Joint Distribution of any Record Value and an Order Statistics

Transcription

1 Interntionl Mthemticl Forum, 4, 2009, no. 22, Joint Distribution of ny Record Vlue nd n Order Sttistics Cihn Aksop Gzi University, Deprtment of Sttistics Teknikokullr, Ankr, Turkey entelpi@yhoo.com Slih Çelebioğlu Gzi University, Deprtment of Sttistics Teknikokullr, Ankr, Turkey scelebi@gzi.edu.tr Abstrct Let X,X 2,...,X n be rndom smple drwn from n bsolutely continuous distribution function F. Let X :n <X 2:n < <X n:n be the order sttistics nd Y p,p be the record vlues of this rndom smple. In this study, the joint distribution of record vlue Y p nd n order sttistic X r:n is given. Mthemtics Subject Clssifiction: 62H0 Keywords: Order sttistics, record vlues, record times Introduction Let X,X 2,...,X n be rndom smple drwn from n bsolutely continuous distribution function F nd let, b be the support of F with <b. Then record times nd the record vlues re defined respectively by L =,Y = X L n + = min { } j : j>ln,x j >X Ln Y n = X Ln n. The record sttistics nd the order sttistcs re frequently used so s to chrcterize some distribution functions. Ngrj nd Nevzorov 997 [8]

2 092 Cihn Aksop nd Slih Çelebioğlu re interested in the distribution of Y 2 given X 2:2 nd thus obtined chrcteriztion of the eponentil distribution. Su et l [0] delt with the distribution of Y given X n:n while Blkrishnn nd Stepnov 2004 [3] re interested in the distribution of Y 2 given X n:n. The results by Ahsnullh 979 [], Blkrishnn nd Blsubrmnin 995 [2], Gupt 978 [4], Hung 975 [5], Kirmni 984 [6], Ngrj 977 [7] nd Nevzorov 2000 [9] re some other studies obtined in similr contet. In this study, the joint distribution of X r:n nd Y p,p re emined. 2 The joint distribution of X r:n nd Y p Let b>y p >y > >y >y 0 =, b nd emine the probbility P {Y p = y p,x r:n = } for the cses y p <, y = nd y p >seprtely. y p < L p <r. Hence we hve P {Y p = y p,x r:n =, L p =k p,y = y,lp = k,...,y 2 = y 2,L2 = k 2,Y = y } = P {X = y,x 2 <y,...,x k2 <y,x k2 = y 2,X k2 + <y 2,......,X k <y p 2,X k = y,x k + <y,...,x k <y, X kp = y p,x r kp:n kp = } = f y i F k i+ k i y i f y p P { X r kp:n kp = } i= = f y i F k i+ k i y i f y p i= n k p! r k p! n r! F r k [ F ] n r f P {Y p = y p,x r:n =, L p =k p,lp = k,...,l2 = k 2 } yp y y2 n k p! = r k p! n r! F r k [ F ] n r f = f y i F k i+ k i y i f y p dy dy p 2 dy i= n k p! r k p! n r! F r k [ F ] n r f p F k y p f y p k i i=2

3 Record vlues nd order sttistic 093 P {Y p = y p,x r:n = } = = r k p=p r k k p=p k = k 3 k 2 =2 F r k [ F ] n r f F k y p f y p n k p! r k p! n r! p i=2 k i n k p! k p r k p! n r! F r k [ F ] n r f F k y p f y p ζ p, k p where ζ p, k p is function independent of the distribution function nd defined by ζ p, k p = k k = k 3 k 2 i=2 k i. The following Tble gives the vlues of the function ζ p, k p corresponding to some p nd k p vlues. These vlues cn be obtined by the Octve progrm code given in Appendi ,5000,8333 2,0833 2,2833 2,4500 2,5929 2,779 2,8290 2,9290 3,099 3,032 3,80 4,0000,4583,8750 2,2556 2,6056 2,9297 3,236 3,545 3,7808 4,0325 4, ,467 0,7083,0208,343,6687,9943 2,374 2,6369 2,9520 3, ,250 0,243 0,3889 0,5568 0,7422 0,946,523,3720, ,0292 0,0639 0,25 0,744 0,2486 0,3342 0,4302 0, ,0056 0,035 0,0260 0,0435 0,066 0,0939 0, ,0009 0,0024 0,0050 0,0090 0,045 0, ,000 0,0004 0,0008 0,006 0,0027 Tble. Some vlues of ζ p, k p 2 The cse y p =. We hve

