Article Multivariate Extended Gamma Distribution

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1 axoms Artcle Multvarate Exteded Gamma Dstrbuto Dhaya P. Joseph Departmet of Statstcs, Kurakose Elas College, Maaam, Kottayam, Kerala , Ida; Tel.: Academc Edtor: Has J. Haubold Receved: 7 Jue 2016; Accepted: 13 Aprl 2017; Publshed: 24 Aprl 2017 Abstract: I ths paper, I cosder multvarate aalogues of the exteded gamma desty, whch wll provde multvarate extesos to Tsalls statstcs ad superstatstcs. By makg use of the pathway parameter β, multvarate geeralzed gamma desty ca be obtaed from the model cosdered here. Some of ts specal cases ad lmtg cases are also metoed. Codtoal desty, best predctor fucto, regresso theory, etc., coected wth ths model are also troduced. Keywords: pathway model; multvarate exteded gamma desty; momets 1. Itroducto Cosder the geeralzed gamma desty of the form where c 1 = γ+1 a Γ( γ+1 g(x = c 1 x γ e ax, x 0, a > 0, > 0, γ + 1 > 0, (1, s the ormalzg costat. Note that ths s the geeralzato of some stadard statstcal destes such as gamma, Webull, expoetal, Maxwell-Boltzma, Raylegh ad may more. We wll exted the geeralzed gamma desty by usg pathway model of [1] ad we get the exteded fucto as where c 2 = g 1 (x = c 2 x γ [1 + a(β 1x ] 1, x 0, β > 1, a > 0, > 0 (2 γ+1 (a( Γ( 1 Γ( γ+1 Γ( 1 γ+1, s the ormalzg costat. Note that g 1 (x s a geeralzed type-2 beta model. Also lm β 1 g 1 (x = g(x, so that t ca be cosdered to be a exteded form of g(x. For varous values of the pathway parameter β a path s created so that oe ca see the movemet of the fucto deoted by g 1 (x above towards a geeralzed gamma desty. From the Fgure 1 we ca see that, as β moves away from 1 the fucto g 1 (x moves away from the org ad t becomes thcker taled ad less peaked. From the path created by β we ote that we obta destes wth thcker or ther tal compared to geeralzed gamma desty. Observe that for β < 1, wrtg β 1 = (1 β Equato (2 produce geeralzed type-1 beta form, whch s gve by g 2 (x = c 3 x γ [1 a(1 βx ] 1 1 β, 1 a(1 βx 0, β < 1, a > 0, > 0 Axoms 2017, 6, 11; do: /axoms

2 Axoms 2017, 6, 11 2 of 12 where c 3 = γ+1 (a(1 β Γ( 1 β 1 γ Γ( γ+1 Γ( 1 1 β +1, s the ormalzg costat (see [2]. Fgure 1. The graph of g 1 (x, for γ = 1, a = 1, = 2, = 1 ad for varous values of β. From the above graph, oe ca see the movemet of the exteded gamma desty deoted by g 1 (x towards the geeralzed gamma desty, for varous values of the pathway parameter β. Beck ad Cohe s superstatstcs belog to the case (2 [3,4]. For γ = 1, a = 1, = 1 we have Tsalls statstcs [5,6] for β > 1 from (2. Several multvarate extesos of the uvarate gamma dstrbutos exst the lterature [7 9]. I ths paper we cosder a multvarate aalogue of the exteded gamma desty (2 ad some of ts propertes. 2. Multvarate Exteded Gamma Varous multvarte geeralzatos of pathway model are dscussed the papers of Matha [10,11]. Here we cosder the multvarate case of the exteded gamma desty of the form (2. For X 0, = 1, 2,...,, let f β (x 1, x 2,..., x = k β x γ 1 1 xγ xγ [1 + (β 1(a 1 x 1 + a 2x a x ], β > 1, > 0, > 0, a > 0, = 1, 2,...,, (3 where k β s the ormalzg costat, whch wll be gve later. Ths multvarate aalogue ca also produce multvarate extesos to Tsalls statstcs [5,12] ad superstatstcs [3]. Here the varables

