Statistics & Probability Letters 14 (1992) 103-105
North-Holland
27 May 1992
Bounds on distribution functions of order
statistics for dependent varrates
G. Carauxand O. Gascuel
Departementd'lnformatique Fondamentale,Montpellier, France
Received July 1991
Abstract: Upper and lower bounds are given for Fx , the distribution function of the rth order statistic from n possibly dependent
random variables. We show that these bounds maybe reached when the random variables have a common distribution function.
For any distribution function F we may construct a set of n exchangeablevariates (with c.d.f. F), whose dependency structure is
such that the bounds are attained.
Keywords:Distribution function of order statistics, upper and lower bounds, dependent variates.
1. Introduction
Let Xl,.,., Xn be identically distributed random
variables with order statistics denoted as Xl:n ~
. ..
~Xj:n
...
~
~Xn:n. Let Fxr:n be the distri-
bution function of the rth smallest order statistics.
It is well known that if Xi. are independent,
with common distribution function F, then
n
Fx.(x)=
r.n
E(~)F(x)j(l-F(X»n-j.
j=r]
(1)
...
..
When identical and independenceconditionsfor
. . .
.
the dlstnbutlonsare droppedbut the vanatesare
h
bl
th
l'
I
exc
angea
e,
e lormu
a
n
Fxr:Jx) = .~ (
)
r -1- 1)( n
j
j-r
(j
-:-1)
J-r
(
j
X Pr kOl {Xk ~x}
)
(2)
Maurer and Margolin (1976) extend the expression (2) to random variables that are not
necessarily exchangeable. The expression (3.2)
that they found, like (2), may be used in practice
only when the dependency structure of the vari-
ates is known.
An extensive literature has developed on inequalities involving linear functions of order
statistics and their expectations (see David
(1988)).Boundson expectations
of order statistics
for
dependent
random
variables
also be
found in Arnold (1988) and Hoover may
(1989).
In this paper we propose upper and lower
b
oun
ds tfor Ft Xr:n( x ) th at h0ld whatevert he depend
ency
.
s roc
ure.
Th
ese
.
b
oun
d
s may
b
e
use
d .
m
statIstIcaltests, for instance,when the dependency
structureis unknown.
These bounds may be reached when the vari.
ates have the same distribution function. This
generalizes some results obtained by Lai and
Robbins (1976) for extremal statistics.
is given by David (1970, p.82).
2. Bounds on Fx r:. (x)
Correspondence
to: Prof. G. Caraux,Dept. d'Informatique
Fondamentale,
LIRMM, 860rue de SaintPriest,34100Mont-
. .
Mmlmum
.
XI:n and maximum
pellier,
France.
treat. Applicationof Bonferroni'sinequalitygives
0167-7152/92/$05.00 @ 1992- Elsevier SciencePublishers B.V. All rights reserved
Xn:n are easy to
103
,
Volume 14, Number 2
F(x) ~FX1:Jx)
and similarly,
STATISTICS & PROBABILITY LE1TERS
~nF(x)
(3)
F(x);;"Fxnn(x);;"l-n(l-F(x».
In the general case, for any r, we have:
27 May 1992
In the general case we define Pj=pr(vjx)=j)o
Developing the let-hand side of (6) we see that
nF(x)=(r-1)
r-I
r-I
LPjJ'=0
L(r-1-j)Pj
J'-0
-
,!.,
Proposition 1. Let XI,...,Xn
be a set ofn random variables which are identically distributed
+n Erpj-
(with c.d.f. F). Then
1) .
(4)
r
rn
-
()
,)
r Vn X ;;" 1\ ~;
,
,..
+ nFxr:Jx)
-.%.
n - r + 1 ( l-F
Proposltlons
..
(x» .
0
1 and 2 can be generalized to
c~ses in which the X j are not identically
tnbuted.
