A note on rescalings of the skew-normal distribution

Transcription

1 Poyeccione Jounal of Mathematic Vol. 31, N o 3, pp , Septembe 01. Univeidad Católica del Note Antofagata - Chile A note on ecaling of the kew-nomal ditibution OSVALDO VENEGAS Univeidad Católica de Temuco, Chile DAVID ELAL-OLIVERO Univeidad de Atacama, Chile and HÉCTOR W. GÓMEZ Univeidad de Antofagata, Chile Received : Decembe 011. Accepted : Mach 01 Abtact In thi aticle, we how that cetain kew-nomal paametic tatitical model ae a eult of ecaling of the kew nomal model tudied by Azzalini (1985). Uing thi pocedue we define a cla of kew-nomal ditibution and we tudy it moment, kewne and kutoi coefficient. At the end of thi aticle we will ue thi cla of ditibution to make ome extenion of the kew-nomal model. Key Wod : Aymmety, Recaling, Skew-Nomal model.

2 198 Ovaldo Venega, David Elal Oliveo and Hécto W. Gómez 1. Intoduction Azzalini (1985) tudied the {SN (λ),λ R} family of Skew-nomal ditibution, with aymmety paamete λ, wheesn(0) i the tandad nomal ditibution. Thi i, X SN(λ) and it denity function i: (1.1) φ λ (x) φ (x) Φ (λx), x,λ R, whee φ and Φ ae the N (0, 1) pobability denity function and cumulative ditibution function, epectively. Some popetie fo thi ditibution ae: λ λ E (X) (1.) ; Va(X) 1 π 1+λ π (1 + λ ) µ M X (t) exp(t λt (1.3) /)Φ 1+λ p β1 1 Ã E! 3/ Ã (4 π) (X) E! (X) (1.4) ; β (π 3), Va(X) Va(X) whee β 1 and β ae the aymmetic and kutoi coefficient, epectively. Fom (4) we get (1.5) p β ; β The following tochatic epeentation of the kew-nomal model wa given by Henze (1986) and Azzalini (1986), i.e., X SN(λ) if (1.6) X a X 1 + bx, λ whee X 1, X ae iid N(0, 1) andom vaiable, with a 1+λ and b 1 1+λ. Henze (1986) ued thi epeentation to compute the odd moment of the kew-nomal denity and gave an extenion of the tuncated nomal ditibution. If X SN(λ) then the kew-nomal ditibution with location and cale i the ditibution of Z µ + σx, µ R and σ>0, and it will be denoted by Z SN(µ, σ, λ). Othe kew-nomal model ha been tudied, fo example by Mukhopadhyay and Vidakovic (1995), Sahu et al. (003) and Nadaajah and Kotz (003). Thee model ae poduct of ecaling of the kew-nomal model of Azzalini (1985).

3 A note on ecaling of the kew-nomal ditibution 199 The pupoe in thee note, i to how how the ecaling affect the epeentation of the kew-nomal model, a well a povide ome extenion of thi ecaled model. Thi aticle i oganized a follow. In ection we give the ecaling of the kew-nomal model, ome popetie, example, moment and aymmetic and kutoi coefficient. In ection 3 ome extenion of thi ecaled model ae peented. In ection 4 peent the appendix.. Recaling Definition 1. We will ay that a andom vaiable X ha a Recaled-Skew- Nomal ditibution with paamete λ, and, with > 0, thatwillbe denoted by X RSN(λ,, ), if it denity function i given by f X (x) µ µ x λ (.1) φ Φ x, whee φ and Φ ae the tandad nomal denity and ditibution function, epectively. Popoition 1. Let U N(0, ) and V N (0, 1) whee U and V ae independent and let X λ U + V whee + λ + λ > 0 and λ R, then the denity function f X of the andom vaiable X i given in (7). Poof Thi poof i in the appendix. Remak 1. a) A ditibution uitable fo fitting poitive data i the half-nomal ditibution. We ay that a andom vaiable X follow a half-nomal ditibution with cale paamete σ if it denity function i given by: f(x; σ) µ x σ φ I{x >0} σ with σ>0. WedenotethibywitingX HN(σ). b) We obeve that in the epeentation we have that > 0, ince it i the vaiance of the nomal model. Thu, we may contuct kewnomal model that ae poduct of ecaling and it mut only atify that > 0. On the othe hand, we can ee that the ditibution of U i the family of ecaled half nomal ditibution.