3 Axoms 2017, 6, 11 3 of 12 are ot depedetly dstrbuted, but whe β 1 we have a result that X 1, X 2,..., X wll become depedetly dstrbuted geeralzed gamma varables. That s, lm f β(x 1, x 2,..., x = f (x 1, x 2,..., x β 1 = kx γ 1 1 xγ xγ e b 1x 1... b x, x 0, b = a > 0, > 0, = 1, 2,...,, (4 where k = γ +1 b Γ( γ +1, γ + 1 > 0, = 1, 2,...,. The followg are the graphs of 2-varate exteded gamma wth γ 1 = 1, γ 2 = 1, a 1 = 1, a 2 = 1, = 2, = 2 ad for varous values of the pathway parameter β. From the Fgures 2 4, we ca see the effect of the pathway parameter β the model. Fgure 2. β = 1.2. Fgure 3. β = 1.5.

4 Axoms 2017, 6, 11 4 of 12 Specal Cases ad Lmtg Cases Fgure 4. β = Whe β 1, (3 wll become depedetly dstrbuted geeralzed gamma varables. Ths cludes multvarate aalogue of gamma, expoetal, chsquare, Webull, Maxwell- Boltzma, Raylegh, ad related models. 2. If = 1, a 1 = 1, = 1, β = 2, (3 s detcal wth type-2 beta desty. 3. If β = 2, a 1 = a 2 =... = a = 1, = =... = = 1 (3, the (3 becomes the type-2 Drchlet desty, D(x 1, x 2,..., x = dx ν x ν x ν 1 [1 + x 1 + x x ] (ν ν +1, x 0, (5 where ν = γ + 1, = 1, 2,...,, ν +1 = (ν ν ad d s the ormalzg costat (see [13,14]. A sample of the surface for = 2 s gve the Fgure 5. Fgure 5. The graph of bvarate type-2 Drchlet wth γ 1 = γ 2 = 1, = 6.

5 Axoms 2017, 6, 11 5 of Margal Desty We ca fd the margal desty of X, by tegratg out X 1, X 2,..., X 1, X +1,..., X 1, X. Frst let us tegrate out X, the the jot desty of X 1, X 2,..., X 1 deoted by f 1 s gve by f 1 (x 1, x 2,..., x 1 = x >0 f β (x 1, x 2,..., x dx = k β x γ 1 1 xγ xγ 1 1 [1 + (β 1(a 1x a 1x 1 x γ [1 + Cx ] dx, x 1 ] (6 where C = (a. By puttg y = [1+((a 1 x a 1 x 1 1 ] Cx ad tegratg we get f 1 (x 1, x 2,..., x 1 = k βγ( γ +1 Γ( γ +1 γ x 1 [a (β 1] γ+1 Γ( 1 xγ xγ 1 1 [ [1 + (β 1(a 1 x 1 + a 2x a 1x 1 1 ] γ+1 ], (7 x 0, = 1, 2,..., 1, a > 0, > 0, = 1, 2,...,, β > 1, > 0, γ +1 > 0, γ + 1 > 0. I a smlar way we ca tegrate out X 1, X 2,..., X 1, X +1,..., X 1. The the margal desty of X s deoted by f 2 ad s gve by [ f 2 (x = k 2 x γ [1 + (β 1a x ] γ+1... γ 1 +1 γ γ ], (8 where x 0, β > 1, > 0, > 0, γ + 1 > 0, γ +1 k 2 = (a ( Γ( γ+1... γ 1 +1 γ γ Γ( γ +1 Γ( γ γ+1, γ γ γ γ 1+1 > 0,... γ +1 > 0. If we take ay subset of (X 1,..., X, the margal destes belog to the same famly. I the lmtg case they wll also become depedetly dstrbuted geeralzed gamma varables. Normalzg Costat Itegratg out X from (8 ad equatg to 1, we wll get the ormalzg costat k β as γ k β = (a 1 (β 1 1 (a 2 (β 1 γ 2 +1 Γ( γ 1+1 Γ( γ Γ( γ +1 > 0, a > 0, γ + 1 > 0, = 1, 2,...,, β > 1, > 0, 4. Jot Product Momet ad Structural Represetatos... (a (β 1 γ+1 Γ( Γ(... γ +1, (9... γ +1 > 0. Let (X 1,..., X have a multvarate exteded gamma desty (3. By observg the ormalzg costat (23, we ca easly obtaed the jot product momet for some arbtrary (h 1,..., h,

