Recurrence Relations for the Product, Ratio and Single Moments of Order Statistics from Truncated Inverse Weibull (IW) Distribution

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1 Recuece Relatios fo the Poduct, Ratio ad Sigle Moets of Ode Statistics fo Tucated Ivese Weiull (IW) Distiutio ISSN Joual of Statistics Vol: 3, No. (2006) Recuece Relatios fo the Poduct, Ratio ad Sigle Moets of Ode Statistics fo Tucated Ivese Weiull (IW) Distiutio M. Alee Astact I this pape we estalished soe ecuece elatios fo the poduct, atio ad sigle oets of ode Statistics fo douly tucated Ivese Weiull distiutio. Key wods Recuece elatios, ode statistics, tucatio, Ivese Weiull (IW) Distiutio. Itoductio Ode statistics aise atually i ay eal life applicatios. The oets of ode statistics have assued cosideale iteest i ecet yeas ad have ee taulated quite extesively fo seveal distiutios. Fo a extesive suvey, see fo exaple, Aold, Balakisha ad Nagaaja (992) ad Balakisha ad Sulta (998). May authos have estalished soe ecuece elatios fo the sigle ad poduct oets see fo exaple, Ali ad Kha (998) ad Kha, Yaqu & Pavez (983),Alee ad Pasha (2003),ad Alee (2004). A ado vaiale X has a douly tucated Ivese Weiull Distiutio with pdf f ( is give y: Dept. of Statistics, Islaia Uivesity Bahawalpu, PAKISTAN daleeiu@hotail.co Recevied Dec. 2006, Accepted Mach 2007

2 2 Alee Let f ( = P Q x + x exp Q x P (.), > 0 Ad the coespodig cdf F( = x F( exp Q Q x P (.2) P Q Whee Q ad -P (0 < Q < P < ) ae espectively the popeties of tucatio o the left ad ight of the Ivese Weiull distiutio, espectively. With Q = logq ad P = log P If we put Q = 0, the distiutio will e tucated to the ight ad fo P =, it will e tucated to the left. Whee as fo Q = 0 ad P = i.e. Q = 0, get the o-tucated distiutio. Let h( = ad the fo (.) ad (.2) we have + x f ( = [ Q + F( ] h( (.3) Whee Q Q = P Q If we put =, it educes to the tucated Ivese Expoetial distiutio. If we put = 2, it educes to the tucated Ivese Rayleigh distiutio. The douly tucated Ivese Weiull Distiutio ay e used to descie soe pheoea that deped o tie stat fot 0. 0

3 Recuece Relatios fo the Poduct, Ratio ad Sigle Moets of Ode Statistics fo Tucated Ivese Weiull (IW) Distiutio 2 Fo Sigle Moets 3 Let ; tucated pdf of X X 2; X ; X ; L e the ode Statistics. The ( ) is give y [ F( ] [ F( ] f ( ) f: ( = C: x a x (2.) whee! C : = ( )!( )! Theoe 2. Fo the tucated ivese Weiull distiutio (.3), Let ( X ) ( J ) J : = E : the ( J ) J ) ( J) ( J ) Q ( J ) ( J ) + : = : + : [ : : ] (2.2) Poof: Usig the pdf of X : fo (2.) with (.3), we have ( J) J Q J : = C : x [ Fx ( )] [ ( )] dx+ C : x [ Fx ( )] [ ( )] dx a x a x J upo itegatig y pats teatig x fo itegatio ad est of the itegad fo diffeetiatio, we get Q J J 2 = [ C : [ ] [ ] [ ] [ ] x Fx ( ) Fx ( ) xdx ) C : x Fx ( ) Fx ( ) J ) J J ] + C + : x [ Fx ( )] [ ( ) xdx ) C : x [ Fx ( )] [ ( ) xdx ) xdx ) J ) ( J ) ( J ) ( J ) ( J ) [ ] + [ : ] ( J ) : = Q J ) : : J ) + : Rewitig the aove equatio, we get ( J ) ( J ) [ ] ( J ) ( J ) ( J ) ( J ) Q + : = : + : : : Hece the Theoe is poved.

