LEVE ~ TECHNICAL DONNA LUCAS JANUARY 1979 U.S. ARMY RESEARCH OFFICE RESEARCH TRIANGLE PARK, NORTH CAROLINA DAAG C-0035

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w,qel /4 I7qpq,fl/ LEVE ( ti ORDERING OF CONCOMITANTS OF ORDER STATISTICS, WITH APPLICATIONS TECHNICAL DONNA LUCAS JANUARY 1979 US ARMY RESEARCH OFFICE RESEARCH TRIANGLE PARK, NORTH CAROLINA

2 w,qel /4 I7qpq,fl/ LEVE ( ti ORDERING OF CONCOMITANTS OF ORDER STATISTICS, WITH APPLICATIONS TECHNICAL DONNA LUCAS JANUARY 1979 US ARMY RESEARCH OFFICE RESEARCH TRIANGLE PARK, NORTH CAROLINA DAAG2977C0035 I L C ) DEPARTMENT OF STATISTICS UNIVERSITY OF NORTH CAROLINA AT CHAPEL HILL CHAPEL HILL, NORTH CAROLINA APPROVED FOR PUBLIC RELEASE DISTRIBUTION UNLIMITED 3 ; r! : 9 04 it 044

w Ordering of Concomitants of Order Statistics, with Applications Donna Lucas* 1 Introduction and siniimary Let (X 1,Y ) Ci l, 2,,n) be n independent random variables from some bivariate distribution

3 w Ordering of Concomitants of Order Statistics, with Applications Donna Lucas* 1 Introduction and siniimary Let (X 1,Y ) Ci l, 2,,n) be n independent random variables from some bivariate distribution When the X s are arranged in ascending order as we denote the corresponding Y s by X <X < <x 1n 2:n n:n [l:n] [2:n] and call these the concomitants of the order statistics David, O Connell and Yang (1977) investigate the probability distribution of R rn the rank of among the n [r:n] Y s We apply these results to a problem in the reconstruction of a broken random sample, first presented by DeGroot, Feder, and Goel (1971) When X and Y are distributed according to Gumbel s bivariate exponential distribution, we show that 11 1n > i ni > > 11 where 1r = P{R s} Additionally, for n 2 and Y stochastically increasing (decreasing) in X, we note that ir > and 22 > (lt u > and 21 > 22 Th1s research was supported by the US Army Research Office under Contract DMG2977C0035 and the National Science Foundation under Contract MSC

w 2 2 Application of the distribution of the rank of the concomitants to a matching problem Suppose a sample of size n is drawn from some bivariate distribution However, before the sample values are

4 w 2 2 Application of the distribution of the rank of the concomitants to a matching problem Suppose a sample of size n is drawn from some bivariate distribution However, before the sample values are observed, each pair in the sample is broken into its two components We observe the X s in some random order and the Y s in some independent random order, thus not knowing the original correspondence of X s and Y s We consider the problem of matching one particular X, rather than reconstructing the entire sample DeGroot, Feder, and Goel (1971) assume that the joint distribution of X and Y can be represented by a probability density function of the form (21) f(x,y) = x) (y)e for (x,y) E R 2, where c and 8 are arbitrary realvalued functions Denote the ordered observations of the unpaired sample by X X l:n 2 X n :n and 2:n < 1 Suppose one wishes to match n:n The posterior probability of obtaining a correct match is maximized by pairing y with X 1 l:n Similarly, this criterion leads to pairing with x n:n n:n A general solution is not pursued We suggest the following procedure for matching one observat ion, not being restricted to bivariate distributions with probability density functions of the form (21) Suppose one wishes to match the r th largest X Then pair with it the k th largest y, where P{R k} = max P{R = s} r,n lsn r,n David, O Connell, and Yang (1977) derive the following expression for P{R = 5) rn (22) P{R r,n a) fl J J c k e l 0 1k 9 slk 0 nrs+l+k f(x y) dy

5 fl w where 3 1 ; I 0 1 (x,y) = P{X < x, Y < y), 0 2 (x,y) P{X < x, Y > y}, 0 3 (x,y) = P{X > x, Y < y}, 0 4 (x,y) = P(X > x, Y > y), t = min(rl, s i), and C k (r,s,n) = ki (rlk)!(s lk) 1 (n rs+l+k)! Numerical results are given for the case in which the joint distribution of X and Y is bivariate normal, n = 9, p 01(0 1)09, 095 It is noted that for small and intermediate values of p, holding r constant, w is not necessarily maximized by s=r However, we observe that for each set of calculations, and 1T 11 >11 12 > > 11 19, 99 > 98 > >11 91 Since z TI r,n+ is (P) (r,s = l,,n), for negative p, and 19 > 18 > > > 92 > > 1T 3 The distribution of the rank of the first concomitant when sampling from Gu,e1 s bivariate exponential distribution We consider the distribution of the rank of (1:n] when the joint distribution of X and Y is Gunbel s bivariate exponential distribution The marginal distributions of both X and Y are standard exponential and the joint probebility density function is

w 4 (31) f( x,y) = e X ) {(l+ox)(l+oy) 0) (x>o, y>o, O0l) as stated by Johnson and lcotz (1972) When 0 = 0, X and Y are independent, and the correlation decreases as 0 increases Also, P{Y > y X = x)

