1 (x μ) Nte that the standard slash randm vectr in (6) is a scale mixture f the nrmal mdel (see eg Fang et al, 199) and s it can be represented as: X (U = u) N p (,u 1/q I p ) with U U(, 1) Next cnsider the linear transfrmatin V = b + AX, where X has the multivariate slash distributin X SL p (μ, Σ,q), b is a vectr in R p, and A is a nnsingular matrix The Jacbian determinant f the transfrmatin is A 1 and hence the pdf f V is A 1 ψ p (A 1 (v b); μ, Σ,q), shwing that V has a multivariate slash distributin SL p (b + Aμ,AΣA T,q) This implies that the slash distributin is invariant under linear transfrmatins and this is summarized in the fllwing prpsitin Prpsitin 1 If X SL p (μ, Σ,q), then its linear transfrmatinv=b+ax SL p (b+ Aμ,AΣA T,q) It can be checked that Prpsitin 1 still hlds when A is nly a full rw rank matrix and therefre marginal distributins f X SL p (μ, Σ,q)are still f the slash typealternatively, this fllws frm the fact that marginal distributins fy N p (,I p ) in (6) are still nrmal Nw we cnsider the mments f X The mments f a unifrm randm variable U U(, 1) are given by E(U k/q ) = q, q k q>k (7) Because Y and U are independent in (6), the mments f X fllw immediately frm the mments f Y and (7) Fr instance, the first tw mments f X fr the multivariate slash distributin are shwn in the fllwing prpsitin

5 J Wang, MG Gentn / Jurnal f Statistical Planning and Inference 136(6) 9 13 Prpsitin If X SL p (μ, Σ,q), then the expectatin and variance f X are given by E(X)=μ if q>1, Var(X)= q q Σ if q> 3 The multivariate skew-slash distributin In this sectin, we define a multivariate skew-slash distributin and derive its pdf We als prvide anther definitin based n a stchastic representatin f the skew-slash distributin, which is useful fr simulatins Bth definitins lead t the same prbability density functin We shw that the multivariate skew-slash distributin is invariant under linear transfrmatins Definitin A randm vectr X R p has a p-dimensinal skew-slash distributin with lcatin parameter μ, psitive definite scale matrix parameter Σ, tail parameter q>, and skewness parameter α, dented by X SSL p (μ, Σ,q,α), if X = Σ 1/ Y + μ, U 1/q where Y SN p (,I p, α) is independent f U U(, 1) (8) When μ = and Σ = I p, X in (8) has a standard multivariate skew-slash distributin The pdf f the randm vectr X in (8) is then η p (x; μ, Σ,q,α) = q 1 u q+p 1 p (ux; uμ, Σ)Φ(uα T Σ 1/ (x μ)) du (9) In the univariate case, ie fr p=1, the distributin f T = X has density ψ(t; q)i(t >) in bth cases when X SL(, 1, q) and when X SSL(, 1, q, α) This invariance prperty hlds als fr the distributin f X in the multivariate case, which fllws directly frm the stchastic representatin in (8) and cnsidering the well knwn result abut the invariance f the distributin f Y, Y SN p (,I p, α), with respect t α This invariance prperty is similar t the ne derived by Gentn et al (1) fr the skew-nrmal distributin, Gentn and Lperfid (5) fr generalized skew-elliptical distributins, and by Wang et al (4) fr skew-symmetric distributins Next cnsider the linear transfrmatin V = b + AX, where X has the multivariate skewslash distributin X SSL p (μ, Σ,q,α), b is a vectr in R p, and A is a nnsingular matrix The Jacbian determinant f the transfrmatin is A 1 and hence the pdf f V is A 1 η p (A 1 (v b); μ, Σ,q,α), shwing that V has a multivariate skew-slash distributin SSL p (b + Aμ,AΣA T,q,A T α) This implies that the skew-slash distributin is invariant under linear transfrmatins and is summarized in the fllwing prpsitin Prpsitin 3 If X SSL p (μ, Σ,q,α), then its linear transfrmatin V = b + AX SSL p (b + Aμ,AΣA T,q,A T α)

