ON BIVARIATE GEOMETRIC DISTRIBUTION. K. Jayakumar, D.A. Mundassery 1. INTRODUCTION

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1 STATISTICA, ao LXVII, 4, 007 O BIVARIATE GEOMETRIC DISTRIBUTIO ITRODUCTIO Probablty dstrbutos of radom sums of deedetly ad detcally dstrbuted radom varables are maly aled modelg ractcal roblems that deal wth certa heomea whch the resectve mathematcal models are sums of radom umber of deedet radom varables A lot of such stuatos arse actuaral scece, queug theory ad uclear hyscs Gedeko ad Korolev (996) gave a umber of stuatos where we usually come across radom summato, esecally geometrc summato ad descrbe the modelg of such stuatos wth resectve hyscal termology Kozubowsk ad Paorska (999) aled the dstrbuto of geometrc sums facal ortfolo modelg Kozubowsk ad Rachev (994) used geometrc sums as a adequate devce to model the foreg currecy exchage rate data I ths aer our am s to obta characterzatos of bvarate geometrc dstrbuto usg geometrc comoudg We troduce ew bvarate geometrc dstrbutos usg the geometrc sums of deedetly ad detcally dstrbuted radom varables These bvarate geometrc dstrbutos are closed uder geometrc summato Therefore the robablty dstrbutos troduced ths study may be arorate modelg bvarate data sets whch are closed uder geometrc summato It s well kow that bvarate aalogues of uvarate dstrbutos ca be obtaed by extedg ther geeratg fuctos arorately Cosder a sequece of deedet Beroull trals whch the robablty of success each tral s, 0< < Let X be the umber of falures recedg frst success The X follows geometrc dstrbuto wth robablty geeratg fucto (gf) Ps ( ) + c( s) where c A atural exteso of Ps ( ) gves the followg bvarate geometrc dstrbuto A o egatve teger valued bvarate radom varable ( XY, ) has bvarate geometrc dstrbuto ( BGD( c, c, θ )) f ts gf s

2 390 π( s, s ) ( ( ))( ( )) ( )( ) + c s + c s θ cc s s () where c, c > 0, 0 θ, s ad s ote that the comoets of ( XY, ) have uvarate geometrc dstrbuto Phatak ad Sreehar (98) cosdered a bvarate geometrc dstrbuto whch could be terreted as a shock model Assume that two comoets are affected by shocks, wth robablty, the frst comoet survves, wth robablty, the secod comoet survves ad wth robablty 0 both comoets fal Let ad be the umber of shocks to the frst ad secod comoets resectvely before the frst falure of the system The (, ) has the followg jot robablty dstrbuto 0 (, ) + P Its gf s π ( s, s) + ( s ) + ( s ) 0 0 ote that (, ) has BGD( c, c,) where, 0 + +,, 0,,, 3, c ad 0 c 0 I order to obta characterzatos of BGD( c, c,) usg geometrc comoudg, we make use of the oerator ' ' defed as follows: Let X be radom varable wth gf π ( s ) X s defed ( dstrbuto) by X the gf π( + s) or X Z where PZ ( ) PZ ( 0), all radom varables j j Z j beg deedet A bvarate exteso of ths result s cosdered If (X, Y) has gf π ( s, s) the the dstrbuto of ( X, Y) s defed by the gf π( + s, + s ) Let {( X, Y ), } be a sequece of deedetly ad detcally dstrbuted radom varables wth gf π ( s, s) Defe ad U X + X + + X V Y + Y + + Y () j j