4 00 Ovaldo Venega, David Elal Oliveo and Hécto W. Gómez c) If 1, then X λ 1+λ U λ V X SN (λ). d) If Z SN ³ 0,, λ,thenx Z RSN(λ,, ). Theefoe, the RSN model i poduct of a ecaling of the SN model. Popetie 1. The following popetie follow immediately fom Definition 1. a) The pdf of the RSN(0,,) i identical to the pdf of the N(0, ). b) Conide X RSN(λ,, ). Then i) A λ, f X (x; λ,, ) tend to HN(0, ). ii) A, f X (x; λ,, ) tend to N(0, ). c) If X RSN(λ,, ), then X RSN( λ,, ). d) If X RSN(λ,, ) and Y N(0, ) then X and Y have the ame pdf. e) If X RSN(λ,, ), thenx Gamma(1/, )..1. Example of the RSN model ³ µ Example 1. If f X (x) φ x λ 1+α 1+α Φ x with α R and 1+β β R then X RSN ³λ, 1+α, p 1+β. Wenotethat 1+α > 1+β ³ 0 and 1+α, p 1+β ]1, [ ]0, [. Such model i tudied by Elal et al. (004). If conide α β λ then the model coepond to the one peented by Sahu et al. (003) in the univaiate cae and, on the othe hand, if α β 0 it coepond to the SN(λ) model by Azzalini (1985). ³ ³ x Example. If f X (x) φ x λ Φ 4+λ then X RSN ³λ,, 4+λ. Obevethatinthicae > 0 4+λ ³ n o and, 4+λ ], [ ]0, [. Thi model i exhibited by Mukhopadhyay and Vidakovic (1995) in an Bayeian analyi.

5 A note on ecaling of the kew-nomal ditibution 01 Example 3. If f X (x) σ φ x ³ σ Φ λδ x whee σ and δ coepond to the tandad deviation of the nomal model, then X RSN (λ, σ, δ) ince σ δ > 0 and (σ, δ) Θ ]0, [. Nadaajah and Kotz(003) intoduced uch model and they called Skew-Nomal-Nomal ditibution... Moment and Moment Geneating Function of the RSN Model Popoition. Let X RSN (λ,, ) then the moment of ode n ae: (.) λ n µ n E [X n ] π ( + λ ) n nx ³ n k k0 ³ λ k Γ ³ n k +1 b k, 0, kodd whee b k E(V k ) k/ Γ( k+1 )/ π, keven. Poof Thi poof i in the appendix. Popoition 3. Let X RSN (λ,, ) then the moment geneating function i: Ã t! Ã! λt (.3) E [exp (tx)] exp Φ. + λ Poof Thi poof i in the appendix..3. Aymmety and Kutoi Coefficient of the RSN model Popoition 4. Let X RSN (λ,, ) then the aymmety and kutoi coefficient ae: p β1 (4 π) 3 λ 3, (π + π λ λ ) 3 β Poof Conideing that 8(π 3) 4 λ 4 (π + π λ λ ) +3. p β1 µ 3 3µ 1 µ +µ 3 1 µ µ 1 3 and β µ 4 4µ 1 µ 3 +6µ 1 µ 3µ 4 1 µ µ 1

