Recurrence Relations for Single and Product Moments of Generalized Order Statistics from Extreme Value Distribution

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1 Aecan Jonal o Appled Matheatc and Stattc, 204, Vol. 2, No. 2, Avalable onlne at Scence and Edcaton Pblhng DOI:0.269/aa Recence Relaton o Sngle and Podct Moent o Genealzed Ode Stattc o Extee Vale Dtbton Kaal Nan Kapoo * Hnd College, Unvety o Delh, Delh, Inda *Coepondng atho: aal.80968@gal.co Receved Janay 08, 204; Reved Mach 04, 204; Accepted Mach 3, 204 Abtact In th pape, we etablh oe ecence elaton ated by ngle and podct oent o Genealzed Ode Stattc o Extee Vale Dtbton. Thee ecence elaton ae ndependent o let tncated pont and theeoe ae alo applcable o Logtc a well a o hal Logtc dtbton tded n Balahnan (985 and Saan and Pandey (202. Fo a patcla cae thee elt vey the coepondng elt o Saan and Pandey (2004 and Ka (200. Keywod: ode tattc, ecod vale, genealzed ode tattc, ngle oent, podct oent, ecence elaton, extee vale dtbton Cte Th Atcle: Kaal Nan Kapoo, Recence Relaton o Sngle and Podct Moent o Genealzed Ode Stattc o Extee Vale Dtbton. Aecan Jonal o Appled Matheatc and Stattc, vol. 2, no. (204: do: 0.269/aa Intodcton Genealzed ode tattc (GOS have been ntodced and extenvely tded n Kap (995 a,b a a ned theoetcal et-p whch contan a vaety o odel o odeed ando vaable wth deent ntepetaton. Exaple o ch odel ae: Odnay ode tattc, Seqental ode tattc, Pogeve type II cenoed ode tattc, Recod vale, th ecod vale and Pee ecod. Thee no natal ntepetaton o genealzed ode tattc n te o obeved ando aple bt thee odel can be eectvely appled n le tetng and elablty analy, edcal and le te data, and odel elated to otwae elablty analy, etc. The coon appoach ae t poble to dene eveal dtbtonal popete at once. The tctal late o thee odel ae baed on the laty o the ont denty ncton. 2. Standad Extee Vale Dtbton A ando vaable X ad to have an tandad extee vale dtbton t pobablty denty ncton o the o x x e (,, 0 x = e e < x< > (2. and the clatve dtbton gven by x e (,, 0 F x = e < x< > (2.2 The extee vale dtbton ed n the analy o data concenng lood, extee ea level and a pollton poble. The clatve dtbton ncton and pobablty denty ncton o ando vaable X, epectvely, tae the o + x ( x = ( F( x. = 0! (2.3 ( x + x + = = x, whee = F( x = 0! = 0! The atheatcal o o pd, a gven n (2.3, vey el to deve the expeon o ecence elaton o ngle and podct oent o GOS. 3. Genealzed Ode Stattc Xn, n be a eqence o aboltely contno, ndependent and dentcally dtbted ando vaable wth cd F( x = P( X x and pd (x. Ae > 0, n n = (,,..., R, Let { } { 2,3, }, 2 n n M =, ch that = + n + M > 0 o all = {, 2,..., n }. Then X( n,,,, =, 2,,n, ae called GOS the ont pd gven by

2 78 Aecan Jonal o Appled Matheatc and Stattc (,,,, ( 2,,,,..., (,,, (,,..., X n X n X nn x x x n n ( ( ( = = = Π Π F x x ( F x ( ( x, n n 2 n (3. whee F (0 + < x x2... xn < F (. By choong appopate vale o paaete, we get the dtbton o a ew vey coon tattc a hown n the table gven below. S.No. Choce o paaete o =, 2,, n GOS becoe = + = = = = and = Jont dtbton o n ode tattc n, 2... n 0 =, = =... = =, N th ecod vale 2 2 n 3 = ( n +, > 0 Seqental ode tattc 4 = +, > 0 Ode tattc wth non ntege aple ze 5 = β, β > 0 Pee ecod vale 6 N, N Pogevely