6 w 4 (31) f( x,y) = e X ) {(l+ox)(l+oy) 0) (x>o, y>o, O0l) as stated by Johnson and lcotz (1972) When 0 = 0, X and Y are independent, and the correlation decreases as 0 increases Also, P{Y > y X = x) decreases as x increases, so we say that Y is stochastically decreasing in X (see Barlow and Proschan (1975)) From (22), we have (32) P{R 1 = s} = n( ) j f f(x,y)dxdy When the joint probability density function of X and Y is (31), (33) (x,y) e (X+y40Xy) and (x,y) = (x+y+oxy) Making the appropriate substitutions, n( : ) i j (x+y+ox)r) sl ) (x+y+oxy) ns (C ) (e Y xy) t(l + Ox) (l + Oy) 0})dxdy n( ) (e ) {( 1+ )(l+ )0}dxdy n( ) (1) k ( s i ) j ( J {(l+ Ox) (1+ey)e}dy)dx Integrating by parts and simplifying, we obtain (34) P{R 1 n a) + e 6 E 1 (! )(n ( ) where E 1 (a) ko (l) k ( 5 ) 2 1), (ns+l+k) L

w 5 We now compare P{R 1 = s} and P{R 1 = si) P{R = s} P{R = si) = { 1 1 + e E 1 ( )(n( ) Z ( i) k ( 5 k0 (ns+l+k) {!

7 w 5 We now compare P{R 1 = s} and P{R 1 = si) P{R = s} P{R = si) = { e E 1 ( )(n( ) Z ( i) k ( 5 k0 (ns+l+k) {! + e 0 E 1 ( ) (n( ) ( 1) k ( s 2 ) 2 i)} k=o (ns+2+k) = ne 0 E 1 ( )[( ) ( 1) k ( Si ) k 0 (ns+i+k) 2 2 ( : ) (1) k ( S2 ) 2 k=0 (n s+ 1+k) n = ne 0 E 1 (! ) ni (1) k ( si ) k=0 k (ns+l+k) 2 1 = ni si ): k si k 1 ki : (n s+ l+k) ns+l e E 1 ( )[( 51 (1) k ( $4 ) (ns+l+k) (35) P{R 1 = s} P{R 1, s 11 n +l e E 1 ( ) Thus, (36) 11 in > 11 1,ni > 11 l,n2 > > 12 > 11 fl We note that any monotonic increasing transformations applied separately to X and Y do not change the values of Due to this fact, (36) holds not only for X and Y having a joint probability density function of the form (31), but for all other variates having distributions which can be derived by such traits formations We conjecture that (36) holds for an even wider class of distributions 1 1

w 6 4 General results for n2 Consider the case in which n 2, and Y is stochasticaily increasing in X Suppose X 1:2 = X i:2 and X 2:2 = X 2 :21 where x l:2 < X 2 :2 Then, (41) P{R 22 = 21X 1:2 = x

8 w 6 4 General results for n2 Consider the case in which n 2, and Y is stochasticaily increasing in X Suppose X 1:2 = X i:2 and X 2:2 = X 2 :21 where x l:2 < X 2 :2 Then, (41) P{R 22 = 21X 1:2 = x 1:2, X 2:2 = X 2 :2 } = (2:2] > Y (l 2] IX l 2 = X 1:21 X 2:2 X 2 :2 } Since (Xi, Y i ) and (X 2, Y ) are independent and identically distributed, the 2 conditional probability density function of Y E?:2] given X r:2 (r: 2] I X = X r: 2 r : = 2 assumption that Y is stochastically increasing in X, that X r:2 is (y Ix = x 2 r = 1,2 It then follows, from the j (42) Pe( (2:2] > y1x 22 X 2:2 ) = ( > y X = x 22 } > P{Y > vix = X 1:2 } P{Y (12] >YIX i:2 x 12 } for all > Hence, > Y 1 2 1X 1:2 = X 12, X 2:2 = x 2 :2 ) = J J C t X x 1: 2 (u Ix x 2: 2 dt du = f F (uix x 12 )f y (UIX X 2:2 du > 1 F (uix x 22 )f y (U1X x 2:2 )du ½ for any x 12 < Thus we have shown (43) P{R 2,2 2) > ½ Since P{R 2,2 1) + P{R 2,2 2)=i, (44) 22 > 21 U

