GENERAL RELATIONS AND IDENTITIES FOR ORDER STATISTICS FROM NON-INDEPENDENT NON-IDENTICAL VARIABLES

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1 A. Ist. Statist. Math. Vol. 44, No. 1, (1992) GENERAL RELATIONS AND IDENTITIES FOR ORDER STATISTICS FROM NON-INDEPENDENT NON-IDENTICAL VARIABLES N. BALAKRISHNAN 1, S. M. BENDRE:* AND H. J. MALIK 2 1Departmet of Mathematics ad Statistics, McMaster Uiversity, Hamilto, Otario, Caada L8S ~K1 2Departmet of Mathematics ad Statistics, Uiversity of Guelph, Guelph, Otario, Caada NIG 2W1 (Received May 18, 1990; revised Novemer 26, 1990) Astract. Some recurrece relatios ad idetities for order statistics are exteded to the most geeral case where the radom variales are assumed to e o-idepedet o-idetically distriuted. I additio, some ew idetities are give. The results ca e used to reduce the computatios cosideraly ad also to estalish some iterestig comiatorial idetities. Key words ad phrases: Order ststistics, recurrece relatios ad idetities, o-idepedet o-idetical variales. 1. Itroductio Let X1, X2,..., X e a sequece of radom variales with correspodig order statistics Xl, <_ X2, <_ "'" <_ X,~. Let Fr,(X) deote the margial distriutio fuctio of Xr, ad F~,s,,~(z, y) deote the joit distriutio fuctio of X~,,~ ad Xs,, 1 <_ r < s <_. The well-kow recurrece relatios whe the radom variales are i.i.d, are give y ad rf~+:,~(x) + ( - r)f~,~(x) --- F,.,,~_a(x), 1 < r < - 1 rfr+l,s+l,(x, y) -1- (8 - r)fr,s+l,(x, y) -I- (Tt - 8)Fr,s,(X, y) = F?,s,-l(x,y), 1 <_ r < s <_. These are proved i terms of raw momets i David ((1981), pp ). The relatios have ee geeralized to the cases whe the radom variales are exchageale (Youg (1967), David ad Joshi (1968)) ad whe they are idepedet ut o-idetically distriuted (Balakrisha (1988)). Recetly, Sathe ad * O leave from Departmet of Statistics, Uiversity of Bomay, Vidyaagari, Bomay , Idia. 177

2 178 N. BALAKRISHNAN ET AL. Dixit (1990) exteded the relatios to the most geeral case where o assumptio is made o the uderlyig joit distriutio fuctio of the variales. The recurrece relatios are give y (1.1) rf~+l,~(x) 4- ( - r)fr,(x) ~- E F~r,_l(X) i=1 ad (1.2) l <r<-1, rfr+l,s+l,(x, y) q- (8 -- r)fr,s+l,(x~ y) q- ( - 8)Fr,s,(X, y) =EF(,i2,_l(x,y), l<_r<s<_-1 i=1 (i) (i) where F~,_l(x! ad F~,8,_ l(x, y) deote the distriutio fuctio of Xr,-1 ad (Xr,-1, Xs,-1) respectively, i a sample of size -- 1 otaied o droppig Xi from the origial sample of size. The eed for recurrece relatios ad idetities is well-estalished i the literature ad for details we refer to the recet moograph y Arold ad Balakrisha (1989). I this paper we geeralize some of the estalished results to the o-idepedet o-idetically distriuted (i.i.d.) variales ad also estalish some ew idetities. The results greatly reduce the amout of direct computatios whe the radom variales are ot ecessarily i.i.d. The results ca also e used to estalish some iterestig comiatorial idetities followig the methods illustrated y Joshi (1973) ad Joshi ad Balakrisha (1981). I order to facilitate the proofs of various results preseted i the followig sectios, we preset elow a lemma which gives a comiatorial idetity satisfied y a complete eta fuctio. LEMMA 1.1. For real positive k ad c ad a positive iteger, where B(.,.) is the eta fuctio. PROOF. Cosider () a~_0(--1)a B(a -i- k, c) = B(k, c + ) (a) k,c) ~-~() a (~)9~01 ~--~(--1) a B(a + = z...~ -1 a ua+k-l(1 - u)c-ldu. a=0 a=0 O chagig the order of summatio ad itegratio we get = ~1 {a~ O(--1)a(:)ua}uk-x(1--u)C-ldu' hece the result. For a alterative proof ased o geeratig fuctios, we refer to Riorda (1968).