6 Axoms 2017, 6, 11 6 of 12 1 xh xh = k β Γ( γ1+h1+1 =... Γ( γ +h +1 γ 1 +h (a 1 (β 1 1 Γ( γ 1+h γ +h +1 Γ(... γ +1 Γ( γ 1+h γ +h (a (β 1 γ+h+1 Γ( h [a (β 1] Γ( γ + h + 1 Γ( γ + 1, (10 γ + h + 1 > 0, γ + h + 1 > 0, β 1 = 1, 2,...,. γ + 1 > 0, γ + 1 > 0, a > 0, β > 1, > 0, Property 1. The jot product momet of the multvarate exteded gamma desty ca be wrtte as 1 xh xh = ( Γ ( Γ γ + h + 1 γ + 1 where Y s are geeralzed gamma radom varables havg desty fucto E(y h, (11 f y (y = c y γ e [a (y ], y 0, β > 1, a > 0, > 0, (12 γ +1 where c = [a (], γ Γ( γ > 0, = 1, 2,...,, s the ormalzg costat. Property 2. Lettg h 2 =... = h = 0, (10, we get γ 1+h 1 +1 =2 β > 1, > 0. (13 s the h th 1 1 = Γ( γ 1+h 1 +1 γ γ +1 γ + 1 > 0, where k 3 s the ormalzg costat. The Γ(... γ +1 h 1 [a 1 (β 1] Γ( γ 1+h Γ( γ 1+1, (13 γ 1 + h > 0, β 1 γ + 1 > 0, γ + 1 > 0, a 1 > 0, momet of a radom varable wth desty fucto of the the form (8, f 3 (x 1 = k 3 x γ 1 1 [1 + (β 1a 1x 1 ] [ 1 = k 3 x γ 1 1 [1 + a 1(β 1x 1 1 ] [ 0 γ 2 +1 γ 2 +1 ]... γ+1, (14... γ+1 ] dx 1 (15 Makg the substtuto y = a 1 (β 1x 1, the t wll be the form of a type-2 beta desty ad we ca easly obtaed the h th 1 momet as (13.

