Ordering of Concomitants of Order Statistics, with Applications
Donna Lucas*
1.
Introduction and s1..DllJllary.
Let (X 1..• Y.)
(i
1.
=
l.2 ••..• n) be n independent random variables from some
bivariate distribution.
When the X's are arranged in ascending order as
xl:n -<
<
2:n-
X
< X
-
n:n
we denote the corresponding V's by
Y[l:n]' Y[2:n]' .... Y[n:n] •
and call these the concomitants of the order statistics.
David,
_~
~
O'Connel~
and Yang (1977) investigate the probability distribution
of R • the rank of Y[
] among the nY's. We apply these results to a
r.n
r:n
problem in the reconstruction of a broken random sample. first presented by
DeGroot, Feder. and Goel (1971).
When X and Yare distributed according to Gumbel's bivariate exponential
distribution, we show that
~ln > ~l.n-l > ••• > ~ll
where
~
rs
= P{Rr,n = s}.
Additionally, for n
(decreasing) in X. we note that
~ll > ~12
and
=2
'
and Y stochastically increasing
~22 > ~2l (~12 > ~ll
and
~2l > ~22)·
*This research was supported by the U.S. Army Research Office under Contract
DAAG-29-77-C-0035 and the National Science Foundation under Contract MSC78-0l434.
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
(2.1)
f(x,y)
= a(x)S(y)e xy
for (x,y)
where a and S are arbitrary real-valued functions.
vations of the un-paired sample by x l : n
Yl:n
~
Y2:n
~
...
~
Yn:n·
~
x2: n
~
...
2
R,
E
Denote the ordered obser~
x : and
n n
Suppose one wishes to match xl : n '
The posterior
probability of obtaining a correct match is maximized by pairing Yl:n with x : '
l n
Similarly, this criterion leads to pairing Yn:n with x :
n n
is not pursued.
A general solution
We suggest the following procedure for matching one observation, not being
restricted to bivariate distributions with probability density functions of the
form (2.1). Suppose one wishes to match the r th largest X.
th
the k
largest Y, where
P{R
r,n
= k} =
max
l~s~n
P{R
r,n
Then pair with it
= s}.
David, O'Connell, and Yang (1977) derive the following expression for
P{R
r,n
(2.2)
= s}.
P{Rr,n
roo
roo
= s } = n J_
J_ oo
oo
k r-l-k s-l-k n-r-s+l+k
Ck 8l 82
83
84
f(x,y)dxdy,
3
where
8 (x,y)
l
= p{X
< x, Y < y}, 8 (x,y)
= p{X
< x, Y > y},
8 (X,y)
3
= p{X
> x, Y < y}, 8 (x,y)
4
= p{X
> x, Y > y},
t
= min(r-l,
2
s-l) ,
~d
(n-l)!
Ck(r,s,n) = k!(r-l-k)!(s-l-k)!(n-r-s+l+k)!
Numerical results are given for the case in which the joint distribution of X
and Y is bivariate normal, n = 9, p = 0.1(0.1)0.9, 0.95.
small and intermediate values of p, holding r
maximized by s=r.
It is noted that for
nrs is not necessarily
However, we observe that for each set of calculations,
.~
const~t,
nIl> n12 > ••• > n19 '
~d
= nr,n+ 1-s (-p) (r,s = l, ••• ,n), for negative p,
3.
The distribution of the rank of the first concomitant when sampling from
Gumbel's bivariate exponential distribution.
We consider the distribution of the
bution of X
~d
r~k
of Y[l:n] when the joint distri-
Y is Gumbel's bivariate exponential distribution.
The marginal
distributions of both X and Yare standard exponential, and the joint probability density function is
4
(3.1)
f(x,y)
= e -(x+y+0xy) {(l+ex)(l+0y) - 0} (x>O, y>O,
as stated by Johnson and Kotz (1972).
When 0
the correlation decreases as 0 increases.
= 0,
0~0~1)
,
X and Yare independent, and
Also, PtY > ylX
= x}
decreases as x
increases, so we say that Y is stochastically decreasing in X (see Barlow and
Proschan (1975».
From (2.2), we have
roo
roo s-l n-s
P{R I ,n = s} = n( n-l
03 04 f(x,y)dxdy.
s- 1) J_.-~ J_.
-~
(3.2)
When the joint probability density function of X and Y is (3.1),
Et(
(3.3)
~
x,y ) -_ e -x - e - (x+y"0xy) ,
and
~(x,y)
= e -(x+y+OYv)
VAJ.
Making the appropriate substitutions,
. (e-(x+y+0xY)t(1 + ex) (1 + 0y) - 0})dxdy
= n(n-l)
S 1
-
s-l
~
(_l)k(sk- l ) In
roo e- nx ( In
roo e-(n-s+l+k) (y+exy) {(1+ex)(1+~')-0}dy)dx.
o
o
k=O
VJ
L
Integrating by parts I and simplifying, we obtain
(3.4)
P{Rl,n
where EI (a)
= s} = ~
-ax
!!.9
+ e
= r; ~ dx.
