RECURRENCE RELATIONS FOR MOMENTS OF RECORD VALUES FROM INVERTED WEIBULL DISTRIBUTION

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1 RECURRENCE RELATIONS FOR MOMENTS OF RECORD VALUES FROM INVERTED WEIBULL DISTRIBUTION B SHAFYA A. AL-HIDAIRAH AND GANNAT R. AL-DAYIAN Depatet o tatitic, Facult o Sciece(Gil), Kig Abdulaziz Uiveit, Jeddah KSA ABSTRACT Recuece elatio o the igle ad double oet o lowe ecod value aiig o iveted Weibull ditibutio (iveted epoetial, ad iveted Raleigh a pecial cae) ae deived. Nueical tud o the ecuece elatio o oet o ecod value aiig o iveted Weibull ad coet ae give. Kewod : Iveted Weibull ditibutio; ecod value; ecuece elatio.. INTRODUCTION Recod value ad the aociated tatitic ae o iteet ad ipotat i a eal lie applicatio, uch a: weathe, educatio, idut, ecooic, ad pot data. The tatitical tud o ecod value tated with Chadle (95) ad o pead i dieet diectio. Fote ad Stuat (954) wee pioee i idig applicatio o ecod coutig tatitic i ieece. The obtaied the ea ad vaiace o the total ube o ecod i a equece o legth. Felle (968) peeted oe eaple o ecod value i gablig poble. Reic (973) ad Shoo (973) dicued the aptotic theo o ecod. A good eleeta eview coceig o ecod value i give b Glic (978). Two oe uve ae give b Nevzoov (987) ad Nagaaa (988). Popetie o ecod value have bee eteivel tudied i the liteatue ee o eaple, Ahaullah (988), Aold ad Balaiha (989), Aold, Bbalaiha ad Nagaaa (99) ad Aold, Bbalaiha ad Nagaaa (998). Becaue o the ipotace o the oet o ecod value i dawig ieece, Balaiha, Ahaullah ad Cha (99) etablihed oe ecuece elatio o the igle ad double oet o lowe ecod value o Guble ditibutio. Balaiha, Cha ad Ahaullah (993) etablihed oe ecuece elatio o igle ad double oet o ecod value o the geealized etee value ditibutio. Balaiha ad Ahaullah (994) etablihed oe ecuece elatio o igle ad double oet o ecod value o the Loa ditibutio, ad the geealized Paeto ditibutio. The oted that i the hape paaete ted to i thei eult, ipl becoe the eult o oet o the tadad epoetial ditibutio, have bee etablihed i the peviou pape i 994.

2 Balaiha ad Ahaullah (995) etablihed oe ecuece elatio o igle ad double oet o uppe ecod value o epoetial ditibutio, the have eteded thei eult to the cae o idepedet oidetical odel, [o oe detail ee Balaiha ad Ahaullah (995)]. Pawla ad Szal (999) deived ecuece elatio atiied b igle ad double oet o the th uppe ecod value o the Paeto, geealized Paeto ad Bu ditibutio. The alo gave oe chaacteizatio o the thee ditibutio.. Pawla ad Szal () deived ecuece elatio o igle ad double oet o GOS ude the cocept o Kap o Paeto, geealized Paeto ad Bu ditibutio. The eult iclude a paticula cae the above elatio o oet o th ecod value Raqab () deived the oet o ecod value o liea epoetial ditibutio. He coputed the ea ad double oet o ecod value o liea epoetial ditibutio ad the ecuece elatio o both igle ad double oet o ecod value. He oted that thee eult ca be ued to etablih iila eult o epoetial ad Raleigh ditibutio a pecial cae. Fiall, he eteded hi eult to iclude the oet o th ecod value. Raqab () deived eact epeio o igle ad poduct oet o ecod tatitic o the geealized epoetial ditibutio, ad ecuece elatio o igle ad poduct oet o ecod value ae obtaied. The ea, vaiace ad covaiace o the ecod value ae coputed o vaiou value o the hape paaete ad o oe ecod value. Thee value ae ued to copute the coeiciet o the BLUE' o the locatio ad cale paaete. Sulta () deived eact eplicit epeio o the igle, double, tiple ad quaduple oet o the uppe ad lowe ecod value o uio ditibutio. Sulta, Mohe ad Child (3) deived eact epeio o the igle, double, tiple ad quaduple oet o lowe ecod value o geealized powe ditibutio. The ued thee epeio to copute the ea, vaiace ad the coeiciet o ewe ad utoi o cetai liea uctio o ecod value. I thi pape, the ecuece elatio o oet o ecod value o iveted Weibull ditibutio ae obtaied. Nueical coputatio ad a iulatio tud ae peeted to illutate the pocedue.. INVERTED WEIBULL DISTRIBUTION The Weibull ditibutio ha bee how to be ueul o odelig ad aali o lie tie data i edical, biological ad egieeig ciece. The ue o the ditibutio i eliabilit ad qualit cotol wo ha bee advocated b Kao i (958), (959). Soe ea o the hito ad alo o the elatio betwee etee-value ad Weibull ditibutio ae i Ma (968). I Figue (3.), it ca be ue Weibull ditibutio a a athe ditibutio. The two ( hape ad cale ) paaete Weibull ditibutio ha the pobabilit deit uctio(pd) give b - - θ () θ e,, (θ, > ),, othewie