3 O bvarate geometrc dstrbuto 39 where s deedet of ( X, Y), ad follows geometrc dstrbuto such that P( ) ( ),,,3,, (3) The the gf of ( U, V ) s gve by U V η ( s, s ) E( s s ) ( π( s, s )) P( ) π( s, s ) ( ) π( s, s ) (4) Block (977) cosdered a comoudg scheme usg bvarate geometrc dstrbuto A radom varable (, ) has bvarate geometrc dstrbuto wth arameters 00, 0, 0 ad f ts survval fucto s F (, ) P ( >, > ) + f f ( 0 ) ( 0 + ) (5) where , 0 + <, 0 + < ad,,, 3, Cosder a sequece of deedetly ad detcally dstrbuted radom varables {( X, Y), } whch s also deedet of (, ) where (, ) follows the bvarate geometrc dstrbuto (5) Defe (6) U X ad V Y Block (977) obtaed the Lalace trasform of ( U, V ) Its dscrete aalogue s η( s, s ) π( s, s )( + η( s,) + η(, s ) + η( s, s )) (7) From (7) we get, ( ) π( s,) η( s,) ( + ) π( s,) 0

4 39 ad η(, s ) ( + ) π(, s ) ( ) π(, s ) 00 0 (8) + 0 Usg geometrc comoudg, secto we obta characterzatos of BGD( c, c,) I secto 3, autoregressve models wth BGD( c, c,) margals are develoed Dfferet bvarate geometrc dstrbutos are troduced secto 4 usg bvarate geometrc comoudg CHARACTERIZATIO OF BIVARIATE GEOMETRIC DISTRIBUTIO The followg theorem gves a characterzato of BGD( c, c,) Theorem Let {( X, Y ), } be a sequece of deedetly ad detcally dstrbuted radom varables ad be deedet of ( X, Y ), Suose that follows the geometrc dstrbuto (3) ad, U ad V are as defed () The ( U, V ) ad ( X, Y ), are detcally dstrbuted f ad oly f ( X, Y), follow BGD( c, c,) Proof Let π ( s, s ) be the gf of ( X, Y), Usg (4), the gf of ( U, V ) s η( s, s ) π( + s, + s ) ( ) π( s, s ) (9) + + Assumg that ( X, Y), follow BGD( c, c,) Substtutg π ( s, s) (9) ad smlfyg, we get η ( s, s) + c ( s ) + c ( s ) Coversely, assume that ( U, V ) follows BGD( c, c,) The from (9) we have π( + s, + s) + c ( s ) + c ( s ) ( ) π( + s, + s ) Solvg we get, π ( s, s) + c ( s ) + c ( s ) ow we obta BGD( c, c, θ ) as a geometrc comoud of deedetly ad detcally dstrbuted radom varables

5 O bvarate geometrc dstrbuto 393 Theorem Cosder a sequece {( X, Y), } of deedetly ad detcally dstrbuted radom varables, also deedet of whch has the geometrc dstrbuto (3) Let U ad V be as defed () The ( U, V ) follows BGD( c, c, q ) where q - f ad oly f ( X, Y), have BGD( c, c,0) Proof Suose that ( X, Y ), have gf π ( s, s) ( + c ( s ))( + c ( s )) From (9), the gf of ( U, V ) s η ( s, s ) ( + c ( s ))( + c ( s )) + + c ( s ) + c ( s ) + c c ( s )( s ) Comarg wth (), we get θ q Hece ( U, V ) follows BGD( c, c, q ) To rove the coverse of the theorem, substtutg the gf of ( U, V ) (9), we get π( + s, + s) + c ( s ) + c ( s ) + c c ( s )( s ) ( ) π( + s, + s ) Solvg, we obta π ( s, s) ( + c ( s ))( + c ( s )) A characterzato of geometrc dstrbuto s obtaed the followg theorem Theorem 3 Suose that {( X, Y), } s a sequece of deedetly ad detcally dstrbuted radom varables accordg to BGD( c, c,) The ( U, V ) ad ( X, Y), are detcally dstrbuted f ad oly f s geometrc where U ad V are as defed () Proof The f art of the theorem s roved theorem I order to rove the oly f art, assume that ( X, Y ), ad ( U, V ) are detcally