type-ii ght cenoed ode tattc The ont pd o t, GOS gven by: (,,,, ( 2,,,,..., (,,, (,,..., X n X n X n x x2 x ( ( = c Π F x ( x = + n + M ( F( x ( x, o (3.2 whee, F (0 + < x x2... x < F (. We now conde two cae: Cae I: = 2 = = n = Cae II: ;,, =, 2,, n -. Fo cae I, the GOS wll be denoted by X(, n,,. The pd o X(, n,, gven by ( n,,, ( x X c (3.3 ( ( = F x ( x g ( F( x, x R (! and the ont pd o X(, n,, and X(, n,,, < n gven by: (,,,, (,,, ( x, y c ( ( F( x ( x X n X n = (! (! ( ( ( ( ( ( g F x h F y h F x F( y ( y, x< y, ( (3.4 whee, c = Π, = + ( n ( +, =,2,..., n, = g( x = h( x h(0, x (0, and + ( x,, h ( x = + log ( x, =. Fo cae II, the pd o X( n,,, X ( n,,, ( x gven by (3.5 ( ( ( ( (3.6 = c x a F x ( x, x R. = X Alo, the ont pd o X(, n,, (, n,,, < n gven by (,,,, (,,, ( x, y X n X n F( y = c a ( = + F( x ( ( ( ( ( x y a F x, = F( x F( y and (3.7 Whee, c = Π, = + n + M, =, 2,..., n.. = Fthe t can be poved that ( a ( ( = Π, n ( = ( a ( ( = Π, + n. ( = + ( = + a( + (v c = c + ( a ( + (v a ( + = 0 = (v ( ( F x ( ( ( a F x = g ( F( x (3.9 = (v = + ( F x (! ( + ( ( F( y = (! F( x ( h ( F( y h ( F( x F( y a ( F ( x (3.0 The oent o ode tattc have geneated condeable nteet n the ecent yea. The expeon o eveal ecence elaton and dentte ated by ngle a well a podct oent o ode tattc have been obtaned by eveal atho n the pat. Thee

3 Aecan Jonal o Appled Matheatc and Stattc 79 elaton help n edcng the qant o coptaton nvolved. Joh (978, 982 etablhed ecence elaton o exponental dtbton wth nt ean and wee the extended by Balahnan and Joh (984 o dobly tncated exponental dtbton. Fo lneaexponental dtbton, Balahnan and Mal(986 deved the la type o elaton whch wee extended to dobly tncated lnea exponental dtbton by Mohe El-Dn et al. (997 and Saan and Phana (999. Nan (200 a, b obtaned ecence elaton o odnay ode tattc and th ecod vale o p th ode exponental and genealzed webll dtbton, epectvely. The ecence elaton o the oent o genealzed ode tattc baed on non dentcally dtbted ando vaable wee developed by Kap (995 a, b. Pawla and Szynal (200 obtaned ecence elaton o ngle and podct oent o genealzed ode tattc o Paeto, genealzed Paeto and B dtbton. Saan and Pandey (2004, 2009 obtaned ecence elaton o ngle and podct oent o genealzed ode tattc o lnea-exponental and B dtbton. Saan and Nan (202 obtaned ecence elaton o ngle and podct oent genealzed ode tattc o dobly tncated p th ode exponental dtbton. Saan and Nan (203 alo obtaned explct expeon o ngle and podct oent o Genealzed Ode Stattc o a new cla o exponental dtbton. In th pape, we have etablhed ecence elaton o ngle and podct oent o GOS o Extee Vale Dtbton. Th dtbton ha any applcaton n analy o data concenng lood, extee ea level and a pollton poble. The elt o obtaned ae genealzed veon o oe o the ecence elaton obtaned by Ka (200, Saan and Pandey (2004. Notaton Fo n =, 2, 3,..., < n, and v, 0,, 2,..., we denote by { } ( = E ( ( ( ( ( :, n, X, n,,, v v ( = ( ( n E, :,, X, n,, X, n,, ( = E ( n :,, X, n,,, v v (v = ( (, :, n, E X, n,, X, n,, 4. Recence Relaton Fo Sngle and Podct Moent Cae I: = 2 =... = n = Theoe. Fo n =, 2, 3,..., < n, and v, { 0,, 2,... } (a :, n, = :, n, :, n, = (4. (b (,, v,:, n, (4.2, v + +, v + + =, :, n,, :, n, = 0 Poo (a: The th ode oent o X(, n,, gven