4 094 Cihn Aksop nd Slih Çelebioğlu P {Y p =, X r:n =, L p =k p,y = y,lp = k,...,y 2 = y 2,L2 = k 2,Y = y } = P {X = y,x 2 <y,...,x k2 <y,x k2 = y 2,X k2 + <y 2,......X k <y p 2,X k = y,x k + <y,...,x k <y X kp =, X r kp:n kp = } = f y i F k i+ k i y i P { X r k:n k = } i= = f y i F k i+ k i y i i= n k p +! r k p! n r! [ F ]r kp F n r f P {Y p =, X r:n =, L p =k p,lp = k,...,l2 = k 2 } y y2 n k p +! = r k p n r! [ F F n r f ]r kp f y i F k i+ k i y i dy dy p 2 dy y i= = n k p +! r k p! n r! [ F ]r kp F n r+k f P {Y p =, X r:n = } = = r k k p=p k = k 3 k 2 =2 p i=2 k i n k p +! r k p! n r! p [ F ] r kp F n r+k f k i=2 i r n k p +! k p r k p! n r! [ F ]r kp F n r+k f ζ p, k p. k p 3 The cse y p >. In ddition, let y t <nd y t+ be stisfied for t =0,,...,p. Let us cll ny nonrecord observtion s n ordinry vlue. In the cse we hve

5 Record vlues nd order sttistic 095 P {Y p = y p,x r:n =, L p =k p,y = y,lp = k,...,y 2 = y 2,L2 = k 2,Y = y } = P {X = y,x 2 <y,...,x k2 <y,x k2 = y 2,X k2 + <y 2,...,X kt <y t,x kt = y t,x kt+ <y t,...,x kt+ <y t, X kt+ = y t+ X kt+ + <y t+,...,x k <y,x kp = y p,x r:n = } t = f y i F k i+ k i y i i= P { X kt+ = y t+,x kt+ + <y t+,...,x k <y,x kp = y p, X r kt+ :n k t+ = }. Now, let us consider the probbility in the lst eqution closely. If we cll the ordinry vlues between two successive record vlues s the set of ordinry vlues, then we hve p t sets of ordinry vlues in the emined probbility. Let the numbers of vlues greter thn in these sets of ordinry vlues be i,i 2,...,i, respectively. For the ske of simple nottion, let I =i,i 2,...,i nd J be the set of indices for the rndom vribles with the ordinry vlues. Thus the emined probbility cn be prtitioned into three prts s follows P { X kt+ = y t+,x kt+ + <y t+,...,x k <y,x kp = y p, X r kt+ :n k t+ = } = P { X kt+ = y t+,x kt+ + <y t+,...,x k <y,x kp = y p, I I X r kt+ :n k t+ = } + P { X kt+ = y t+,x kt+ + <y t+,...,x k <y, j J X kp = y p,x kp = y p,x r kt+ :n k t+ =, X j = } + P { X kt+ =, X kt+ + <,...,X k <y,x kp = y p, I I 3 X r kt+ :n k t+ = } where

6 096 Cihn Aksop nd Slih Çelebioğlu I = I 2 = I 3 = { i,i 2,...,i m {0,k p r p + t} j= min {k p k t+ +,n r} p t} { i,i 2,...,i m {0,k p r p + t } min {k p k t+,n r} p t} { i,i 2,...,i m {0,k p r p + t} min {k p k t+2 +,n r} p t }. j=2 i j j= i j i j In this prtition, the first sum represents the cse where none of the ordinry nd record vlues re not equl to while the second sum points out the cse rndom vrible indeed by j of ordinry vlues is equl to. The lst sum denotes the cse where one of the record vlues is equl to, nd is only possible for t<. Define Ik t = j=k i j nd emine these sums, respectively: the first sum gives P { X kt+ = y t+,x kt+ + <y t+,...,x k <y,x kp = y p, I I X r kt+ :n k t+ = } = kj+ k f y j j [F y j F ] F k j+ k j I I j=t+ f y p P { X r kp+p t+i t :n kp = } kj+ k j = I I j=t+ f y j [F y j F ] f y p n k p! r k r k p + p t + I t t+! n r p + t It!F [ F ] n r p+t It f nd the second sum gives