6 14 J Wang, MG Gentn / Jurnal f Statistical Planning and Inference 136(6) 9 density N SN SSL x -4-4 Fig 1 Density curves f the univariate nrmal, skew-nrmal, and skew-slash distributins Here again, it can be checked that Prpsitin 3 still hlds when A is nly a full rw rank matrix and therefre marginal distributins f X SSL p (μ, Σ,q,α) are still f the skew-slash type Alternatively, this fllws frm the fact that marginal distributins f Y SN p (,I p, α) in (8) are still skew-nrmal, see Azzalini and Dalla Valle (1996) The multivariate skew-slash distributin SSL p (μ, Σ,q,α) reduces t the skew-nrmal distributin SN p (μ, Σ, α) when q, t the slash distributin SL p (μ, Σ,q)when α =, and t the nrmal distributin N p (μ, Σ) when bth α = and q Thus, the skew-slash distributin includes a wide variety f cntur shapes T illustrate the skewness and tail behavir f the skew-slash, we draw the density f the univariate skew-slash distributin SSL 1 (, 1, 1, 1) tgether with the densities f the standard nrmal distributin N 1 (, 1) and skew-nrmal distributin SN 1 (, 1, 1) Fig 1 depicts the three density curves (the skewnrmal and skew-slash have been centered at zer and rescaled) Nte that the densities f the skew-nrmal and skew-slash distributins are psitively skewed and that the skew-slash distributin has much heavier tails than the nrmal and skew-nrmal distributin Actually, this skew-slash distributin des nt have finite mean and variance, see Prpsitin 4 Fig depicts cntur plts f the standard bivariate skew-slash pdf fr the parameter values q = 5, α = (1, 1) T (left panel) and q = 1, α = (5, ) T (right panel) The right panel shws a case with heavier tails than the left panel, but bth exhibit skewness Next we discuss an equivalent definitin f the multivariate skew-slash distributin based n an apprach described by Arnld and Beaver () Fr simplicity f the expsitin, we set μ = and Σ = I p Cnsider the cnditinal distributin f W given α T W >W, where α R p, and (W,W T ) SL p+1 (,I p+1,q) with W = (W 1,W,,W p ) T The

7 J Wang, MG Gentn / Jurnal f Statistical Planning and Inference 136(6) Fig Cntur plts f the standard bivariate skew-slash pdf fr q = 5, α = (1, 1) T (left panel) and q = 1, α = (5, ) T (right panel) jint prbability density functin f (W,W) cnditinal n α T W >W is given by ψ p+1 (w, w;,i p+1,q)1(α T w >w ) P {W α T, w R, w R p W < } It fllws frm the symmetry f the slash distributin that P {W α T W < }= 1 Integrating ut w yields the density f W which is given by ψ p+1 (w, w;,i p+1,q)1(α T w >w ) dw = q(π) (p+1)/ α T w 1 u p+q p ( exp( u w /) exp u = q 1 i=1 ) wi / du dw u p+q 1 p (uw;,i p )Φ(uα T w) du, w R p, which is exactly the density η p (w;,i p,q,α) in (9) f the multivariate skew-slash distributin SSL p (,I p,q,α) This result prves the equivalence between Definitin and the definitin based n cnditining Cnsequently either f these tw definitins can be used t simulate frm the multivariate skew-slash distributin, see Sectin 41 We nw cmpute the mments f the skew-slash distributin Using the same argument as fr the multivariate slash distributin, the mments f X SSL p (μ, Σ,q,α) fllw readily frm (7) and the mments f Y SN p (,I p, α), see eg Gentn et al (1) Fr instance, the first tw mments f X fr the multivariate skew-slash distributin are shwn in the fllwing prpsitin

8 16 J Wang, MG Gentn / Jurnal f Statistical Planning and Inference 136(6) 9 Prpsitin 4 If X SSL p (μ, Σ,q,α), then the expectatin and variance f X are given by E(X)=μ + q q 1 π δ if q>1, Var(X)= q q Σ ( ) q δδ T if q>, π q 1 where δ = Σα/ 1 + α T Σα Nte that when there is n skewness, ie α = δ =, the expectatin and variance in Prpsitin 4 reduce t the expressins given in Prpsitin, whereas when q (n heavy tail behavir) they reduce t the expressins given by (4) and (5) 4 Applicatins In this sectin, we present three applicatins f the skew-slash distributin The first ne illustrates the use f the skew-slash distributin in simulatin studies, whereas the ther tw invlve the statistical analysis f real data sets 41 Skew-slash distributins in simulatin studies The skew-slash distributin can be used in simulatin studies as a challenging distributin fr a statistical prcedure As an illustratin, we perfrm a small simulatin t study the behavir f tw lcatin estimatrs, the sample mean and the sample median, in fur different univariate settings We cnsider tw symmetric distributins, a standard nrmal N 1 (, 1) and a slash SL 1 (, 1, ), and tw asymmetric distributins, a skew-nrmal SN 1 (, 1, 3) and a skew-slash SSL 1 (, 1,, 3) The lcatin means f the asymmetric distributins are adjusted t zer, s that all fur distributins are cmparable Thus, this setting represents fur distributins with same mean, but with different tail behavir and skewness Actually, with q =, the variance f the slash and skew-slash distributins is infinite We simulate 5 samples f size n = 1 frm each f these fur distributins On each sample, we cmpute the sample mean and sample median and reprt the bxplt fr each distributin in Fig 3In the left panel, we bserve that all bxplts f the estimated means are centered arund zer but have larger variability fr the heavy tailed distributins (the slash and the skew-slash) In the right panel, we see that the bxplt f the estimated medians has a slightly larger variability than the bxplt fr the estimated means at the nrmal distributin, but has a much smaller variability at the slash distributin This indicates that the median is a rbust estimatr f lcatin at symmetric distributins On the ther hand, the median estimatr becmes biased as sn as unexpected skewness arises in the underlying distributin, see the bxplt f the estimated medians f the skew-nrmal distributins This effect is even mre severe under bth skewness and heavy-tails represented by the skew-slash distributin Althugh this simulatin is very simple, it illustrates the use f nn-nrmal distributins t challenge the behavir f statistical prcedures