6 394 dstrbuted as BGD( c, c,) Wthout loss of geeralty, takg c c The gf of ( U, V ) s η( s, s ) ( π( + s, + s )) P( ) From the assumto, we get P ( )( + ( s) + ( s)) + ( s) + ( s) Exadg wth resect to ad comarg the coeffcets of (( s ) + ( s )) j, we get ( + )( + )( + j ) P ( ) Cosder j! j, for j,,3, 3 E( t ) t t t + E ( ) + E ( ( + )) + E ( ( + )( + )) +!! 3! t ( ) ( ) t But E( t ) ( t) P( ) Therefore, P( ) ( ), for,, 3, Theorem 4 Let ad be two deedet radom varables followg geometrc dstrbuto gve (3) wth arameters a ad b resectvely ad {( X, Y ), } be a sequece of deedetly ad detcally dstrb- uted radom varables whch are also deedet of ad Let U X + X + + X ad V Y + Y + + Y, for, The

7 O bvarate geometrc dstrbuto 395 ( a U, a V ) ad ( b U, b V ) are detcally dstrbuted f ( X, Y), have BGD( c, c,) Proof Suose that ( X, Y), have BGD( c, c,) Therefore from (9) we obta the gf of ( a U, a V ) as η ( s, s) + c ( s ) + c ( s ) Smlarly, we ca show that ( b U, b V ) has also the same gf Hece the roof I the ext theorem we obta characterzato of BGD( c, c, θ ) usg bvarate geometrc comoudg Theorem 5 Let {( X, Y), } be a sequece of deedetly ad detcally dstrbuted radom varables Suose that (, ) has bvarate geometrc dstrbuto gve (5) ad deedet of ( X, Y), Take 00 0, 0 µ, 0 µ ad ( µ + µ ) The ( U, V ) follows c BGD c,,0 f ad oly f ( X, Y), follow BGD( c, c,) where U µ µ ad V are as gve (6) Proof Assume that ( X, Y), follow BGD( c, c,) Substtutg ts gf ad the values of 00, 0, 0, (7) The gf of ( U, V ) s ( s, s ) ( ) ( ) η + c s + c s c( s) c( s) µ + + µ + + ( µ µ ) η( s, s) µ µ Solvg we get, η( s, s) c( s) c( s) + + µ µ Coversely, assume that η( s, s) c( s) c( s) + + µ µ

8 396 Therefore from (7) c( s) c( s) + + µ µ ( s, s ) π c( s) c( s) c( s) c( s) µ µ µ µ Substtutg the values of 00, 0, 0 ad we get π ( s, s) + c ( s ) + c ( s ) Theorem 6 Cosder a sequece of deedetly ad detcally dstrbuted radom varables {( X, Y), } Suose that (, ) s deedet of ( X, Y ), ad follows the bvarate geometrc dstrbuto (5) such that, +, + ad 00 0 The 0 0 ( q U, q V ) has BGD( c, c,0) f ad oly f ( X, Y), have BGD( c, c,) where q Proof Usg (7) the gf of ( q U, q V ) s 0( ) π( s,) 0( ) π(, s ) π( s, s) + ( 0 + ) π( s,) ( 0 + ) π(, s ) η( s, s) π( s, s ) where π ( s, s) s the gf of ( X, Y), Assume that ( X, Y), have BGD( c, c,) The we get qq cq s cq s η ( s, s) + ( ) + ( ) + + cq( s) + cq( s ) O smlfcato, we get

9 O bvarate geometrc dstrbuto 397 η ( s, s) ( + c ( s ))( + c ( s )) Coversely suose that (7) we have ( q U, q V ) follows BGD( c, c,0) The from π( q + qs, q + qs) ( + c ( s ))( + c ( s )) π ( q + q s, q + q s ) q q + + c( s) + c( s) Solvg, we get π ( s, s) + c ( s ) + c ( s ) Theorem 7 Let {( X, Y), } be a sequece of deedetly ad detcally dstrbuted radom varables Suose that (, ) s deedet of ( X, Y ), ad has bvarate geometrc dstrbuto, as stated theorem 6 The ( q U, q V ) has gf BGD( qq,,0) f ad oly f ( X, Y ), have gf BGD( q, q,) where q - Proof Suose that ( X, Y), have BGD( q, q,) Substtutg (7), we get the gf of ( q U, q V ) as η( s, s ) qq q q( s ) + q q( s ) + q q( s ) + q q( s ) + O smlfcato we get η ( s, s) ( + q( s ))( + q( s )) Coversely, take that ( q U, q V ) follows BGD( qq,,0) From (9), we have ( + q( s ))( + q( s )) π( q+ qs, q+ qs) q q + π( q+ qs, q+ qs) + q( s) + q( s)