7 Record vlues nd order sttistic 097 P { X kt+ = y t+,x kt+ + <y t+,...,x k <y,x kp = y p, j J X r kt+ :n k t+ =, X j = } = p {{[ k t+2 k t+! f y z k z=t+ t+2 k t+ i 2!i! ] [F y t+ F ] i F k t+2 k t+ i 2 f [ ] kt+3 k t+2 [F y i t+2 F ] i 2 F k t+3 k t+2 i 2 2 [ kp k [F y i F ] i F kp k i ]} {[ ] kt+2 k + t+ [F y i t+ F ] i F k t+2 k t+ i [ k t+3 k t+2! k t+3 k t+2 i 2 2!i 2! [F y t+2 F ] i 2 Defining F k t+3 k t+2 i 2 2 f ] [ kp k [F y i F ] i F kp k i ]} + } n k p r k p + p t + I t F r kp+p t+it [ F ] n r p+t I t. B ul = {,u= 0, otherwise the second sum cn be written in the following form = u= l= k t+l+ k t+! k t+l+ k t+l i l B ul!i l! f y t+l [F y t+l F ] i l f y p n k p r k p + p t + I t F r k t+ [ F ] n r p+t It f. For the lst sum, it is obvious tht

8 098 Cihn Aksop nd Slih Çelebioğlu I I 3 j=t+2 kj+ k f y j j [F y j F ] f y p n k p +! r k p + p t + I2 t! n r p + t I2 t +! F r kt+ [ F ] n r p+t It 2 + f. On the other hnd, it is lso stisfied tht P {Y p = y p,x r:n =, L p =k p,lp,k,...,l2 = k 2 } yp yt+2 yt y2 t = f y i F k i+ k i y i t=0 = t=0 i= P { X kt+ = y t+,x kt+ + <y t+,...,x k <y,x kp = y p, X r kt+ :n k t+ = } dy dy 2 dy t dy t+ dy yp yt+2 t+ i=2 k i F k t+ P { X kt+ = y t+,x kt+ + <y t+,...,x k <y,x kp = y p, X r kt+ :n k t+ = } dy t+ dy. Prtition the integrl under the lst sum into three prts ectly s in without considering the coefficient t+ i=2 F kt+ k i for one moment nd evlute ech prt seprtely: yp yt+2 I I j=t+ kj+ k j f y i j [F y j F ] j t f y p dy t+ dy n k p! r k p + p t + I t! n r p + t I! F r k t t+ [ F ] n r p+t It f. Mking the trnsformtion F y j F =u j, j = t +,...,p gives f y j dy j = du j nd y j+ F y j+ F 0 = u j+ 0, nd results

9 Record vlues nd order sttistic 099 = I I j=t+ j=t+ kj+ k j k= k l= i l + k [F y p F ] It + f y p n k p! r k p + p t + I t! n r p + t I! F r k t t+ [ F ] n r p+t It f = kj+ k j j t I I l= i l + j t [F y p F ] It + f y p n k p! r k p + p t + I t! n r p + t I! F r k t t+ [ F ] n r p+t It f. The evlution of second integrl is s follows: yp yt+2 u= l= k t+l+ k t+l! k t+l+ k t+l i l B ul!i l! f y t+l [F y t+l F ] i l f y p dy t+ dy n k p r k p + p t + I t F r k t+ [ F ] n r p+t It f = u= l= k t+l+ k t+l! k t+l+ k t+l i l B ul!i l! [F y p F ] It + k j= i j + k f y p k= = u= n k p r k p + p t + I t l= F r k t+ [ F ] n r p+t It k t+l+ k t+l! k t+l+ k t+l i l B ul!i l! l j= i j + l [F y p F ] It + f y p n k p r k p + p t + I t F r k t+ [ F ] n r p+t It f = u= l= f y p k t+l+ k t+l! k t+l+ k t+l i l B ul!i l! n k p r k p + p t + I t [F y p F ] It + l j= i j + l F r k t+ [ F ] n r p+t It f.