7 Axoms 2017, 6, 11 7 of 12 Property 3. Lettg h 3 =... = h = 0, (10, we get 1 xh 2 2 = Γ( γ 1+h 1 +1 γ 2+h 2 +1 Γ(... γ +1 =3... γ +1 Γ( γ 1+h 1 +1 Γ( γ 2+h 2 +1 h 1 [a 1 (β 1] 1 [a 2 (β 1] h 2 2 Γ( γ 1+1 Γ( γ 2+1, (16 γ 1+h 1 +1 γ 2+h 2 +1 γ + 1 > 0, β 1 γ + 1 > 0, β > 1, γ + h + 1 > 0, γ + 1 > 0, a > 0, > 0, = 1, 2, whch s the jot product momet of a bvarate exteded gamma desty s deoted by f 4 ad s gve by f 4 (x 1 x 2 = k 4 x γ 1 1 xγ 2 2 [1 + (β 1(a 1x 1 + a 2x 2 ] [ γ ]... γ+1, (17 where k 4 s the ormalzg costat. (17 s obtaed by tegratg out X 3,..., X from (3. By puttg h 4 =... = h = 0, (10, we get the jot product momet of trvarate exteded gamma desty ad so o. Theorem 1. Whe X 1,..., X has desty (3, the E{x h xh [1 + (β 1(a 1 x a x ] h } Γ( γ 1+h 1 +1 = k 1... Γ( γ +h +1 β γ 1 +h (a 1 (β 1 1 = Γ( h γ 1+h γ +h +1 Γ( Γ( h γ 1+h γ +h +1 Γ(... γ +1 Γ( h... (a (β 1 γ+h+1 Γ( h { [a (β 1] Γ( γ + h + 1 h Γ( γ + 1 }, (18 γ h + h + 1 > 0, γ + h + 1 > 0, > 0, > 0, = 1, 2,...,. β 1 γ + 1 > 0, γ + 1 > 0, a > 0, β > 1, Corollary 1. Whe X 1,..., X has desty (3, the E{[1 + (β 1(a 1 x a x h γ + h + 1 > 0, 4.1. Varace-Covarace Matrx ] h } = Γ( β 1 h > 0, Γ( h γ 1+h γ +h +1 Γ( β 1 > 0, = 1, 2,...,.... γ +1 Γ( h, (19 γ + 1 > 0, a > 0, β > 1, > 0, Let X be a 1 vector. Varace-covarace matrx s obtaed by takg E[(X E(X(X E(X ]. The the elemets wll be of the form

8 Axoms 2017, 6, 11 8 of 12 where ad Var(x 1 Cov(x 1, x 2... Cov(x 1, x E[(X E(X(X E(X Cov(x 2, x 1 Var(x 2... Cov(x 2, x ] =.... Cov(x, x 1 Cov(x, x 2... Var(x Cov(x, x j = E(x x j E(x E(x j,, j = 1, 2,...,, = j (20 Var(x = E(x 2 [E(x ] 2, = 1, 2,...,. (21 E(x x j s are obtaed from (10 by puttg h = h j = 1 ad all other h k = 0, k = 1, 2,...,, k =, j. E(x s ad E(x 2 s are respectvely obtaed from (10 by puttg h = 1 ad h = 2 ad all other h k = 0, k = 1, 2,...,, k =. Where 4.2. Normalzg Costat E(x 1 x 2 = 0 0 x 1 x 2 f 2 (x 1, x 2 dx 1 dx 2. (22 Itegrate out x from (8 ad equate wth 1, we wll get the ormalzg costat K α as γ K α = +1 1 (a 1 (α 1 1 (a 2 (α 1 γ 2 +1 Γ( γ 1+1 Γ( γ 2+1 Γ( γ Regresso Type Models ad Lmtg Approaches (a (α 1 γ+1 Γ( Γ( α 1 γ 1+1 γ +1 α 1. (23 The codtoal desty of X gve X 1, X 2,..., X 1, X +1,..., X s deoted by f 5 ad s gve by f 5 (x x 1, x 2,..., x 1, x +1,..., x = γ = +1 [a (β 1] Γ( Γ( γ +1 Γ( γ +1 [1 + xγ f β (x 1, x 2,..., x f 6 (x 1, x 2,..., x 1, x +1,..., x (β 1a x 1 + (β 1(a 1 x 1 + a 2x a 1x a +1x a x ] [1 + (β 1(a 1 x 1 + a 2x a 1x a +1x a x ] γ +1, (24 where f 6 s the jot desty of X 1, X 2,..., X 1, X +1,..., X. Whe we take the lmt as β 1 Equato (24, we ca see that the codtoal desty wll be the form of a geeralzed gamma desty ad s gve by lm f 5(x x 1, x 2,...,, x 1, x +1,..., x = (a β 1 Γ( γ +1 γ +1 x 0, > 0, > 0, γ + 1 > 0. xγ e a x, (25