3 the, the ditibutio o /X i eeed a the ivee o iveted weibull ditibutio. Iveted Weibull (IW) ditibutio have bee ecetl deived a a uitable odel to decibe degadatio pheoea o echaical copoet Kelle ad Kaath (98) uch a the daic copoet (pito, cahat, etc.) o dieel egie. Eto (989) dicued the popetie o thi ditibutio ad it potetial ue a lietie odel. A lot o wo ha bee doe o IW ditibutio; o eaple, Calabia ad Pulcii (99) tudied the aiu lielihood ad leat-quae etiatio o IW paaete. Calabia ad Pulcii (994) tudied Bae -aple pedictio o IW ditibutio. Mahoud, Sulta, ad Ae (3) coideed the ode tatitic aiig o IW ditibutio, ad deived the eact epeio o the igle oet o ode tatitic. The vaiace ad covaiace. The obtaied baed o the oet o ode tatitic, the bet liea ubiaed etiate BLUE' o the locatio ad cale paaete o IW ditibutio. Sulta, Iail, ad AL-Moihee (6) ivetigated the itue odel o two iveted Weibull ditibutio, ad dicued oe popetie o thi odel. Let X be a ado vaiable ditibuted a IW (θ, ) with cale paaete θ > ad hape paaete >, deoted b X IW (θ, ), ha pd, cuulative ditibutio uctio (cd), eliabilit uctio (SF), ad hazad ate uctio(hrf), give epectivel b - (+ ) θ () θ e,, (θ, > ). (.), othewie ad F () e θ S() - e,, θ,, >,(θ, > ). (.) othewie >,(θ, > ). (.3) othewie θ h() - e (+ ) e θ θ, >, (θ, > ). (.4) It a be obeved that i < <, the HRF i a ootoe deceaig, o the othe had, i > the HRF i iceaig ad the deceaig. The liit o the HRF, a ted to iiit, equal zeo. The cuve o ou IW (θ, ) populatio ad the coepodig HRF' ae plotted i Figue (.). The it populatio, i Fig. (..a), i whe < θ, ( θ.5,.,.5), the ecod populatio, i Fig. (..b), i whe, ( θ.,.5,. ), the thid ad outh populatio, i Fig. (..c) ad Fig. (..d), ae whe > θ, ( θ.,.,. ) ad ( θ.,.5, 3. ), epectivel.