9 J Wang, MG Gentn / Jurnal f Statistical Planning and Inference 136(6) N SL SN SSL N SL SN SSL - Fig 3 Bxplts f the sample mean (left panel) and sample median (right panel) n 5 samples f size n = 1 frm fur distributins: N 1 (, 1); SL 1 (, 1, ); SN 1 (, 1, 3); SSL 1 (, 1,, 3) (the means f SN 1 and SSL 1 are adjusted t zer) 4 Fiber-glass data set This applicatin is cncerned with a unidimensinal data set f n = 63 breaking strengths values f 15 cm lng glass fibers Jnes and Faddy (3) andazzalini and Capitani (3) fit tw frms f skew-t distributins t these data They bth nte skewness n the left as well as heavy tail behavir We instead fit a skew-slash distributin The fitted parameters, btained by maximizing the likelihd functin, are ˆμ = 181, ˆσ = 333, ˆq = 333, and ˆα = 3 The negative value f ˆα indicates skewness n the left and the small value f ˆq indicates a heavy tail behavir Fig 4 depicts a histgram f the data and the fitted skew-slash prbability density functin, which has a finite mean and variance 43 Australian athletes data set This applicatin deals with the Australian athletes data set analyzed by Ck and Weisberg (1994) in a nrmal setting and Azzalini and Dalla Valle (1996) with the skewnrmal distributin It cnsists f several variables measured n n= athletes and we fcus n bdy mass index (BMI) and lean bdy mass (LBM) We fit a bivariate skew-slash t these data in rder t investigate the pssible heavier-than-nrmal tail behavir f (BMI, LBM) The parameters, estimated by maximizing the likelihd functin, are ˆμ = (4, 6) T, ˆσ 1,1 = 38, ˆσ 1, = 7, ˆσ, = 9, ˆα = (395, 11) T, and ˆq = 898, where σ i,j is the (i, j)-entry f the matrix Σ 1/ The skewness parameter ˆα indicates apparent skewness, as can be seen in Fig 5, but the parameter ˆq des nt indicate a serius heavy-tail behavir We use a likelihd rati test fr the null hypthesis H : q =, that is t test that a skew-nrmal distributin is enugh The uncnstrained lg likelihd functin value

10 18 J Wang, MG Gentn / Jurnal f Statistical Planning and Inference 136(6) fib Fig 4 Fitted skew-slash prbability density functin (slid line) t the fiber-glass data is and the cnstrained lg likelihd functin value is 11953, which yields a likelihd rati test statistic f 616 and a p-value f 567 with an asympttic χ 1 distributin under the null hypthesis There is nt enugh evidence in the data t reject H, and therefre a skew-nrmal distributin wuld be apprpriate fr this example 5 Discussin We have intrduced a multivariate skew-slash distributin, a flexible distributin that can take skewness and heavy tails int accunt This distributin is useful in simulatin studies where it can intrduce distributinal challenges in rder t evaluate a statistical prcedure It is als useful in analyzing data sets that d nt fllw the nrmal law We have used the fiber-glass data set and the Australian athletes data set fr illustratin Additinal flexibility can be intrduced in the skew-slash distributin by allwing higher rder dd plynmials in

