Introducing a Novel Bivariate Generalized Skew-Symmetric Normal Distribution

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1 Joural of mathematics ad computer Sciece 7 (03) 66-7 Article history: Received April 03 Accepted May 03 Available olie Jue 03 Itroducig a Novel Bivariate Geeralized Skew-Symmetric Normal Distributio Behrouz Fathi-Vajargah Departmet of Statistics, Uiversity of Guila, Rasht, Ira fathi@guilaacir Parisa Hasaalipour Departmet of Statistics, Uiversity of Guila, Rasht, Ira phasaalipour699@gmailcom Abstract We itroduce a geeralizatio of the bivariate geeralized skew-symmetric ormal distributio [5] We deote this distributio by SGN (, ) We obtai some properties of SGN (, ) ad derive the momet geeratig fuctio Keywords: Geeralized-skew-ormal distributio, SGN (, ), Coditioal distributio Itroductio The skew-ormal distributio itroduced by Azzalii [] This desity has bee studied ad geeralized by some researchers For example, Azzalii ad Dalla Valle [4], Azzalii ad Capitaio [3], Arellao-Valle [], Jamalizadeh [7], Sharafi ad Behbodia [8], Hasaalipour ad sharafi [6] ad Yadegari [9] Fathi ad Hasaalipour [5] cosidered a geeralizatio of SGN (, ) distributio ad they called it the bivariate geeralized skew-symmetric ormal distributio Its probability desity fuctio is give by xy f ( x, y ;, ) ( x ) ( y ) ( ), x, y R, R, 0 () ( xy ) This distributio deoted by ( X, Y ) ~ SGN (, ) I this paper, we itroduce a ew family of skewormal distributio which geeralizes () while preservig most of its properties I sectio, we preset the defiitio ad some properties of SGN (, ) class ad sectio 3, gives some importat theorems about coditioal distributios of SGN (, ) 66

2 B Fathi-Vajargah, P Hasaalipour/ J Math Computer Sci 7 (03) 66-7 A ovel bivariate geeralized skew-symmetric ormal distributio I this sectio, we defie the SGN (, ) class ad obtai some its properties SGN (, ) Defiitio Vector ( XY, ) has SGN (, ) distributio if ad oly if for every it has the followig desity xy f ( x, y ;, ) c(, ) ( x ) ( y ) ( ), x, y R, () ( xy ) where R ad 0 The coefficiet c (, ), which is a fuctio of ad the parameters, is give by c (, ), (3) xy ( x ) ( y ) ( ) dx dy ( xy ) with this properties: c lim (, ) 4 for all 0 c c (, ) (, ) We deote this by ( X, Y ) ~ SGN (, ) Some simple properties of SGN (, ) We ow preset some properties of this ovel distributio SGN (0, ) ( x ) ( y ), for all 0 SGN (,0) ( x ) ( y ) ( xy ) xy X Y y c ( y ) ( x ) ( ) ~ GBSN ( y ) 3 4 ( xy ) xy Y X x c ( x ) ( y ) ( ) ~ GBSN ( x ) ( xy ) [6] [6] 5 If ( X, Y ) ~ SGN (, ), the ( X, Y ) ~ SGN (, ), ( X, Y ) ~ SGN (, ) ad ( X, Y ) ~ SGN (, ) [] 67

3 B Fathi-Vajargah, P Hasaalipour/ J Math Computer Sci 7 (03) lim f ( x, y ;, ) 4 ( x ) ( y ) Ix0, y0 7 lim f ( x, y ;, ) 4 ( x ) ( y ) Ix0, y0 8 X Y X Y lim f ( x, y ;, ) f ( x, y ;, ) ( x ) 9 Y X Y X lim f ( x, y ;, ) f ( x, y ;, ) ( y ) Some theorems about coditioal distributios of SGN (, ) Theorem If X, Y, Z,, Z are iid N (0,) distributio the we have: XY ( X, Y ) Z ( ) ~ SGN (, ) (4) ( XY ) Where Z max Z,, Z ( ) XY Proof: Suppose A ( Z( ) ) The, we write ( XY ) f ( X, Y ) A P ( A X x, Y y ) f ( x, y ) ( x, y A ) PA ( ) XY P ( Z ( ) X x, Y y ) ( x ) ( y ) ( XY ) XY P( Z( ) ) ( XY ) xy xy P ( Z,, Z ) ( x ) ( y ) ( xy ) ( xy ) XY XY P ( Z,, ) ( ) ( ) c (, ) ( x) y Z XY XY xy ( ) ( ) ( xy ) For radom umber geeratio, it is more efficiet to use sigle variat of this result, amely to put 68