4 Figue (.): The pd ad HRF' o IW (θ, ) Ditibutio.3 ( ) ( ) h( ). ( ) ( ) h( ).5 h( ). h( ) 3 3 Fig. (..a) θ.5,.,.5. Fig. (..b) θ.,.5,...6 ( ) ( ) h( ).4 ( ) ( ) h( ).5 h( ). h( ) 3 3 Fig. (..c) θ.,.5,.. Fig. (..d) θ.,.5, 3.. Figue (.) how that Whe < θ, the cuve o the pd ad HRF ae ootoe deceaig. Whe the cuve o the pd tae the hape o the iveted epoetial ditibutio. Whe the cuve o the pd tae the hape o the iveted Raleigh ditibutio. Whe > θ the cuve o the pd ad HRF ae iceaig ad the deceaig. The th oet o the IW (θ, ) ditibutio i give b The ea o the IW (θ, ) ditibutio i give b E() Γ, >. (.5) θ The vaiace o the IW (θ, ) i give b V() Γ - Γ, >, (.6) θ θ

5 whee Γ(.) i the gaa uctio. The quatile o the IW (θ, ) ditibutio i give b q [ θ ( log q ) ], < q <, (.7) ad the pecial cae a be obtaied b uig (.3.7) uch a it ad thid quatile, whe q. 5 ad q. 75, epectivel. Alo, i q. 5, we obtai the edia o, which i give b edia [ θ ( log ) ] edia. (.8) It a be obeved, o (.), that the pd i a ootoe deceaig whe < <, i which cae the ode i zeo. O the othe had, the pd i a ootoe iceaig the deceaig i the cae o >, i which cae the ode i obtaied b aiizig the pd. It i give b, < < ode + [ θ ( + )]. (.9) ode ( ), > Figue(.): Relatiohip Betwee Weibull Ditibutio ad Othe Ditibutio Iveted Weibull Ditibutio Iveted Epoetial Ditibutio /X /X /X Iveted Raleigh Ditibutio Weibull Ditibutio Epoetial Ditibutio - log ( θ) Raleigh Ditibutio Etee Value Ditibutio

6 3. RECURRENCE RELATION FOR MOMENTS OF RECORD VALUES 3. Recod value FROM INVERTED WEIBULL DISTRIBUTION Recod value ca be viewed a ode tatitic o a aple whoe ize i deteied b the value ad the ode o occuece o the obevatio. Let X, X,, X be a equece o i.i.d ado vaiable with cuulative ditibutio uctio ( cd ) F() ad pd (). Set Y a(i){ X, X,, X },. We a X i a uppe ( lowe ) ecod o thi equece i Y > ( < ) Y -, >. B deiitio, X i a uppe a well a lowe ecod value. Oe ca tao o uppe ecod value to lowe ecod b eplacig the oigial equece o ado vaiable b {-X, }o b {/X i, i } ( i P( X i > ) o all i ); the lowe ecod value o thi equece will coepod to the uppe ecod value o the oigial equece. Sice the tud will ivolve both lowe ad uppe ecod value ( depedig o the populatio ude coideatio), we hall ue the ollowig otatio o coveiece: X U() o the th uppe ecod ad X L() o the th lowe ecod. The idice at which the uppe ecod value occu ae give b the ecod tie {U(), }, whee U() i{ > U( -),X > X U(-) },, with U(). Alo, the idice at which the lowe ecod value occu ae give b the ecod tie {L(), L( -), X X L(-), >, with L(). We will coie ou attetio to ecod value o cotiuou ado vaiable. Fo oe detail o both cotiuou ad dicete ecod value, [ee Ahaullah (995)], ad [Aold, et al.(998)]. }, whee L() i{ > < } 3. Pobabilit deit uctio o ecod value The oit pd o the it uppe ecod value X U(), X U(),, X U() i give b, [ee Aold, et al.(998)] ( ) ( ) ( U(i) ),,..., U(), U(),..., U() U(), F( ) i < < <... < <, (3.) U() ad the pd o the th uppe ecod value X U() i obtaied to be ( ) { log[ F()]} (), Γ() < <,,,..., (3.) ad the oit pd o X U() ad X U() ( < ) i obtaied to be (), (,) { log[ F()]} Γ()Γ( ) F() { log[ F()] + log[ F()]} (), < < <,,,..., <, (3.3) whee X U() ad X U(). The oit pd o the it lowe ecod value X L(), X L(),..., X L() i give b U() U() U(i)