10 398 Smlfyg, we get π ( s, s) + q ( s ) + q ( s ) 3 BIVARIATE AUTOREGRESSIVE GEOMETRIC PROCESS There are may stuatos whch dscrete tme seres arse, ofte as couts of evets, objects or dvduals cosecutve tervals For examle, umber of road accdets that occur o atoal hghways of a coutry o a day, umber of customers watg a tcket bookg couter that recorded every oe hour durato, umber of calls receved a fre rescue cetre a week, etc Moreover such data ca also be obtaed by dscretzato of cotuous varate tme seres Recetly, much focus s gve o develog teger valued statoary autoregressve rocesses (see, Jayakumar (995)) Here we develo frst order autoregressve rocess wth margals as bvarate geometrc dstrbuto Cosder a autoregressve rocess ( X, Y ),, wth structure ( X0, Y0) d( U, V ) ad for,, 3, ( X, Y ) ( ρ X + U, ρ Y + V ), 0< ρ < (0) where ( U, V), are deedetly ad detcally dstrbuted radom varables satsfyg ( U, V) d( Iε, Iψ ) { I, } ad {( ε, ψ), } are two deedet sequeces of deedetly ad detcally dstrbuted radom varables ad ( I, ) have Beroull dstrbuto wth PI ( 0) PI ( ) ρ The followg theorem gves a ecessary ad suffcet codto for a statoary autoregressve rocess to have BGD( c, c,) margals Theorem 3 Let ( X, Y), be a frst order bvarate autoregressve rocess wth structure (0) The rocess s statoary wth BGD( c, c,) margals f ad oly f ( ε, ψ), follow BGD( c, c,) Proof The gf of (0) s π ( s, s ) π ( ρ + ρs, ρ + ρs ) π ε ψ ( s, s ) () X, Y X, Y, Suose that the rocess s statoary wth BGD( c, c,) margals, the we have π + c ( s ) + c ( s ) + c ρ( s ) + c ρ( s ) ( s, s ) UV,

11 O bvarate geometrc dstrbuto 399 Solvg we get, UV, ( s, s ρ π ) ρ + + c ( s ) + c ( s ) To rove the coverse assume that ( ε, ψ, ) follow BGD( c, c,) Takg (), we get π ( s, s ) π ( ρ + ρs, ρ + ρs ) π ( s, s ) X, Y X0, Y0 U, V Uder the assumto, π ρ (, ) ρ + ( ) ( ) ( ) ( ) X, Y s s + cρ s + cρ s + c s + c s + c ( s ) + c ( s ) Hece the rocess s statoary wth margals follow BGD( c, c,) ow cosder a frst order radom coeffcet autoregressve model wth followg structure ( X, Y ) d( ε, ψ ) ( X, Y ) ( V X + V ε, V Y + V ψ ), () where {( ε, ψ, )} ad { V, } are two deedet sequeces of deedetly ad detcally dstrbuted radom varables such that V, have uform dstrbuto o (0, ) The followg theorem gves a ecessary ad suffcet codto for the rocess () to be statoary Theorem 3 Cosder a autoregressve rocess ( X, Y), gve () It s a statoary frst order autoregressve rocess wth BGD( c, c,) margals f ad oly f ( ε0, ψ 0)d BGD( c, c,) Proof The gf of () s V X V V Y V X, Y s s E s + ε s + ψ π (, ) ( ) πx, Y ( v + vs, v + vs ) π, ( v + vs, ε ψ v + vs) dv 0 If ( X, Y ), s statoary, we have