9 Axoms 2017, 6, 11 9 of 12 Theorem 2. Let (X 1, X 2,..., X have a multvarate exteded gamma desty (3, the the lmtg case of the codtoal desty f β (x x 1, x 2,...,, x 1, x +1,..., x wll be a geeralzed gamma desty (25. Best Predctor The codtoal expectato, E(x x 1,..., x 1, s the best predctor, best the sese of mmzg the expected squared error. Varables whch are preassged are usually called depedet varables ad the others are called depedet varables. I ths cotext, X s the depedet varable or beg predcted ad X 1,..., X 1 are the preassged varables or depedet varables. Ths best predctor s defed as the regresso fucto of X o X 1,..., X 1. E(x x 1,..., x 1 = x =0 x f 7 (x x 1,..., x 1 dx = [a (β 1] γ+1 Γ( Γ( γ +1 Γ( [1 + (β 1(a 1x a 2x x =0 γ a 1x 1 x γ +1 [1 + (β 1(a 1 x 1 + a 2x a x ] dx. 1 γ+1 ] + (26 We ca tegrate the above tegral as the case of Equato (6. The after smplfcato we wll get the best predctor of X at preassged values of X 1,..., X 1 whch s gve by E(x x 1,..., x 1 = [a (β 1] 1 Γ( γ +2 Γ( γ +2 Γ( γ +1 Γ( γ +1 [1 + (β 1(a 1 x 1 + a 2x a 1x 1 1 ] 1, (27 > 0, a > 0, β > 1, x > 0, = 1, 2,..., 1, γ +2 > 0, γ + 1 > 0. We ca take the lmt β 1 (27. For takg lmt, let us apply Strlg s approxmatos for gamma fuctos, see for example [15] Γ(z + a (2π 1 2 z z+a 1 2 e z, for z ad a s bouded (28 to the gamma s (27. The we wll get lm E(x x 1,..., x 1 = β 1 Γ( γ +2 (a 1 Γ( γ +1 (29 whch s the momet of a geeralzed gamma desty as gve ( Multvarate Exteded Gamma Whe β < 1 Cosder the case whe the pathway parameter β s less tha 1, the the pathway model has the form g(x = Kx γ [1 a(1 βx ] 1 β, β < 1, a > 0, > 0, > 0, (30

10 Axoms 2017, 6, of 12 1 a(1 βx 0, ad K s the ormalzg costat. g(x s the geeralzed type-1 beta model. Let us cosder a multvarate case of the above model as g β (x 1, x 2,..., x = K β x γ 1 1 xγ xγ [1 (1 β(a 1 x 1 + a 2x a x ] 1 β, β < 1, > 0, > 0, a > 0, = 1, 2,...,, 1 (1 β(a 1 x 1 + a 2x a x 0. where K β s the ormalzg costat ad t ca be obtaed by solvg K β... x γ xγ [1 (1 β(a 1 x a x ] 1 β dx 1... dx = 1 (32 Itegrato over x yelds the followg, K β x γ 1 1 xγ xγ 1 1 [1 (1 β(a 1x a 1x 1 1 ] u 1 β x γ [1 + C 1 x ] dx, (33 0 where u = [ 1 (1 β(a 1 x a 1 x 1 1 a (1 β ] 1 ad C 1 = (1 βa (31 [1 (1 β(a 1 x a 1 x 1 1 ]. Lettg y = C 1x the the above tegral becomes a type-1 Drchlet tegral ad the ormalzg costat ca be obtaed as, K β = j=1 γ j +1 [ j ((1 βa j j ]Γ(1 + 1 β + γ γ + 1 Γ( γ Γ( γ +1 Γ(1 + 1 β (34 Whe β 1, (31 wll become the desty of depedetly dstrbuted geeralzed gamma varables. By observg the ormalzg costat (34, we ca easly obtae the jot product momet for some arbtrary (h 1,..., h, 1 xh xh = K β = j=1 h j [((1 βa j j=1 γ j +h j +1 [ j ((1 βa j Γ( γ1+h Γ( γ +h +1 Γ(1 + 1 β j ]Γ(1 + 1 β + γ 1 + h γ + h + 1 Γ(1 + 1 β + γ γ +1 Γ( γ 1+h Γ( γ +h +1 j ]Γ(1 + 1 β + γ 1 + h γ + h + 1 Γ( γ Γ( γ + 1, (35 γ + h j + 1 > 0, γ j + 1 > 0, a j > 0, β < 1, j > 0, j = 1, 2,...,. Lettg h 2 =... = h = 0, (35, we get 1 = Γ(1 + h 1 [(1 βa 1 ] 1 ]Γ(1 + 1 β + γ γ +1 Γ( γ 1+h β + γ 1+h γ γ +1 Γ( γ 1+1 γ 1 + h > 0, γ j + 1 > 0, a 1 > 0, β < 1, j > 0, > 0, j = 1, 2,...,., (36