7 ( ) ( ) L(), L(),..., L() L() i ( L(i),,...,, F( L(i) ) < < <... < <, (3.4) L() L( ) ad the pd o the th lowe ecod value X L() i obtaied to be: ( ) { logf()} (), Γ() < <,,,..., (3.5) ad the oit pd o the lowe ecod value X L() ad X L() ( < ) i obtaied to be, whee X L() ad X L(). (,) { log[f()]} Γ()Γ( ) { log[f()] + log[f() ]} (), < < <,,,..., <, (3.6) L() ) () F() 3.3 Moet o ecod value Let u deote the igle oet o the th uppe ecod value E(X U() ) b µ, the double oet o the uppe ecod value E(X X ) oet o the th lowe ecod value E(X L() ) b lowe ecod value E(X X ) L() L() b uppe ad lowe ecod value ae give a ollow: U() U() b, µ,, the igle µ (), ad the double oet o, µ (),(). The igle ad double oet o both (i) Sigle oet o the uppe ecod value X U() ae give b { log[ F()]} ()d,,,... Γ() (3.7) (ii) The double oet o the uppe ecod value X U() ad X U() ( < ) ae give b, (), { log[ F() ]} Γ()Γ( ) F() { log[ F()] + log[ F()]} ()d d, < < <,,,..., <. (3.8) (iii) The igle oet o lowe ecod value X L() ae give b () { logf()} ()d,,,... Γ() (3.9) (iv) The double oet o the lowe ecod value X L() ad X L() ( < ) ae give b

8 µ Γ() Γ(-), (), () { logf() + logf()} { logf() } ()d d, () F() < < <,,,..., <. (3.) 3.4 Pobabilit deit uctio o the lowe ecod value o iveted Weibull ditibutio Let X L(), X L(),, X L() be the it lowe ecod value aiig o IW(θ, ) i (.), ad (.). B uig the pd o the th lowe ecod value X L() give i (3.5), the pd o the th lowe ecod value o IW (θ, ) a θ {( θ ) } e, θ (+ ) () Γ() >, (θ, > ),. (3.) Siilal b uig the oit pd o the lowe ecod value X L(), X L() i (3.6) ad () ad F() a give, epectivel, b (.) ad (.), the oit pd o X L() ad X L() o lowe ecod value o IW(θ, ) i give b, ( -+ ) θ (, ) Γ() Γ( ) (+ ) e θ (+ ) - - {( θ ) } { }, < < <, (θ, > ), <. (3.) 3.5 Recuece elatio o oet o ecod value I thi ectio, we ue the elatio betwee pd ad cd o IW (θ, ) to etablih oe ecuece elatio o the igle ad double oet o lowe ecod value. Fo (.) ad (.), we have () { log F() }F(). (3.3) Thi elatio will be ued i the ollowig eult to etablih ecuece elatio.

9 (i) Relatio o the igle oet Reult 3. Fo, ad, the elatio o igle oet o lowe ecod value o the IW (θ, ) ditibutio i give b: Poo ( + ) ( ), >. (3.4) Fo ad, the igle oet o the lowe ecod value o IW ca be witte a - () { log F() } () d. Γ() B uig (3.3) i the above elatio, we obtai - () { log F() } F() d. Γ() Itegatig b pat teatig o itegatio ad the et o dieetiatio, the the itegal ca be witte a ollow ( + ) µ (), >. (ii) Relatio o double oet Oce agai the elatio (3.3) will be ued to deive ecuece elatio o the double oet o the lowe ecod value o the IW (θ, ) ditibutio auig thei eitece. Reult 3. Fo ad,,,... the elatio o double oet o lowe ecod value o the IW(θ, ) ditibutio i give b, + (),( + ) (+ ), >. (3.5) ad o, we have,, ( + ),() (),(), >. (3.6) Poo The double oet o the lowe ecod value o IW (θ, ) ditibutio a be witte a, (), () g() I()d, < <, Γ()Γ( ) (3.7) whee