10 00 Cihn Aksop nd Slih Çelebioğlu The evlution of lst integrl is similr to the first integrl which results I I 3 j=t+2 kj+ k j j t l=2 i l + j t [F y p F ] It 2 + f y p n k p +! r k p + p t + I2 t! n r p + t I2 t +! F r k t+2 [ F ] n r p+t It 2 f. Thus in the cse y p >, we hve P {Y p = y p,x r:n =, L p =k p,lp = k,...,l2 = k 2 } t+ = k t=0 i=2 i F kt+ { kj+ k j [F yp F ] It + j t l= i f y p l + j t I I j=t+ n k p! r k p + p t + I t! n r p + t I! F r k t t+ [ F ] n r p+t It f + u= l= [F y p F ] It + l j= i f y p j + l k t+l+ k t+l! k t+l+ k t+l i l B ul!i l! n k p r k p + p t + I t p 2 t+ + k t=0 i=2 i F kt+ kj+ k j I I 3 j=t+2 } F r k t+ [ F ] n r p+t It f [F yp F ] It 2 + j t l=2 i f y p l + j t n k p +! r k p + p t + I2 t! n r p + t I2 t +! F r k t+2 [ F ] n r p+t It 2 f. Hence in the cse y p >,we hve the joint distribution of pth record vlue

11 Record vlues nd order sttistic 0 Y p, nd rth order sttistic X r:n s follows: P {Y p = y p,x r:n = } k n p k 3 t+ = k p=p k = { I I j=t+ k 2 =2 t=0 i=2 kj+ k j k i F k t+ [F yp F ] It + j t l= i f y p l + j t n k p! r k p + p t + I t! n r p + t I! F r k t t+ [ F ] n r p+t It f + + n u= l= k t+l+ k t+l! k t+l+ k t+l i l B ul!i l! n k p r k p + p t + I t k p k p=p k = I I 3 j=t+2 k 3 p 2 t+ k 2 t=0 i=2 kj+ k j [F y p F ] It + l j= i f y p j + l } F r k t+ [ F ] n r p+t It f k i F k t+ [F yp F ] It 2 + j t l=2 i f y p l + j t n k p +! r k p + p t + I2 t! n r p + t I2 t +! F r k t+2 [ F ] n r p+t It 2 f. 3 Appendi. Octve Code for Clculting ζ p, k p Define the vlues of p nd k p for which ζ p, k p is to be clculted. toplm in the lst line is the vlue of ζ p, k p. # p= # kp= for i=2:p- ki=i; end toplm=0;

12 02 Cihn Aksop nd Slih Çelebioğlu devm=; i=; while i<p crpim=; for j=2:p- crpim=crpim*/kj-; end toplm=toplm+crpim; i=2; while i<p ki=ki+; if ki>ki+- ki=i; i=i+; else brek end end end toplm References [] M. Ahsnullh, Chrcteriztion of eponentil distribution by record vlues, Snkhy, Ser. B [2] N. Blkrishnn, K. Blsubrmnin, A chrcteriztion of geometric distribution bsed on record vlues, Journl of Applied Sttisticl Science [3] N. Blkrishnn, A. Stepnov, Two chrcteriztions bsed on order sttistics nd records, Journl of Sttisticl Plnning nd Inference [4] R. C. Gupt, Reltions between order sttistics nd record vlues nd some chrcteriztion results, Journl of Applied Probbility [5] J. S. Hung, Chrcteriztion of distributions by the epected vlues of order sttistics, Annls of Institute Sttisticl Mthemtics [6] S. N. U. A. Kirmni, M. I. Beg, On chrcteriztion of distributions by epected records, Snkhy, Ser. A

13 Record vlues nd order sttistic 03 [7] H. N. Ngrj, On chrcteriztion bsed on record vlues, Austrli Journl of Sttistics [8] H. N. Ngrj, V. B. Nevzorov, On chrcteriztions bsed on record vlues nd order sttistics, Journl of Sttisticl Plnning nd Inference [9] V. B. Nevzorov, A chrcteriztion of eponentil distributions by correltions between records. Mthemticl Methods in Sttistics [0] J. Su, N. Su, W. Hung, Chrcteriztions bsed on record vlues nd order sttistics. Journl of Sttisticl Plnning nd Inference Received: September, 2008