4 B Fathi-Vajargah, P Hasaalipour/ J Math Computer Sci 7 (03) 66-7 XY ( X, Y ) Z ( ) ( XY ) Z ( Z, Z ) (5) XY ( X, Y ) Z ( ) ( XY ) This make a importat poit for SGN (, ) distributio, comparig with acceptace-rejectio method simulatio of idepedet ormal distributio Theorem If ( X, Y ) ~ SGN (, ), the ( X Y ) L () as, where () shows chi-square radom variable with oe degree of freedom Proof: Let ( X Y ) Z The desity of Z is z y z y ( ) ( ) z zy zy f Z ( z, y ;, ) e c( y ) z f ( z ) a ( z, y ;, ) ; z 0 () with z y z y ( ) ( ) zy zy a( z, y ;, ) c( y ) Sice c ( y ) as, we coclude that a ( z, y ;, ), as Therefore, the desity f Z ( z, y ;, ) coverges to the distributio of () coverges to the distributio of, ie L () Z ( X Y ) (), as Hece, the distributio of Z Theorem 3 If ( X, Y ) ~ SGN (, ) ad Z ~ N (0,), the X Y ad Z are idetically distributed, D X ie, lim Z ~ HN (0,) Y, where HN (0,) deotes the stadard half-ormal distributio X Proof: We kow that Z has desity ( z) Iz 0 The desity W is Y 69

5 B Fathi-Vajargah, P Hasaalipour/ J Math Computer Sci 7 (03) 66-7 f ( w ) f ( w ) f ( w ) W X Y X Y w y w y c ( y ) ( w ) ( ) c ( y ) ( w ) ( ) ( wy ) ( wy ) w y w y c ( y ) ( w ) ( ) ( ) ( wy ) ( wy ) ( w ) b ( w, y ;, ) Now, we ca show that b ( w, y ;, ) as, the X lim Z Y D lim W ( w) for w 0 ad we have Theorem 4 The momet geeratig fuctio ( X, Y ) ~ SGN (, ) is t t WK M X, Y ( t, t ) c (, ) e E E ( ) ( WK ) where W ~ N ( t,), K ~ N ( t,) Proof: Refereces M t t E e (, ) ( ) X, Y t X t Y c (, ) e ( x ) ( y ) ( ) dx dy t c e E E t x t y xy t WK (, ) ( ) ( WK ) ( xy ) [] R B Arellao-Valle, H W Gomez ad F A Quitaa, A ew class of skew-ormal distributio Commu Stat Theory Methods, 33(7), (004) [] A Azzalii, A class of distributios which icludes the ormal oes Scad J Stat,7-78 (985) [3] A Azzalii ad A Capitaio, Statistical applicatio of the multivariate skew-ormal distributio J R Statist Soc B, 6, (999) [4] A Azzalii ad A Dalla-Valle, The multivariate skew-ormal distributio Biometrika 83, (996) [5] B Fathi ad P Hasaalipour, Simulatio ad theory of bivariate geeralized skew-symmetric ormal distributio Uder review (0) [6] P Hasaalipour ad M Sharafi, A ew geeralized Balakrisha skew-ormal distributio Statistical Papers, 53, 9-8 (00) 70

6 B Fathi-Vajargah, P Hasaalipour/ J Math Computer Sci 7 (03) 66-7 [7] A Jamalizadeh, J Behboodia ad N Balakrisha, A two-parameter geeralized skew-ormal distributio Statistics ad Probability Letters 78, 7-76 (008) [8] M Sharafi, J Behboodia, The Balakrisha skew-ormal desity Statistical Papers 49, (008) [9] I Yadegari, A Gerami ad MJ Khaledi, A geeralizatio of the Balakrisha skew-ormal distributio Statistics ad Probability Letters 78, (008) 7