6 0 Ovaldo Venega, David Elal Oliveo and Hécto W. Gómez and fom Popoition 3 we have λ µ 1 π + λ, µ, µ 3 4 λ 3 + λ, µ π ( + λ ) Thi complete the eult. Remak. Since the RSN model i a eult of a ecaling of the SN model, the aymmety and kutoi ange of the RSN model ae equal to the SN model, in othe wod, the inteval coincide with the given in (5). Onthe othe hand, any ecaling of the andom vaiable that ha a kew-nomal ditibution, do no affect the eult of the aymmety and kutoi, a well a, the expeion given in (4). Ifin(9) we conide 1,weobtain (3). The b k in the Popoition ae the moment of the tandad nomal ditibution, ee Johnon et al. (1995). 3. Extenion of the RSN model In thi ection we how extenion of the kew-nomal model uing the ecaled kew-nomal model. Popoition 5. Let X RSN (λ,, ), Y N 0,σ,withX and Y independent, and H X + Y.Then +σ +σ H RSN ³λ,, p + σ + λ σ. Poof Thi poof i in the appendix. Definition. Let X N(0,σ ),weaythatt ha a Tuncated-Nomal ditibution with paamete c, σ, will be denoted by T Υ σ (c), if 1 it denity function, f T (t), i given by: f T (t) 1 F X (c) f X(t), c R, c x<, withf X and f X ditibution and denity function of the vaiable X, epectively. ³ ε Popoition 6. Let T Υ, ε R, λ R, + λ > 0, (, ) Θ ]0, [,andletv N (0, 1) with T and V independent. If (3.1) X g(x) ³ Φ λ + λ T + + λ V, then X have a denity function to g(x), whee 1 (3.) Φ µ x ε φ + λ µ λx + ε.

7 A note on ecaling of the kew-nomal ditibution 03 Poof Thi poof i in the appendix. If X i conideed a in Popoition 6, then we will ay that X ha a extended ecaled-kew-nomal ditibution with paamete λ,,, ε and, will be denoted by X ERSN (λ,,, ε). Definition 3. Let X ERSN (λ,,, ε). The family of ditibution with location and cale i defined a the ditibution of Z µ + σx, µ R and σ>0, wheeθ (µ, σ, λ,,, ε) and will be denoted by ZERSN (θ). Popoition 7. Let µ X W w N (w, 1) and WERSN (0, 1,λ,,,0). Then X ERSN 0, 1,λ, 1+, (1+ ) + λ (1+ ), 0. Poof By a diect computation in f X (x) R f X W (x w)f W (w)dw i obtained the eult. Coollay 1. Let X W ³ w N (w, 1) and WERSN (0, 1,λ,,,0). Then W X x ERSN x 1 1+,, λ,,, λx (1+ ). Popoition 8. Let X V versn (0, 1,w,,,0) and V N (θ 1,θ ). Then f X (x) φ µ x Φ θ 1 x. +θ x Poof By a diect computation in f X (x) eult i obtained. Z f X V (x v)f V (v)dv the Remak 3. Popoition 5 how that the addition of a andom vaiable ecaled kew nomal and a nomal to cale i a ecaled kew nomal. The Popoition 6 extend the eult by Henze (1986) of the extended kew nomal model. When 1in Popoition 7, we obtain the model of the Example. In Bayeian context, the Coollay 1 i a eult of conjugate pioi. When in Popoition 8, we obtain the Skew-Genealized- Nomal model to cale intoduced by Aellano-Valle et al. (004). 4. Appendix PoofofthePopoition1: Let a λ + λ and b + λ then

8 04 Ovaldo Venega, David Elal Oliveo and Hécto W. Gómez P [X x] P [a U + bv x] Z P V x au U u f b U (u)du 0 Z 0 µ x au Φ f b U (u)du By uing a + b and diffeentiating with epect to x we have Z µ 1 x au f X (x) 0 b φ f b U (u)du " b π π exp 1 µ # x Z " exp 1 µ # u ax 0 b du Taking y u ax b, and uing the elation a b λ, the eult i obtained. PoofofthePopoition: Let a λ and b + λ + λ E [X n ] E [(a U + bv ) n ] Ã! nx n a n k E k k0 h Conideing that E U n ki 1 ³ π n k h U n ki h b k E V ki. ³ Γ n k+1 and E(V k ) ae the k-th moment of the tandad nomal andom vaiable then the eult i obtained. PoofofthePopoition3: E [exp (tx)] Z exp [tx] µ µ x λx φ Φ dx Z " # µ exp tx x λx π Φ dx " t # Z exp exp 1 Ã x! t Φ π Taking y x t we have " t # Z E [exp (tx)] exp " t # " exp E Φ Φ Ã! λ y + λt φ(y)dy Ã!# λ T + λt µ λx dx with TN (0, 1)