12 400 π XY, ( s, s) π XY, ( v + vs, v + vs ) dv 0 Let πxy, s s γxy, s s γ (, ) (, ) The we get XY, s s γxy, 0 (, ) ( vs, vs ) dv (3) Takg s j δ s for j, j XY, s XY, 0 γ (( δ, δ ) ) γ (( δ, δ ) sv) dv If sv t, the s XY, XY, 0 sγ (( δ, δ ) s) γ (( δ, δ ) t) dt XY, Dfferetatg wth resect to s ad the dvdg by γ (( δ, δ ) s ), we get ' XY, XY, sγ (( δ, δ ) s) + γ (( δ, δ ) s ) γ (( δ, δ ) s ) XY, Wrtg γxy, (( δ, δ) s ) + ω(( δ, δ ) s ), we get ω(( δ, δ ) s) µ s where µ s a fucto of ( δ, δ ) That s, γxy, (( δ, δ) s ) + µ s + cs + cs Thus π XY, ( s, s ) + c( s) + c( s) ad hece we obta ( ε0, ψ 0)d BGD( c, c,) Coversely, suose that ( X0, Y0) d( ε0, ψ 0) ad ( ε0, ψ 0) has BGD( c, c,) From (3), we have

13 O bvarate geometrc dstrbuto 40,, 0 γx Y( s, s ) γ ε ψ ( vs, vs ) dv cs + cs Therefore, ( s, s ) ( ) ( ) π X, Y + c s + c s Thus ( X, Y ) s dstrbuted as bvarate geometrc dstrbuto By ducto we get ( X, Y), follow BGD( c, c,) Hece the rocess s statoary 4 BIVARIATE GEOMETRIC DISTRIBUTIO I ths secto we costruct varous bvarate geometrc dstrbutos usg the bvarate geometrc comoudg We dscuss the dscrete aalogues of may mortat bvarate exoetal dstrbutos lke, Marshall-Olk s (967) bvarate exoetal, Dowto s (970) bvarate exoetal ad Hawkes (97) bvarate exoetal Theorem 4 Let {( X, Y), } be a sequece of deedetly ad detcally dstrbuted radom varables accordg to BGD( c, c,) ad (, ) follow, deedet of ( X, Y),, the bvarate geometrc dstrbuto gve (5) Choose 00 µ, 0 µ, 0 µ µ ad µ µ + µ + µ Defe U ad V as gve (6) The ( U, V ) follows bvarate geometrc dstrbuto whch s the dscrete aalogue of Marshall-Olk s bvarate exoetal Proof Let ( X, Y), have gf π ( s, s) Substtutg the values of 00, 0, 0, ad π ( s, s) (7), we get η ( s, s ) ( µ + µ η ( s,) + µη (, s ) + ( µ ) η( s, s )) + c( s) + c( s) Solvg for η ( s,) ad η (, s ), we get (4) η( s,) µ + µ µ + µ + c ( s ) (, ) µ + µ ( ) ad η s µ + µ + c s

14 40 Substtutg η ( s,) ad η (, s ) (4), we get ( s, s ) ( ) ( ) η + c s + c s µ ( µ + µ ) µ ( µ + µ ) µ ( µ ) η( s, s) µ + µ + c( s) µ + µ + c( s) O smlfcato, c( s) c( s) η ( s, s) µ + µ + + µ + µ + c( s) + c( s) µ µ I the ext theorem, we gve the bvarate geometrc dstrbuto whch s aalogues to Dowto s bvarate exoetal dstrbuto Theorem 4 Suose that ( X, Y), are deedetly ad detcally dstrbuted radom varables wth gf c( s) c( s) π( s, s) + +, µ > 0 ad deedet of + µ + µ (, ) where (, ) follows the bvarate geometrc dstrbuto gve (5) Takg 00 ( + µ ), ad µ ( + µ ) The( U, V ) has bvarate geometrc dstrbuto whch s the dscrete aalogue of Dowto s bvarate exoetal dstrbuto where U ad V are as defed (6) Proof Suosg that ( X, Y), have gf π ( s, s) From (7) the gf of ( U, V ) s c( s) c( s) s s η ( s, s ) (( µ ) µ ( µ ) η(, )) + µ + µ c( s) c( s) ( + µ ) + µ + µ + + µ + µ ( µ ) ( µ ) ( ) ( µ ) ( ) ( )( ) µ ( µ ) c s + + c s + cc s s + ( + c ( s ))( + c ( s )) c c ( s )( s ) µ +µ