10 () I() { log F() } { log F() + log F() } d. F() (i) Fo +, we get, (), ( ) ()I ()d, < <, Γ() + (3.8) whee () Io () { log F() } d. F() Upo uig the elatio i (3.3) i the above epeio, ield I () o { log F() } d. Itegatig b pat teatig o itegatio ad the et o the itegad o dieetiatio i each pat o the ight had ide, the itegal, I o (), ca be witte a ollow () Io () { log F() } { log F() } d. + F() Upo ubtitutig the above epeio o I o () itead o I() i (..7), we obtai, + (),( + ) { log F() } () d Γ() () + { log F() } () d d. Γ() F() The elatio (3.5) i poved ipl upo uig the deiitio o the igle ad double oet o lowe ecod value i the above equatio ad ipliig the eultig epeio to get, + (),( + ) (+ ), >. (ii) Fo () I() { log F() } { log F() + log F() } d. F() Upo uig the elatio i (..4) i the above epeio, ield { log F() } { log F() + log F() } d. I() (3.9) Fo the it pat I(), upo itegatig b pat teatig - o itegatio ad the et o the itegad o dieetiatio, we obtai I() { log F() } { log F() + log F() } () d F() ( ) { log F() } { log F() + log F() } () d. F() Upo ubtitutig the above epeio o I() ito (..7), we obtai

11 , (),() Γ()Γ( () () d F() ( ) Γ()Γ( ) () () d F() ) d d. { log F() } { log F() + log F() } { log F() } { log F() + log F() } The elatio i (3.6) i poved ipl upo uig the deiitio o the igle ad double oet o lowe ecod value i the above equatio ad ipliig the eultig epeio to get,, ( + ),() (),(), >. Rea (a) I, we get the ecuece elatio o igle ad double oet o lowe ecod value o the iveted epoetial ditibutio. ( + ) (), >. (3.), + (),( + ) (+ ), >. (3.),, ( + ),() (),(), >. (3.) (b) I, we get the ecuece elatio o igle ad double oet o lowe ecod value o the iveted Raleigh ditibutio. ( + ) (), >. (3.3), + (),( + ) (+ ), >. (3.4),, ( + ),() (),(), >. (3.5)

12 4. SOME NUMERCAL RESULT oe ueical tud with the coet o the ecuece elatio o igle ad double oet o lowe ecod value o iveted Weibull ditibutio (3.4), ad (4.58), i Table (4.), ad (4.). Table (4.): Sigle Moet o the Lowe Recod Value o Doubl Tucated Iveted Weibull Ditibutio, θ , , , , , , , , ,., θ ,..74( 3 ).38( 3 ).( 3 ) ,..87( 3 ).58( 3 ).8( 3 ).98( 3 ) ,..33( 3 ).787( 3 ).549( 3 ).394( 3 ).8( 3 ).97( 3 ).8( 3 ).5,.5 3.5, , , ,. 5, Coet: It i oted that, o Table (4.), a iceae the igle oet iceae. While, a iceae the igle oet' deceae. Ad, a θ iceae the igle oet deceae. I iceae the igle oet' iceae.

13 Table(4.): Double Moet o the Lowe Recod Value o Iveted Weibull Ditibutio K ad S, θ.5,. 3.5,. 5,..5,.5 3.5,.5 5,.5.5,. 3.5,. 5,

14 Coet: It i oted that, o Table (4.), a ad iceae the double oet iceae. While, a ad θ iceae the double oet' deceae. CONCLUSIONS Soe cotibutio baed o ecod value have bee ade. The cotibutio icluded wo o the igle, ad double oet o ecod value. Soe popetie o iveted Weibull ditibutio wee dicued. Seveal ecuece elatio wee etablihed o the oet o ecod value o iveted Weibull ditibutio. Geeal cocluio o thi tud ca be uaized a ollow: pd' ad oit pd' ae deied o lowe ecod value aiig o iveted Weibull ditibutio. The coectio betwee the cd ad the pd o the iveted Weibull ditibutio ha bee etablihed to ititute the ecuece elatio o oet o ecod value. Seveal ecuece elatio ae etablihed o the igle ad double oet o lowe ecod value aiig o iveted Weibull ditibutio ( iveted epoetial, ad iveted Raleigh ditibutio a pecial cae). The veiicatio o legitiac o the ecuece elatio o igle ad double oet o lowe ecod value o iveted Weibull ditibutio, b uig Mathcad poga wa copleted. Ad we checed the ecuece elatio o oet, the we ade ue wee accepted. 4