by :, n, c ( ( (4.3 = x F x ( x g ( F( x dx. (! β Sbtttng (x o (2.3 we have :, n, c (4.4 + ( ( = x F x g ( F( x dx. (! = 0 β Integatng by pat, tang x + a the pat to be ntegated, we obtan c :, n, = (! = x { ( ( F x g ( F( x ( x (4.5 β ( + 2 ( } ( F( x g F( x ( x On btttng + =, c = c2 and = + n (4.5, we hall deve the ecence elaton a tated n (4.. Poo (b. By denton whee, = c, v,:, n, = (! (! ( ( ( ( ( ( ( :,,, x F x x dx, β g Fx J x v ( :,,, v y h( F( y h( F( x x ( F( y ( y J x v Sbtttng (y o (2.3 we have: ( :,,, J x v v + h ( F( y y h ( F( x ( F( y ( y = c = 0 x (4.6 (4.7 dy. (4.8 dy. Integatng by pat, tang y v + a the pat to be ntegated, we obtan:

4 80 Aecan Jonal o Appled Matheatc and Stattc x ( :,,, J x v = c { ( = 0 ( ( ( ( v + + y h F y h F x ( ( ( } ( F y ( y ( 2 h( F( y h( F( x + F y y dy Ate ng + = and c = c2, we get: ( :,,, J x v ( : + +,,, ( J x v =. = 0 ( J x:,,, On btttng J(x : v,,, o obtaned n (4.6, we hall deve the ecence elaton a tated n (4.2. Cae II: ;,, =, 2,, n -. Lea. (a (b c ( ( ( a F x = ( ( ( c a F x = = c 2 ( ( ( a F x = F( y c a ( = + F( x F( y c a ( = + F ( x = F( y c2 a ( = + F ( x (4.9 (4.0 Poo (a: Deentatng both de o (3.9, wth epect to x, we get: ( ( ( a F x = ( ( F x g ( F( x = (! 2 / ( ( F( x g ( F( x g( F( x ( ( F x g ( F( x = (! + ( F( x g F( x ( 2 ( / ( g ( F( x = ( F( x ( ( F x g ( F( x = (! 2 ( ( F( x g ( F( x ( ( ( a F x = =, F( x a ( ( ( F x = whch on ng the elaton c = c2 lead to (4.9. Poo (b: Deentatng both de o (3.0, wth epect to y, we get: = = + a ( ( F( y ( F( x ( + ( ( F( x (! ( F( x ( ( F y ( h ( F( y h ( F( x 2 / ( h ( F( y h ( F( x h ( F( y ( F( x (! ( F( x ( ( ( F y ( + ( = ( ( F( y h( F( y h( F( x 2 h ( F( y ( ( F( y + h ( F( x / ( h ( F( y = ( F( y ( + ( ( F( x (! ( F( y F( y ( h( F( y h( F( x F( x = ( + ( 2 ( F( x ( 2! ( F( y F( y 2 ( h( F( y h( F( x F( x F( y a ( = + F ( x =, ( F( y F( y a ( = + F ( x whch on ng the elaton c = c2 lead to (4.0. Theoe 2. Fo n =, 2, 3,..., < n, and { }, 0,, 2,... v (a

5 Aecan Jonal o Appled Matheatc and Stattc :, n, = :, n, :, n, = (4. (b, v + + v,,: n,,,: n,, = (4.2, 0 v v + + = + +, : n,, Poo (a: The th ode oent o X( n,,, gven by: :,, ( ( ( n = c x a F x ( x dx β = (4.3 Sbtttng the vale o (x o (2.3, we have + = c x a ( ( F ( x dx. (4.4 = 0 β = :, n, Integatng by pat, tang x + a the pat to be ntegated, we obtan: :, n, = c = x a ( ( ( F x ( x β = (4.5 Ate ng (4.9, we hall deve the ecence elaton gven n (4.. Poo (b: We now that v,,:,, ( ( ( n = x a F x β = (4.6 ( x J ( x : v,,, dx, F( x ( :,,, J x v whee v F( y ( y. = c y a ( dy F ( x F ( y x = + Sbtttng (y o (2.3 we have: ( :,,, J x v v+ F( y (4.7 = c y a ( dy. = 0 F( x x = + Integatng by pat, tang y v + a the pat to be ntegated, we obtan: ( :,,, J x v y c ( ( = dy 0 F y = a x ( ( y ( ( = + F x ( y y F( y = 0 ( ( dy v F y = + + x c a ( ( ( = + F x F( y = c y a ( = 0 F( x x = + F( y ( y c2 y a ( dy x = + F( x F( y ( : + +,,, ( J x v =. = 0 J x:,,, (4.8 ( by ng 22 On btttng the above expeon o J( x: v,,, n (4.6 we get v,,: n,, = ( ( ( ( x x a F x = F( x dx = 0 J( x: v,,, β + + J( x:,,, whch on ng (4.6, lead to ( Conclon In the tdy peented above, we deontate the ecence elaton o ngle and podct oent o GOS o Extee Vale Dtbton. Thee elt genealze