3 RELATIONS AND IDENTITIES FOR ORDER STATISTICS 179 Remark 1. ad we get The lemma holds true for icomplete eta itegrals i geeral ~=o(-1)a(a)ip (a + k, c) = Ip(k, c + ) where Ip(a, ) is defied as the icomplete eta itegral give y Ip(a,) = L P u~-l(1 - u)-ldu, p c (0,1). 2. Relatios for margial distriutio fuctios Let FIChu "#"-m](x), 1 < r < m _< deote the distriutio fuctio of r-th order statistic i a sample of size m otaied o droppig Xil, Xi2,.., X~,_,, from the origial sample of size. Further, let H~,m ( x ) = ~ F),m[~l... i,_m] ( x ). l <il <i2<'"<i-m < For m =, H~,~(x) = Fr,(X), 1 < r <. Also, whe the variales are idetically distriuted, H~,m(x) = ( ) Fr,m(x). Result 2.1. For l<r<, (2.1) (2.2) F~,(x) = Z (-1)J+'~-~+l Hx,j(x), j=--r+l F~,~(x) = E(-1) j+~ 1 Hjj(z). j=r PROOF. We prove (2.1) ad (2.2) follows o same lies. From (1.1), we have F~,.(x) - -r+ 1 ~[i~] ~, r-~lfr-l,(x)+~_l rr-l,-llx}. i1=1 Upo usig (1.1) to the r.h.s, of the aove equatio, we get Fr,(X ) ~- (-r+2)(~-r+ 2 _(~ -- ~ + 1! (r - l)(r - 2) 1)F~-2'~(x) - (r - 1)(r - 2) E ~[il]r-2' 1 ~12[il,i2]{~, + (r - 1)(r - 2) " "-~"~-~'~'" ix=l i2=1 i1#i2 i1=1-1'~}(~

4 t80 N. BALAKRISHNAN ET AL. By repeatig this process of usig (1.1) for the expressio o the r.h.s, r- 1 times ad simplifyig the resultig equatio, we derive the relatio i (2.1). Remark 2. The relatios i (2.1) ad (2.2) have already ee estalished y Balakrisha (1988) for idepedet oidetical variales ad they reduce to the well-kow idetities of Srikata (1962) for i.i.d, radom variales. Result 2.2. (2.3) (2.4) 1F~,~(x) ZB(r, r+l)hl,~(x), r 1 F~ (x) = Z B(r, - r + 1)H~ r(x). -r+l " PROOf'. We prove (2.3) ad (2.4) follows o same lies. From (2.1), we have ilf, ~ 1 1 (-1)J+-r+l (J - r) j=-r+l O iterchagig the order of summatio ad makig trasformatio, the r.h.s. reduces to = (_1) / j t (-j+l-1) H~,j(x). j=l Prom Lemma 1.1, the term iside the rackets {} is B(j, -j+ 1), thus estalishig (2.3). Remark 3. y Joshi (1973). idetities. For idetical variales, the result reduces to the idetities give The result also explais the presece of the factor 1/r i Joshi's For i = 1, 2,..., defie for a fixed (2.5) Ci+k-1 f ( + i)( + i + 1)..- ( + i + k- 2), k = 2,3,... 1, k=l. The, y adoptig a method similar to the oe used i provig Result 2.2, we ca prove the followig two results. Result 2.3. For i,k = 1,2,..., (2.6) ~ fr,(x)/{(? ~ ~- i - 1)(r -~- i)-.-(r -t- i -/- k - 2)} Ci+k-1 k - 1

5 - - E RELATIONS AND IDENTITIES FOR ORDER STATISTICS 181 (2.7) E Fr'(x)/{(- r + i)(- r + i + 1)---(- r + i + k- 1)} Remark 4. 1Ci+k_l ~(k+r-2) - 1 For i = 1 ad i case of idetical radom variales, (2.6) ad (2.7) reduce to expressios (2) ad (3) respectively, of Balakrisha ad Malik (1985). Result 2.4. For k = 1,2,..., (2.8) EFr,~(x)/{r(r + 1)... (r + k - 1)(- r + 1)..-(- ec2k~(r+2k-2) r + k)} ad for k, 1 = 1, 2... (2.9) EF~.~(x)/{r(r + 1).--(r + k- 1)(- r + 1)... ( - r + l)} 1 B(r,- r + l) CT+l ~=1.{(r+k+l- ~ (r+k+l-2)h~#(x)} k Hl,r(x) + l - 1 where C2k ad Ck+z are as defied i (2.5) with i = Relatios for joit distriutio fuctios Let F[ i1'#~,~,,~... ](x,y), 1 _< r < s _< m _<, deote the joit distriutio fuctio of r-th ad s-th order statistics from a sample of size m, otaied o droppig Xil, Xi2,., Xi... from the origial sample of size. Further, let Hr,,m(X, y) = FI lr,...,m y), 1_<il <i2 <--"<i-m <_ where, for m =, HT,s,(x, y) = FT.s.~(x, y), 1 <_ r < s <_.