4 4 Alee Coollay 2. By epeated applicatio of the ecuece elatio i (2.2), we otai fo 0 ad J=0,,2, J J J [ Q { }] J J ) + : ( p) p: p+ : p= 0 = p: 3. Fo Poduct Moets Let X : X 2: L X : e the Ode Statistics. the douly tucated joit pdf of X : Ad ( < s ) is give y X s : [ Fx ( )] [ F( ( )] [ F ( a x< y f, s : ( x, = C, s : (3.) whee! C, s : = ( )!( s )!( s)! Theoe 3. Fo tucated ivese Weiull distiutio (.3), let ( J, J K, s : = E [ x :. x s : ], the ( J k, ) J ) s) ( J s, ) Q ( J k, ) ( J, +, =, +, [, s :, s : ] (3.2) Poof: Usig the joit pdf of X : ad X s : fo (3.) with (.3), we have J k, = C, s : Q x y [ Fx ( )] [ F( ( )] [ F ( ydxdy ) + C, s : x a x< y J k y a x< y [ F( ] [ F( ( )] [ F ( dxdy Upo Itegatig R.H.S with espect to X y pats takig X J-- fo itegatio ad est of the itegad fo diffeetiatio, we get.

5 C Recuece Relatios fo the Poduct, Ratio ad Sigle Moets of Ode Statistics fo Tucated Ivese Weiull (IW) Distiutio = Q C J ) J k, x y a x< y J k s, : x y a x< y 2 [ Fx ( )] [ F( ( )] [ F ( ] ydxdy ) C J ) 2 [ Fx ( )] [ F( ( )] [ F ( ( ydxdy ) + [ J k 2 J k x y [ Fx ( )] [ F( ( )] [ F ( ydxdyc ) s, : x y [ Fx ( )] a x< y a x< y [ F( ( )] [ F ( ydxdy ) ] 5 +, s : ( J, ( J, ( J, ( J, ) [ ] + [, ] Q k, = J ), s, +, J ) +, Rewitig the aove equatio, we get ( J, J ) s) ( J, s) Q k +, =, s : +, s :,, ( J, ( J, ) [ ] Hece the theoe is poved. Coollay 3. By epeated applicatio of the ecuece elatio i (3.2) we otai fo 0 < s ad J,k = 0,,2, ( J ) ( J, ) [ p, Q { }] ( J k, ) J ) k +, = ( p) p+, p, p= 0 Coclusio ad Reaks The ecuece elatios fo the sigle ad poduct oets of ode statistics ae estalished fo tucated IW distiutio. These elatios ay e used to copute the sigle, poduct ad atio oets i a siple ecusive ae fo ay saple size. Soe eaks that coect ou esults with elevat liteatue esults ae listed elow:. Settig = i (2.2), (2.3), (3.) & (3.2) we get the coespodig ecuece elatios fo the sigle & poduct oets of ode statistics i the case of IE distiutio (Alee 2003).

6 6 Alee 2. Puttig =2 i (2.2), (2.3), (3.) & (3.2) we get the coespodig esults fo IR distiutio (Mohsi 200) 3. If Q=0 & P= the (2.2), (2.3), (3.) & (3.2) educes to the o - tucated case (Alee 2003, 2005). Refeece. Ahsaullah, M.& Novzoov, V.B.(200), Odeed Rado Vaiales. Nova Sciece Pulishes, Ic. New Yok. 2. Alee, M. ad Pasha, G. R. (2005), Recuece Relatios fo the Poduct, Ratio ad Sigle Moets of Ode Statistics fo Ivese Weiull (IW) Distiutio (Suitted fo Pulicatio). 3. Alee,M.(2004).Classical Popeties, Ratio, Poduct ad Ivese Moets of Ode Statistics ad Chaacteizatio Fo Ivese Weiull Distiutio. J. of Stats. Vol. XI(), Alee,M. ad Pasha, G.R. (2003).Ratio, Poduct ad Sigle Moets of Ode Statistics fo Ivese Weiull distiutio, J. of Stats. Vol. X (2) pp Ali, M.A (999). Recuece Relatio fo Poduct Moets of Ode Statistics fo Tucated Geealized Liea Expoetial Distiutio. Poc.ICCS-VI, vol., Aold, B.C.Balackisha N.R Nagaaga H.N(993), A fist couse i Ode Statistics, Joh Willy & sos, Ic. NewYok. 7. Ghaaph, M.K (993). Copaiso of Estiatos of Locatio Measues of a Ivese ayleigh Distiutio. The Egyptia Statistics Joual. 37, Mohsi, M.(200). Soe Distiutioal popeties of Lowe Recod Statistics fo Ivese Rayleigh Distiutio.(Upulished M.phil Theses) Uivesity of Lahoe, Pak. 9. Mukajee, S.P. ad Maiti, S.S.(996). A Pecetile Estiato of the Ivese Rayleigh Paaete. IAPQR Tasactios, 2, Mukajee, S.P. ad Sae L.K.(984). Bivaiate Ivese Rayleigh Distiutio i Reliaility Studies, Joual of Idia Statistical Associatio, 22, Voda, V.Gh.(972). O the Ivese Rayleigh Distiuted Rado Vaiales, Repot i Statistical Applied eseach JUSE, 9, 3-2.