11 Axoms 2017, 6, of 12 (13 s the h th 1 momet of a radom varable wth desty fucto, g 1 (x 1 = K 1 x γ 1 1 [1 (1 βa 1x 1 ] where K 1 s the ormalzg costat. Lettg h 3 =... = h = 0, (35, we get 1 β + γ γ+1, (37 = 1 xh j=1 h j [((1 βa j Γ(1 + 1 β + γ γ +1 Γ( γ 1+h 1 +1 Γ( γ 2+h 2 +1 j ]Γ ( β + γ + h j=3 γ j + 1 j Γ( γ Γ( γ 2 + 1, (38 γ 1 + h > 0, (γ 2 + h > 0, γ j + 1 > 0, a 1 > 0, a 2 > 0 β < 1, j > 0, γ j + 1 > 0, j = 1, 2,...,. If we proceed the smlar way as Secto 4.1, here we ca deduce the varace-covarace matrx of multvarate exteded gamma for β < Coclusos Multvarate couterparts of the exteded geeralzed gamma desty s cosdered ad some propertes are dscussed. Here we cosdered the varables as ot depedetly dstrbuted, but whe the pathway parameter β 1 we ca see that X 1, X 2,..., X wll become depedetly dstrbuted geeralzed gamma varables. Jot product momet of the multvarate exteded gamma s obtaed ad some of ts propertes are dscussed. We ca see that the lmtg case of the codtoal desty of ths multvarate exteded gamma s a geeralzed gamma desty. A graphcal represetato of the pathway s gve Fgures 1 4. Ackowledgmets: The author ackowledges gratefully the ecouragemets gve by Professor A. M. Matha, Departmet of Mathematcs ad Statstcs, McGll Uversty, Motreal, QC, Caada ths work. Coflcts of Iterest: The author declares o coflct of terest. Refereces 1. Matha, A.M. A Pathway to matrx-varate gamma ad ormal destes. Lear Algebra Appl. 2005, 396, Joseph, D.P. Gamma dstrbuto ad extesos by usg pathway dea. Stat. Pap. Ger. 2011, 52, Beck, C.; Cohe, E.G.D. Superstatstcs. Physca A 2003, 322, Beck, C. Stretched expoetals from superstatstcs. Physca A 2006, 365, Tsalls, C. Possble geeralzatos of Boltzma-Gbbs statstcs. J. Stat. Phys. 1988, 52, Matha, A.M.; Haubold, H.J. Pathway model, superstatstcs Tsalls statstcs ad a geeralzed measure of etropy. Physca A 2007, 375, Kotz, S.; Balakrshma, N.; Johso, N.L. Cotuous Multvarate Dstrbutos; Joh Wley & Sos, Ic.: New York, NY, USA, Matha, A.M.; Moschopoulos, P.G. O a form of multvarate gamma dstrbuto. A. Ist. Stat. Math. 1992, 44, Furma, E. O a multvarate gamma dstrbuto. Stat. Probab. Lett. 2008, 78, Matha, A.M.; Provost, S.B. O q-logstc ad related models. IEEE Tras. Relab. 2006, 55,

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