11 J Wang, MG Gentn / Jurnal f Statistical Planning and Inference 136(6) 9 19 bmi lbm Fig 5 Cnturs f the fitted bivariate skew-slash prbability density functin t the (BMI, LBM) variables f Australian athletes data the skewing functin Φ( ) in (9) Fr instance, an dd plynmial f rder three wuld yield a distributin that can mdel bimdality, see Ma and Gentn (4) fr further discussins n this tpic Jnes and Faddy (3) discuss different versins f skew-t distributins Their wn prpsal is develped in the univariate setting and they acknwledge that its extensin t the multivariate setting (Jnes, 1) is f questinable usefulness Families f clsely related multivariate skew-t distributins are cnstructed by Azzalini and Capitani (3), and Sahu et al (3) Hwever, Jnes and Faddy (3) pint ut that dd mments f these families are analytically intractable, which results in the unavailability f a Fisher scring algrithm fr likelihd maximizatin Anther family f univariate skew-t distributins is prpsed by Fernández and Steel (1998) They use Bayesian fitting techniques because standard asympttic likelihd thery is nt applicable due t the discntinuity f even derivatives f their skew-t density at the rigin The multivariate skew-slash distributin intrduced in this article is clearly an alternative t skew-t distributins because it can mdel bth skewness and heavy tails One interesting advantage f the multivariate skewslash distributin is that its mments can be cmputed analytically by taking advantage f the mments f the multivariate skew-nrmal distributin, see the discussin in Sectin 3 Anther attractive feature is that simulatins frm the multivariate skew-slash distributin

12 J Wang, MG Gentn / Jurnal f Statistical Planning and Inference 136(6) 9 are straightfrward frm sftwares that permit simulatins frm the multivariate skewnrmal r nrmal distributin Acknwledgements The authrs thank tw annymus referees fr helpful cmments n this article References Andrews, DF, Bickel, PJ, Hampel, FR, Huber, PJ, Rgers, WH, Tukey, JW, 197 Rbust Estimates f Lcatin: Survey and Advances Princetn University Press, Princetn, NJ Arnld, BC, Beaver, RJ, The skew-cauchy distributin Statist Prbab Lett 49, 85 9 Arnld, BC, Beaver, RJ, Skewed multivariate mdels related t hidden truncatin and/r selective reprting Test 11, 7 54 Azzalini, A, 1985 A class f distributins which includes the nrmal nes Scand J Statist 1, Azzalini, A, Capitani, A, 1999 Statistical applicatins f the multivariate skew nrmal distributin J Ry Statist Sc Ser B 61, Azzalini, A, Capitani, A, 3 Distributins generated by perturbatin f symmetry with emphasis n a multivariate skew-t distributin J Ry Statist Sc Ser B 65, Azzalini, A, Dalla Valle, A, 1996 The multivariate skew-nrmal distributin Bimetrika 83, Branc, MD, Dey, DK, 1 A general class f multivariate skew-elliptical distributins J Multivariate Anal 79, Ck, RD, Weisberg, S, 1994 An Intrductin t Regressin Graphics Wiley, New Yrk Fang, K-T, Ktz, S, Ng, K-W, 199 Symmetric Multivariate and Related Distributins Chapman & Hall, New Yrk Fernández, C, Steel, MJF, 1998 On Bayesian mdelling f fat tails and skewness J Amer Statist Assc 93, Gentn, MG, Lperfid, N, 5 Generalized skew-elliptical distributins and their quadratic frms Ann Inst Statist Math, t appear Gentn, MG, He, L, Liu, X, 1 Mments f skew-nrmal randm vectrs and their quadratic frms Statist Prbab Lett 51, Grss, AM, 1973 A Mnte Carl swindle fr estimatrs f lcatin J Ry Statist Sc Ser C Appl Statist, Jnes, MC, 1 Multivariate t and beta distributins assciated with the multivariate F distributin Metrika 54, Jnes, MC, Faddy, MJ, 3 A skew extensin f the t distributin, with applicatins J Ry Statist Sc Ser B 65, Kafadar, K, 198 A biweight apprach t the ne-sample prblem J Amer Statist Assc 77, Ma, Y, Gentn, MG, 4 A flexible class f skew-symmetric distributins Scand J Statist 31, Mrgenthaler, S, Tukey, JW, 1991 Cnfigural Plysampling: A Rute t Practical Rbustness Wiley, NewYrk Msteller, F, Tukey, JW, 1977 Data Analysis and Regressin Addisn-Wesley, Reading, MA Rgers, WH, Tukey, JW, 197 Understanding sme lng-tailed symmetrical distributins Statist Neerlandica 6, 11 6 Sahu, SK, Dey, DK, Branc, MD, 3 A new class f multivariate skew distributins with applicatins t Bayesian regressin mdels Canad J Statist 41, Wang, J, Byer, J, Gentn, MG, 4 A skew-symmetric representatin f multivariate distributins Statist Sinica, t appear