9 A note on ecaling of the kew-nomal ditibution 05 Thu, E [exp (tx)] exp PoofofthePopoition5: H whee a + σ X + + σ Y h i ³ t λ Φ t + λ λ + λ and b + λ. Thu, [a U + bv ]+ + σ. + σ Y, H a + σ U + + σ (bv + Y ) a + σ U + b + σ W with W N (0, 1) + σ Now, we have the hypothee of Popoition 1, then letting A a +σ and B b +σ, and by uing +σ A +B and A B λ + σ + λ σ follow that H SHN ³λ,, + σ + λ σ. PoofofthePopoition6: Let a λ + λ, b and c ε + λ + λ P [X x] P [at + bv x] Z P V x at T t f T (t)dt b c Z c µ x at Φ f T (t)dt b dp [X x] dx Z p µ p / x at φ( /)t φ ε c b b Φ( )dt + λ 1 µ Z " x φ πbφ( ε ) exp 1 + λ c µ t ax b # dt, taking y t ax b, the eult i obtained. Acknowledgment The autho acknowledge helpful comment and uggetion by the efeee which ubtantially impoved the peentation. The eeach of O. Venega wa uppoted by Gant DGIP The eeach of D. Elal-Oliveo wa uppoted by Gant DIUDA The eeach of H. W. Gómez wa uppoted by FONDECYT (Chile)

10 06 Ovaldo Venega, David Elal Oliveo and Hécto W. Gómez Refeence [1] Aellano-Valle, R. B., Gómez, H. W. and Quintana, F. A. A New Cla of Skew-Nomal Ditibution. Communication in Statititic: Theoy and Method. 33(7), pp , (004). [] Azzalini, A. A cla of ditibution which include the nomal one. Scand. J. Statit. 1, pp , (1985). [3] Azzalini, A. Futhe eult on a cla of ditibution which include the nomal one. Statitica. 46, pp , (1986). [4] Elal, D., Elal, S. and Gómez, H. W. La Ditibución SN(α, β). Revita de la Facultad de Ingenieía. 18, pp. 9-31, (004). [5] Henze, N. A pobabilitic epeentation of the kew-nomal ditibution. Scand. J. Statit. 13, pp , (1986). [6] Johnon, N., Kotz, S. L. and Balakihann, N. Continuou univaiate ditibution, Second Edition. New Yok: Willey and Son, (1995). [7] Mukhopadhyay, S. and Vidakovic, B. Efficiency of Linea Baye Rule fo a Nomal Mean : Skewed Pio Cla. The Statitician. 44 (3), pp , (1995). [8] Nadaajah, N., Kotz, S. Skewed ditibution geneated by the nomal kenel. Statitic & Pobability Lette. 65, pp , (003). [9] Sahu, S. K., Dey, D. K. and Banco, M. A new cla of multivaiate kew ditibution with application to Bayeian egeion model. Canad. J. Statit. 31, pp , (003). Ovaldo Venega Depatamento de Ciencia Matemática y Fíica, Facultad de Ingenieía, Univeidad Católica de Temuco, Chile, ovenega@uct.cl

11 A note on ecaling of the kew-nomal ditibution 07 David Elal-Oliveo Depatamento de Matemática, Facultad de Ingenieía, Univeidad de Atacama, Avenida Copayapu 485, Copiapó, Chile, and Hécto W. Gómez Depatamento de Matemática, Facultad de Ciencia Báica, Univeidad de Antofagata, Antofagata, Chile,