EI(:)(n(::~)
s-I
k!O (_I)k (skI)
1
-(n-_-s-+-I+-k-)~2
- I) ,
5
We now compare P{R I ,n = s} and P{R I ,n = s-l}.
n
S
n
n-l
I
P{R 1, n = s} - P{R l,n = s-l } = {n + e EI(S)(n(s_l)
{~
n
+
eS
E
I
(~) (n(~=;) sI
2
(_l)k
k=O
(si/)
--1-
(n-s+2+k)
2
- I)}
n
S
n n-l s-l
k
1
= ne EI(S) [(s-l) l (-1) (S~l)
2
k=O
(n-s+l+k)
s-l
+ (n-l) t (_l)k s-l
k
s-l k__LI
( k ) n-s+l
1
]
2
(n-s+l+k)
n
s-l
= n e S E (n) [(n-l) t (l)k (S-l)
1
]
n-s+l
1 S s-l k~O k
(n-s+l+k) .
n
P{R
(3.5)
l,n
= s}
- P{R
I ,n
=s
1 e S E (n)
- n-s+l
1S
- I} -
Thus,
(3.6)
~ln > ~l,n-l > ~I,n-2 > ••• > ~12 > ~ll
We note that any monotonic increasing transformations applied separately to X
and Y do not change the values of
~rs.
Due to this fact, (3.6) holds not only
for X and Y having a joint probability density function of the form (3.1), but
for all other variates having distributions which can be derived by such transformations.
We conjecture that (3.6) holds for an even wider class of
distributions.
6
4.
General results for n=2.
Consider the case in which n=2, and Y is stochastically increasing in X.
Suppose Xl : 2 = xl : 2 and X2 : 2 = x2 : 2, where xl : 2 < x2 : 2 .
Then,
Since (Xl'Yl) and (X 2,Y 2) are independent and identically distributed, the
conditional probability density function of Y[r:2] given Xr : 2 = xr : 2 is
(ylX .2 = x .2) = fy(ylx = x .2)' r = 1,2. It then follows, from the
[r:2]
r.
r.
r.
assumption that y is stochastically increasing in X, that
fy
e·
Hence,
P{y[2:2] > y[I:2] IX1 : 2
= x 1 : 2,
X2 : 2
= x 2 : 2}
= ~oo f~oo fy(tlX = x1 : 2)fy (uIX
= ~oo
Fy(uiX
> ~oo Fy(uiX
for any x1 : 2 < x2 : 2 .
Since P{R 2,2
(4.4)
= x1 : 2)fy (uIX = x2:~dU
= x2: 2)fy (uIX = x2 : 2)du = ~ ,
Thus we have shown
P{R2 ,2 = 2} >
(4.3)
= I}
+
P{R2, 2
= x 2: 2)dt du
= 2} = 1,
~ •
7
Due to the relationships P{R 2, 2
P{R , 2
1
= 1}
+
P{R1 , 2
= 2} = 1,
= 1}
+
P{R1, 2
= 1} = 1
and
we also have
(4.5)
It can be shown similarly that for Y stochastically decreasing in X,
(4.6)
References
Barlow, R.E., and Proschan, F. (1975).
Life Testing.
Statistical Theory of Reliability and
Holt, Rinehart, and Winston, Inc.
David, H.A., O'Connell, M.J., and Yang, S.S. (1977).
Distribution and expected
value of the rank of a concomitant of an order statistic.
-e
Ann. Statist. 5,
216-223.
DeGroot, M.H., Feder, P.I., and Goe1, P.K. (1971).
Matchmaking.
Ann. Math.
Statist. 42, 578-593.
Johnson, N.L., and Kotz,
s.
(1972).
Multivariate Distributions.
Distributions in Statistics: Continuous
Wiley, New York.
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REPORT NUMBER
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TITLE (an<l Subtitle)
GOVT ACCESSION NO.
Ordering of Concomitants of Order Statistics,
with Applications
RECIPIENT'S CATALOG NUMBER
S.
TYPE OF REPORT /I PERIOD COVERED
TECHNICAL
6.
_....•
B.
AU THORr,)
7
3.
PERFORMING O'G. REPORT NUMBER
Mimeo Series No.1204
CONTRACT OR GRANT NUMBER(.)
Donna Lucas
9.
DAAG-29-77-C-0035
10. PROGRAM ELEMENT. PROJECT. TASK
PERFORMING ORGANIZATION NAME AND ADDRESS
I\,REA /I WORK UNIT 'lUMBERS
Department of Statistics
University of North Carolina
Chapel Hill, North Carolina 27514
II.
~.
CONTROLLING OFFICE NAME AND ADDRESS
, 2.
U.S. Army Research Office
Research Triangle Park, NC
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REPORT DATE
January 1979
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KEY WORDS (C&nl/nue on ,everse .. ide If necessary and Idemllf.y by block number)
en'"red In Block
it diller"'"
Irom R"por')
Concomitants of Order Statistics, Broken Random Sample
20.
ABST1ACT (Continue on reverse sIde If neces.ary lind Idenllfy by block number)
--
Let Xj'Y i ) be n independent rv's from some bivariate distribution. Let X .
denote tne rth ordered X-variate, and Y[r:n] the V-variate paired with Xr:n·r.n
The distribution
of Rr , n' the rank of Y[ r.. nJ' is applied to a matching problem.
. .
~lso, 1t 1S shown that TI 1n > TI 1 ,n-1 > ••• > TIll when sampling from Gumbel's
bivariate exponential distribution, where TI = P{R
= s}
rs
r,n
.
DO
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1 JAN 73
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