15 REFERENCES Ahaullah, M. (995). Recod Statitic, Nova Sciece Publihe, Ic., New Yo. Ahaullah, M., ad Bho, D.S. (996). Recod value o etee value ditibutio ad a tet o doai o attactio o Tpe I etee value ditibutio, Id. J. Statit., Seie B, 58(),5-58. AL-Saleh, J.A., ad Agawal, S.K. (6). Eteded Weibull Tpe ditibutio ad iite itue o ditibutio, Statit. Metho.,3, Aold, B.C., Balaiha, N., ad Nagaaa, H.N. (998). Recod, Joh Wile & So, New Yo. Bai, L.J. (978). Statitical Aali o Reliabilit ad Lie-Tetig Model ( Theo ad Method), Macel Dee, Ic., New Yo. Balaiha, N., ad Ahaullah, M. (994). Recuece elatio o igle ad poduct oet o ecod value o the Loa ditibutio, ad geealized Paeto ditibutio, Cou. Statit.-Theo. Meth.,3() Balaiha, N., ad Ahaullah, M. (995). Relatio o igle ad poduct oet o ecod value o epoetial ditibutio, J. Appl. Statit. Scie. (), Balaiha, N., ad Cha, P.S. (998). O the oal ecod value ad aociated ieece, Statit. Pobab. Lett., 39, Balaiha, N., Ahaullah, M., ad Cha, P.S. (99). Relatio o igle ad poduct oet o ecod value o Gubel ditibutio, J. Appl. Statit. Scie., 5, 3-7. Balaiha, N., Cha, P.S. ad Ahaullah, M., (993). Recuece elatio o ecod value o geealized etee value ditibutio, Cou. Statit.-Theo. Meth.,(5), Joho, N.L., Kotz, S., ad Balaiha, N. (995). Cotiuou uivaiate ditibutio,, d ed., JohWile & So, Ic., New Yo. Kudu, D., ad Raqab, M.Z. (5). Geealized Raleigh ditibutio: dieet ethod o etiatio, Coput. Statit. Da. Aal.,49, 87-. Mohaad, H.H. (3). A tud o the oet o ecod value ad aociated ieece, M. D. Thei, AL-Azha Uiveit Gil Bach, Caio, Egpt. 5

16 Mahoud, M.A.W., Sulta, K.S., ad Ae, S.M. (3). Ode tatitic o ivee Weibull ditibutio ad aociated ieece, Coput. Statit. Da. Aal.,4, Nadaaah, S. (5). O the oet o the odiied Weibull ditibutio, Reliab. Egie. St.,9, 4-7. Naa, M.M, ad Eia, F.H. (3). O the epoetiated Weibull ditibutio, Cou. Statit.-Theo. Meth.,3(7), Pawla, P., ad Szal, D. (999). Recuece elatio o igle ad poduct oet o th ecod value o Paeto, geealized Paeto, ad Bu ditibutio, Cou. Statit.-Theo. Meth.,8(7), Pawla, P., ad Szal, D. (). Recuece elatio o igle ad poduct oet o th ecod value o Weibull ditibutio, ad a chaacteizatio, J. Appl. Statit. Scie.(), 7-6. Pawla, P., ad Szal, D. (). Recuece elatio o igle ad poduct oet o geealized ode tatitic o Paeto, geealized Paeto, ad Bu ditibutio, Cou. Statit.-Theo. Meth.,3(4), Solia, A.A., Abd Ellah, A.H., ad Sulta, K.S. (5).Copaio o etiate uig ecod tatitic o Weibull odel: Baeia ad o-baeia appoache, Coput. Statit. Da. Aal.(i pe). Stewat, J. (994). Calculu, 3 d editio, Boo/Cole, New Yo. Sulta, K.S. (). Moet o ecod value o uio ditibutio ad aociated ieece, Egpt. Statit. J. ISSR, UNIV.,44(), Sulta, K.S., Iail, M.A., ad AL-Moihee, A.S. (6). Mitue o two ivee Weibull ditibutio: popetie ad etiatio, Copu. Statit. Da. Aal.( i pe). Sulta, K.F., Mohe, M.E., ad Chil, A. (3). Recod value o geealized powe uctio ditibutio ad aociated ieece, J. Appl. Statit. Scie. Joual o Facult o Coece AL-Azha Uiveit Gil Bach,8, Raqab, M.Z., ad Ahaullah, M. (3). O oet geeatig uctio o ecod o etee value ditibutio, Pa. J. Statit.,9(), -3. 6