, 7 Due to the relationships P{R 2 2 = l} + P{R 1 2 1) = 1 and P{R 1 2 = i}+ P{R 1 2 = 2) = 1, we also have (45) 11 > 12 It can be shown similarly that for Y stochasticaily decreasing in X, (46) 21 >

9 , 7 Due to the relationships P{R 2 2 = l} + P{R 1 2 1) = 1 and P{R 1 2 = i}+ P{R 1 2 = 2) = 1, we also have (45) 11 > 12 It can be shown similarly that for Y stochasticaily decreasing in X, (46) 21 > 22 and 12 > 11 References Barlow, RE, and Proschan, F (1975) Statistical Theory of Reliability and jesti g Molt, Rinehart, and Winston, Inc David, HA, O Connell, MJ, and Yang, SS (1977) Distribution and expected value of the rank of a concomitant of an order statistic Ann Statist 5, DeGroot, MH, Feder, P1, and Goel, PIC (197i) Matchmaking Ann Math Statist 42, Johnson, NL, and Icotz, S (1972) Distributions in Statistics: Continuous Multivariate Distributions Wiley, New York I I I

w UN CLASSI F IED S E C U R I T Y C L A S S I F I C A T I O N OF T H I S P A G E (W in Di EnIe,ed) REAl ) INS TK UCT I N ; REPORT DOCUMENTATION PAGE I R E P O RT NUMBER T I T L E ( e I SubIIII ) 4 NO

10 w UN CLASSI F IED S E C U R I T Y C L A S S I F I C A T I O N OF T H I S P A G E (W in Di EnIe,ed) REAl ) INS TK UCT I N ; REPORT DOCUMENTATION PAGE I R E P O RT NUMBER T I T L E ( e I SubIIII ) 4 NO V 3 OF R E P O R T & PEHIOD C O V E R E D 5 I3EF ORE COMPLETING FORM R E C I p I E N T S C A T A L O G NUMBER lng of Concomi tants of Order Statistics, Applications, QE TECHNICAL pt1) 6 1 p c 7 AUT HORI 8 Mimeo Ser i esno1204 C O N T R A C T OR G R A N T NU M B E R ( 3 ) DAAc a/ Luca 9 II PERFORMING OP G A N I Z A T I O N NAME AND A D D R E S S Department of Statistics University of North Carolina Chapel Hjil, North Carolina ONTN OLL ING C C F I CE NAME A N D ADD R ESS / k US Army Research Offi ce Research T r i a n g l e Park, NC NON ITO PING A G I N V N A M E A A t DW S ( I l Ii1 f nv iron, C nu l I I n ( 4 1 1, IA ) 977C 35,, 5J N I, 1 ir I1 i RpB T P*e BRA, rt A R E A A WORK UNIT N U M B E R S jj IS JarnPLr l979 / 7 S E C U R I T Y C L A S S (of Ihia ref0rt ) UNCLASSIFIED IS I6 DE C L A S S I F ICA T IO N SC H E D U L E DOWNGRADING D I S T P IB UT I O N c T, r E M E N T (of Ih114 Rp rr ) Approved for Public Release: Distribution Unlimited I dli frr i l iron H porl) DISTRIB UTION S T A T EMENT (o lfh hpt,4, 1 l BSo, k 20 I I? O J IA S U P P L E M E N T A R Y NOTES 9 KEY WO RDS (C&UInia on vta ild If nc aary d IdnSlfy by block numbr) Concomitan ts of Order Stati sti cs, Broken Random Sample (C A BST Ilnu rv ai It nca d l*nuty by block nb,) Let rx,v be n i ndependent rv s from,, me bivariate distribution Let denote tie rth ordered Xvariate, and YCr:n)tbe Vvariate paired with The distribution of the rank of V is applied to a matching problem Also, it Is shown tht when sampl ing from bivarlate exponential DD, )473o?Iow or I No v u so u o E T E CLASS IFICAt ION OF T P A GE (Un D 1J R IL L U I

176 5 t h Fl oo r. 337 P o ly me r Ma te ri al s

176 5 t h Fl oo r. 337 P o ly me r Ma te ri al s A g la di ou s F. L. 462 E l ec tr on ic D ev el op me nt A i ng er A.W.S. 371 C. A. M. A l ex an de r 236 A d mi ni st ra ti on R. H. (M rs ) A n dr ew s P. V. 326 O p ti ca l Tr an sm is si on A p ps