15 O bvarate geometrc dstrbuto 403 By comarg wth (), we ote that ( U, V ) follows BGD( c, c, θ ) where µ θ + µ The bvarate geometrc form of Hawkes bvarate exoetal dstrbuto s gve the followg theorem Theorem 43 Let {( X, Y), } be a sequece of deedetly ad detcally dstrbuted radom varables wth gf s π( s, s) ( + µ ( s))( + µ ( s)) Assume that (, ) follow bvarate geometrc dstrbuto gve (5) ad deedet of ( X, Y), Choose 0 0, γ ad γ The the dstrbuto of ( U, V ) s the bvarate geometrc form of Hawkes bvarate exoetal dstrbuto Proof From (8), we have η( s,) π( s,)( γ + ( γ ) π( s,)) Substtutg π ( s,) ad solvg η ( s,), we get η( s,) µ ( s) + γ Smlarly η(, s ) µ ( s ) + γ Aga from (7) η( s, s) ( + µ ( s ))( + µ ( s )) (, ) ( ) + s ( s) π µ µ s s + + γ γ µ ( s) µ ( s) µ µ ( s ) ( s) γ γ γ γ µ µ + ( s) + ( s) (( + µ ( s))( + µ ( s)) ) γ γ

16 404 Thus ( U, V ) have the bvarate geometrc form of Hawkes bvarate exoetal dstrbuto Deartmet of Statstcs Uversty of Calcut, Ida K JAYAKUMAR DAVIS ATOY MUDASSERY REFERECES H W BLOCK, (977), A famly of bvarate lfe dstrbutos Theory ad Alcatos of Relablty: Wth Emhass o Bayesa ad oarametrc Methods, C P Tsokos ad I Shm, eds, Academc Press, F DOWTO, (970), Bvarate exoetal dstrbutos relablty theory Joural of Royal Statstcal Socety, Seres B, 3, B V GEDEKO, V KOROLEV, (996), Radom summato: Lmt theorems ad alcatoscrc Press, ew York A G HAWKES, (97), A bvarate exoetal dstrbuto wth alcatos to relablty Joural of Royal Statstcal Socety, Seres B, 34, 9-3 K JAYAKUMAR, (995), The statoary soluto of a frst order teger valued autoregressve rocesses Statstca, LV, -8 T J KOZUBOWSKI, A K PAORSKA, (999), Multvarate geometrc stable dstrbuto facal alcatos Mathematcal ad Comuter Modelg, 9, 83-9 T J KOZUBOWSKI, S T RACHEV, (994), The theory of geometrc stable laws ad ts use modelg facal data Euroea Joural of Oeratos Research, 74, A W MARSHALL, I OLKI, (967), A multvarate exoetal dstrbuto Joural of the Amerca Statstcal Assocato, 6, A G PHATAK, M SREEHARI, (98), Some characterzatos of bvarate geometrc dstrbuto, Joural of Ida Statstcal Assocato, 9, 4-46 O bvarate geometrc dstrbuto SUMMARY Characterzatos of bvarate geometrc dstrbuto usg uvarate ad bvarate geometrc comoudg are obtaed Autoregressve models wth margals as bvarate geometrc dstrbuto are develoed Varous bvarate geometrc dstrbutos aalogous to mortat bvarate exoetal dstrbutos lke, Marshall-Olk s bvarate exoetal, Dowto s bvarate exoetal ad Hawkes bvarate exoetal are reseted