Kangweon-Kyungki Math Jou 2 24, No, pp 5 22 RCURRNC RLATION FOR QUOTINTS OF TH POWR DISTRIBUTION BY RCORD VALUS Min-Young Lee and Se-Kyung Chang Abtact In thi pape we etablih ome ecuence elation atified by quotient moment of uppe ecod value fom the powe ditibution Let { n, n } be a equence of independent and identically ditibuted andom vaiable with a common continuou ditibution functioncdf F and pobability denity functionpdf f Let Y n ma{, 2,, n } fo n We ay j i an uppe ecod value of { n, n }, if Y j > Y j, j > The indice at which the uppe ecod value occu ae given by the ecod time {un}, n, whee un min{j j > un, j > un, n 2} and u Suppoe P OW,, then + + un un un and + un + + um un+ + un + + un Intoduction Let { n, n } be a equence of independent and identically ditibuted andom vaiable with a common cdf F and pdf f Sup- Received Novembe, 23 2 Mathematic Subject Claification: 6, 62, 62H Key wod and phae: Powe ditibution, Recod Value, Recuence Relation, pectation
6 Min-Young Lee and Se-Kyung Chang poe Y n ma{, 2,, n } fo n We ay j i an uppe ecod value of thi equence if Y j > Y j, j > We define the ecod time un by u and un min{ j j > un, j > un, n 2 } The ecod time of the equence { n, n } ae andom vaiable and ae the ame a thoe fo the equence {F n, n } We know that the ditibution of un doe not depend on F Hence, the ditibution of un can be detemined by conideing the unifom ditibution F We will call the andom vaiable P OW,, if the coeponding pobability cumulative function F of i of the fom if < < and > F othewie Some chaacteization of the powe ditibution ae known But mainly ome ecuence elation atified by the ingle and poduct moment of ecod value Such eult have been etablihed by Ahanullah[] Balakihnan, Ahanullah and Chan[4, 5], and Balakihnan and Ahanullah[2, 3] fo the eteme value, eponential, Paeto and genealized eteme value ditibution And Dalla[6] chaacteize the powe and Paeto ditibution uing the conditional epectation In thi pape, we will give ome ecuence elation atified by the quotient moment of uppe ecod value fom the powe ditibution 2 Main Reult Theoem 2 Fo m n2,,, 2, and, 2,, + + un un un
Recuence elation fo quotient of the powe ditibution by ecod value 7 Poof Fit of all, we have that fo the powe ditibution in, f [ F ] The joint pdf of and un i f m,n, y ΓmΓn m Rm [Ry R] nm fy Let u conide fo m n2,,, 2, and, 2,, + un un <<y< ΓmΓn m <<y< ΓmΓn m y+ y f m,n, ydyd y + yrm [Ry R] nm fydyd R m ΓmΓn m R m y + y[ry R]nm fydy d y + [Ry R]nm [ F y]dy d Uing integating by pat teating fo integation and [Ry y+ R] nm [ F y] fo diffeentiation on the econd integation, we
8 Min-Young Lee and Se-Kyung Chang get y + [Ry R]nm [ F y]dy [ ] y [Ry R]nm [ F y] n m + Then we have + un un <<y< ΓmΓn m <<y< y [Ry R]nm2 fydy y [Ry R]nm fydy ΓmΓn m y Rm [Ry R] nm2 fydyd y Rm [Ry R] nm fydyd <<y< <<y< un y f m,n, ydyd y f m,n, ydyd un Hence + un Thi complete the poof un + un
Recuence elation fo quotient of the powe ditibution by ecod value 9 Coollay 22 Fo m,,, 2, and, 2,, + um+ + um+ Poof Upon ubtituting n m+ in Theoem 2 and implifying, then we have + um+ + um+ Theoem 23 Fo m n 2 and,,, 2,, + un [ + + um + un+ un + + ] un Poof In the ame manne a Theoem 2, let u conide fo m n 2 and,,, 2,, + un un <<y< ΓmΓn m + y y f m,n, yddy <<y< [Ry R] nm fyddy y Rm
2 Min-Young Lee and Se-Kyung Chang Since the powe ditibution, f F Upon ubtituting the above epeion and implifying the eulting equation, we obtain that + un un a mn y fy y R m [Ry R] nm d dy, whee a mn ΓmΓn m Uing integating by pat teating fo integation and R m [Ry R] nm fo diffeentiation on the econd integation, we get y R m [Ry R] nm d [ + + R m [Ry R] nm y m + R m2 [Ry R] nm d + n m y + + R m [Ry R] nm2 d + Then we have + un un + Γm Γn m <<y< [Ry R] nm fyddy + ΓmΓn m <<y< [Ry R] nm2 fyddy ] y + y Rm2 + y Rm
Recuence elation fo quotient of the powe ditibution by ecod value 2 Hence + un [ + + <<y< + + + y f m,n+, yddy <<y< + um un+ + um + un+ un + + y f m,n, yddy + + + un ] un Thi complete the poof Coollay 24 Fo m and,,, 2,, + um+ + [ + um um+2 + + ] + um+ Poof Upon ubtituting n m+ in Theoem 23 and implifying, then we have + um+ + [ + um um+2 + + ] + um+
22 Min-Young Lee and Se-Kyung Chang Refeence M Ahanuallah, Recod Statitic, Nova Science publihe, Inc, Commack NY, 995 2 N Balakihnan and M Ahanuallah, Recuence elation fo ingle and poduct moment of ecod value fom genealized paeto ditibution, Comm Statit Theoy and Method 23, 994, 284-2852 3 N Balakihnan and M Ahanuallah, Relation fo ingle and poduct moment of ecod value fom eponential ditibution, J Appl Statit Sci, 993b 4 N Balakihnan, P S Chan and M Ahanuallah, Recuence elation fo moment of ecod value fom genealized etme value ditibution, Commun Stait- Theo Meth 225, 993, 47-482 5 N Balakihnan, M Ahanuallah and P S Chan, Relation fo ingle and poduct moment of ecod value fom Gumbel ditibution, Stat and Pob Lette, 5, 992, 223-227 6 A C Dalla, A chaacteization uing conditional vaiance, Meika 28, 98, 5-53 7 J Galambo, Chaacteization of pobability ditibution, No 675, Spinge- Velag, New Yok, 978 Min-Young Lee Depatment of Applied Mathematic Dankook Univeity Cheonan, Chungnam 33-74, Koea -mail: leemy@dankookack Se-Kyung Chang Depatment of Applied Mathematic Dankook Univeity Cheonan, Chungnam 33-74, Koea -mail: kchang@dankookack