the coepondng elt o Ka (200 and Saan and Pandey (2004. Acnowledgeent The atho wh to than the eeee o valable coent, whch led to an poveent n the peentaton o th pape. Reeence [] Atha, H., Kheaa S. K. and Nayabddn (202. Expectaton dentte o Paeto dtbton baed on genealzed ode tattc. Aecan Jonal o Appled Matheatc and Matheatcal Scence,, [2] Balahnan, N. (985. Ode tattc o the Hal logtc dtbton. J. Statt. Cop. Sl., 20, [3] Balahnan, N. and Joh, P.C. (984. Podct oent o ode tattc o dobly tncated exponental dtbton. Naval Re. Logt. Qat, 3, [4] Balahnan, N. and Mal, H. J. (986. Ode tattc o lnea exponental dtbton, Pat I: Inceang hazad ate cae. Con. Statt. Theo. Meth. 5, [5] Joh, P. C. (978. Recence elaton between oent o ode tattc o exponental and tncated exponental dtbton. Sanhya, Se. B, 39,

6 82 Aecan Jonal o Appled Matheatc and Stattc [6] Joh, P. C. (982. A note on xed oent o ode tattc o exponental and tncated exponental dtbton. J. Statt. Plann. In., 6, 3-6. [7] Kap, U. (995a. A concept o genealzed ode tattc. B. G. Tebne, Stttgat. [8] Kap, U. (995b. A concept o genealzed ode tattc. J. Statt. Plann. In., 48, -23. [9] Ka, D. (200. Recence elaton o ngle and podct oent o genealzed ode tattc o p th ode exponental dtbton and t chaactezaton, J. Statt. Re. Ian 7, [0] Mohe El-Dn, M. M., Mahod, M. A. W., Ab-Yoe, S. E. and Sltan, K. S. (997. Ode tattc o the dobly tncated lnea exponental dtbton and t chaactezaton. Con. Statt.- Sl. Copt. 26, [] Nan, K. (200 a. Recence elaton o ngle and podct oent o th ecod vale o genealzed Webll dtbton and a chaactezaton. Intenatonal Matheatcal Fo, 5, No. 33, [2] Nan, K. (200 b. Recence elaton o ngle and podct oent o odnay ode tattc o p th ode exponental dtbton. Intenatonal Matheatcal Fo, 5, No. 34, [3] Pawla, P. and Szynal, D. (200. Recence elaton o ngle and podct oent o genealzed ode tattc o Paeto, Genealzed Paeto and B dtbton. Con. Statt. Theo. Meth., 30, [4] Saan, J. and Nan, K. (202a. Recence elaton o ngle and podct oent o genealzed ode tattc o dobly tncated p th ode Exponental Dtbton, JKSA 23. [5] Saan, J. and Nan, K. (202b. Relatonhp o oent o th ecod vale o dobly tncated pth ode exponental and genealzed Webll dtbton. PobStat Fo., 05, [6] Saan, J. and Nan, K. (202c. Relatonhp o oent o th ecod vale o dobly tncated pth ode exponental and genealzed Webll dtbton. PobStat Fo., 05, [7] Saan, J and Nan, K (203. Explct Expeon o Sngle and Podct Moent o Genealzed Ode Stattc o a New Cla o Exponental Dtbton Chaactezaton, JKSA 24, [8] Saan, J. and Pandey, A. (2004. Recence elaton o ngle and podct oent o genealzed ode tattc o lnea exponental dtbton. Jonal o Appled Stattcal Scence, 3, [9] Saan, J. and Pandey, A. (2009. Recence elaton o ngle and podct oent o genealzed ode tattc o lnea exponental and B dtbton. Jonal o Stattcal Theoy and Applcaton, 8, No. 3, [20] Saan, J. and Pandey, A. (20. Recence elaton o agnal and ont oent geneatng ncton o dal genealzed ode tattc o nvee Webll dtbton. Jonal o Stattcal Stde, 30, [2] Saan J. and Pande V.(202. Recence elaton o oent o pogevely type-ii ght cenoed ode tattc o hal logtc dtbton. J. Stattcal Theoy and Applcaton,, [22] Saan, J. and Phana, N. (999. Moent o ode tattc o dobly tncated lnea exponental dtbton. J. Koean Statt. Soc., 28,