17 Appedi Reult 4. Fo eal, with,, ad o itege,, the elatio o igle oet o GOS withi a cla o DT ditibutio(..3) i give b ( λ ( X ;,, )) ;,, ;,, { P E [ η( X ;,, ) e ] E [ ( X ;,, ) ] η }, (4.37) whee η( X Poo ;,, ) λ ( ) i a cotiuou uctio. λ ( ) Let X ha the pd (4.6), the o (4.3) i the double tucated cae, we obtai [ F ()] () d. C P ;,, g ()) Γ() (i) Q Itegatig b pat teatig [ F ()] () o itegatio ad the et o dieetiatio, the the itegal ca be witte a ollow C P ;,, g ())[ F ()] d Γ() Q ( ) C + Γ() which ca be witte a C ;,, Γ() P Q P Q C + Γ( ) Subtitutig o (4.), i (ii), we get C P ;,, -;,, Γ() Q P g P Q λ () λ () g g g ()) () d. λ () Fo (4.7), we ca wite P equal to P e λ () Subtitutig the above eult i (iii), we obtai Hece, ;,, -;,, PC Γ() C Γ() P Q P Q λ () e λ () λ () g λ () ()) ()) λ () λ() g [ F ()] () d. [ F ()] [ F ()] ()) d - [ F ()] ()d. [ F ()] (). ()) ()) [ F ()] ()d [ F ()] ()d. (ii) (iii) 7

18 ;,, -;,, λ () PE e λ () λ() λ () E. λ () Reult 4. Fo eal, with,, ad o itege >,, the elatio o double oet o GOS withi a cla o DT ditibutio(..3) i give b,i,i i ( λ(y ) µ ;,,,;,, µ ;,,, ;,, P E[ X (Y )e ] E[ X (Y ) ],, ) ;,, η ; ;,, ;,, ;,, η Poo whee η( Y ;,, ) i { } λ ( ) i a cotiuou uctio. λ ( ) Fo (4.3), i the double tucated cae, we obtai whee C P,i ;,,,;,, [F ()] ()g ()) I() d, Γ()Γ( ) Q Q < <, (v) P < P { h ()) h ())} [F ()] ()d. (iv) i I() (vi) Itegatig b pat teatig [ F ()] () o itegatio ad the et o dieetiatio, the the itegal ca be witte a ollow i P i- I() { h ()) h ())} [F ()] [F ()]d (vii) ( - -) P i + { h ()) h ())} [F ()] () d. Subtitutig o (4.), i (vii), we get i P i- λ () I() { h ()) h ())} [F ()] P () d λ () ( - -) P i + { h ()) h ())} [F ()] ()d. Hece, o deiitio o P, we obtai I() ip i P P i- λ () λ () i- λ () λ () ( ) e λ { h ()) h ())} { h ()) h ())} ( - -) P i + { h ()) h ())} [F ()] ()d. Upo ubtitutig the above epeio o I() ito (v), we obtai [F [F ()] ()] ()d ()d 8

19 9 { } { } { } d. ()d ()] [F ()) h ()) h ()) ()g ()] [F ) Γ()Γ( -)C ( - ()d d ()] [F ()) h ()) h ()) ()g ()] [F () λ () λ ) Γ()Γ( i C ()d d ()] [F ()) h ()) h ()) ()g ()] [F e () λ () λ ) Γ()Γ( PC i µ P Q P i P Q P i- P Q P λ() i-,i ;,,,;,, - + Hece, () λ () λ E e () λ () λ PE i µ µ i λ() i,i ;,, ;,,,,i ;,,,,, ;. < < < P Q.