6 i 182 N. BALAKRISHNAN ET AL. Result 3.1. For l<r<s_< (3.1) s-1 Fr, s,(x,y)-~e E (-1)m+-r-s+l j=r m=-s+j+l "(~-ll)(m!:l)hj,j+l,m(x,y), (3.2) 8--1 Fr,s,(X,y)-- E (--1)-m-r+l j=s-r m=-s+j+l ( j-ls_r_l )(m~j-1) gl'j+l'm(x'y)'s (3.3) --?~ i F~,8,~(x,y) = Z (-1) m+8 j=s--r r=r+j (j-1)(m-j-i) s - r - 1 r - 1 Hm-j,m,m(X, y). The aove give idetities ca e proved y startig with the recurrece relatio i (1.2) ad y repeatedly usig it i a way similar to the oe used i provig Result 2.1. These exted the idetities give y Srikata (1962) to those for joit distriutio fuctios ad also to the case of i.i.d, radom variales For i.i.d. variales, (3.2) has ee estalished y Srikata. Now, y startig with the idetities i Result 3.1 ad followig the lies as used i provig Result 2.2, we ca prove the followig result. Result 3.2. (3.4) (3.5) (3.6) - 1 E s=r+l --1 s=rq-1 --F~,8,(x,y) 8--r --E E "(*-*, II -- S q- 1)Sr,r+l,s(X, y), s=r+l E s=r+l --1 ~ 1 E s=r+l i -s+l --1 =EE s=r+l B(s - 1, - s + 1)Hl,~+l,8(x,y), F,.,.(x, y) B(s - 1, - s+ 1)H~,s,s(x,y).

7 RELATIONS AND IDENTITIES FOR ORDER STATISTICS 183 Remark 5. I case of idetical radom variales, the result reduces to 1 -X ~_~ 1 Fr,s,(X,y) = ~ ~ s(s 1 L 1)Fr,r+l,s(X, y), s=r+l 8 -- r s=r~-i Fr,s,(X ' y) ~ ~ 8(8 1 - l)fl,r+l,s(x, Y)' s=r+l r s=r+l 1-1 ~ 1 ) --i ~..~ 1 -E =E r:l s:r+l -- s -~- 1Fr's'(x'Y s=r+l 8(8 -- 1) Fr's's(x'y)' which exteds the idetities give y Joshi (1973) for the joit distriutios. As rightly poited out y Joshi, these idetities ca e effectively used i checkig the computatios of the product momets of order statistics. Remark 6. It should e poited out, however, that if the relatio i (1.1) (or equivaletly (2.1) or (2.2)) ad (1.2) (or equivaletly (3.1), (3.2) or (3.3)) are employed i the computatios of sigle momets ad product momets respectively, the all the idetities estalished i this paper will e automatically satisfied; see Arold ad Balakrisha (1989) for details. Ackowledgemets The authors would like to thak Mrs. Eda Pathmaatha for the excellet typig of the mauscript. NSERC is to e thaked for providig fiacial assistace for Dr. Bedre's visit to McMaster Uiversity through the operatig research grats of Drs. Balakrisha ad Malik. The authors would also like to thak two referees for makig some suggestios which led to a cosiderale improvemet i the presetatio of this paper. REFERENCES Arold, B. C. ad Balakrisha, N. (1989). Relatios, ouds ad approximatios for order statistics, Lecture Notes i Statist., 53, Spriger, New York. Balakrisha, N. (1988). Recurrece relatios for order statistics from idepedet ad oidetically distriuted radom variales, A. Ist. Statist. Math., 40, Balakrisha, N. ad Malik, H. J. (1985). Some geeral idetities ivolvig order statistics, Comm. Statist. Theory Methods, 14, David, H. A. (1981). Order Statistics, 2d ed., Wiley, New York. David, H. A. ad Joshi, P. C. (1968). Recurrece relatios etwee momets of order statistics for exchageale variates, A. Math. Statist., 39, Joshi, P. C. (1973). Two idetities ivolvig order statistics, Biometrika, 60, Joshi, P. C. ad Balakrisha, N. (1981). Applicatios of order statistics i comiatorial idetities, J. Comi. Iform. System Sci., 6, Riorda, J. (1968). Comiatorial Idetities, Wiley, New York. Sathe, Y. S. ad Dixit, U. J. (1990). O a recurrece relatio for order statistics, Statist. Proa. Lett., 9, 1 4. Srikata, K. S. (1962). Recurrece relatios etwee the PDF's of order statistics, ad some applicatios, A. Math. Statist., 33, Youg, D. H. (1967). Recurrece relatios etwee the P.D.F.'s of order statistics of depedet variales, ad some applicatios, Biometrika. 54,