£... l(xj<x}
j=1
be the number of Xj which are ~Xo Then vn(x)
is a non-negative random variable and, according
to Markov's inequality, we have, for A > 0,
P(
Fxr:Jx»)
n
Fx.(x);;"lr,n
;-
-j)Pj
Since .%;;" o it follows that
Proof. Let l(Xi< x} be the indicator of the event
{X.] ~ x} and let
vjx)
j~r(n
= (r - 1)(1-
Fx. (x) ~ inf ( ~F(X),
,..,
n
E(vn(x»)
()
S
A,
Propositio~ 3. Let XI,...,Xn
be a set ofn random varlables
with distribution
functions
FXl'. o. ,Fxn. Then
(01
sup,-
If A.= r (r E {1,. o. ,n}),th~n the left-hand side in
(S)IS Fxr:n(x). Moreover, smce E(l(xj"x})=F(x),
then
E( vn(x») = nF( x),
(6)
d h
an t us
dis-
E7=1(1-FXj(X»)
n-r+1
)
0 ( E7= IFx} x)
~Fxr:Jx)~mf
r,l.
0
)
The proof of this proposition follows the same
lInesasthat of Propositions1 and 2. We just have
to changethe equality(6) to
n
n
Fxr:Jx)~ -;F(x).
E(vn(x»)= L Fx(x).
Since Fxr:n(x) ~ 1, the inequality
'-I
J-
(4) is proven.
J
0
Proposition 2. Let XI, 0.0' Xn be a set of n ran-
3. Optimality
dom variables which are identically
(with cod.f. F). Then
.
In ~hIS part, we construct a set of exchangeable
vanates, the dependency structure of which is
such th~t ~he bound (4) is attained. Using the
FxrJ x) ;;" snp ( o, 1 - _-.!:
.
distributed
(1 - F( x»
n- r+ 1
).
of the bounds
sameprIncIple,we mayalsoprovethat the bound
(7)
(7) may be reached.
Proof. If x is a continuity point of F then it is
Proposition4. Let F be any distributionfunction.
clear that the inequality
since
Then, for any integer rand n (1 ~ r ~ n), a set ofn
dependent random variables Xl,..., Xn can be
found, wit~ distribution function F, for which the
(7) is contained in (4)
( -x )
FXrn(x) = 1 - F -Xn-r+l:n.
104
bound (4) lS reached.
(j
j
Volume 14, Number 2
STATISTICS & PROBABILITY LElTERS
Proof. Let 'l/ be a random variable
distributed
in the range [0, 1] and let
uniformly
Each random
tion P,
r
variable
X.
J
has distribution
func-
Also,
Uj*=-'l/
VjE{I,.."r},
7-1
1
Uj*=-+-'l/ n
n
Then
27 May 1992
(8a)
Xr:n=p-I(Ur:n)
VjE{r+l,...,n}.
Uj * are uniformly
(8b)
distributed
Pxr:.(x)=PUr:n(P(x».
'.
' II y, usIng ( 10,)
Fma
in the range
( n
pxr.n(x)= inf -P(x),
,
[O,rln]
and
when}E{I,...,r},
.
1.)
r
0
or in the range
.1 ]
[( . - 1)1
in,}
h
n
.
w en}E
{
I
r+,...,n.
}
Similarly,
we may prove
Now let u = {UI' U2"."
un} be a random permutation of the set {I, 2,... , n} defined as a trial
consisting of drawing, without replacement,
each
. .
. '.
.
Proposition
5.. Let P be any dlStnbutlon functlon.
Then, for any Integer r a~d n (1 ~ r ~ n), a set of n
element of the set {I, 2,.,., n}. In this scheme, ~
is the number associated with the outcome of the
jth draw.
dependen~ ra~do.m ~anables. X I"."
Xn ~an be
found, with dlstrlbutlon functlon P, for which the
bound (7) is reached.
Let
U. = U*.
J
(9)
Uj
Since Ur:n = Ur*, we see from (8) that
n
= inf(-U,
r
pur.n(u)
.
1)
Note
distributed.
Vu E [0,1].
(10)
Using
4 and 5 do not apply
In this case Xr:n =p-I(Ur:n)
cannot
be constant
and equal
to j. The dependency
structure may not vary. Then Xr:n = rand
n
Pxr:!'(x) =? .when x <
({u. = k} n {Uk* ~ u}) )
Pr(U. ~ u) = pr (
J
k= I
J
easIly
n
E
Propositions
be written. Moreover,
we may provide examples
in which the bounds may not be fulfilled,
whatever the dependency
structure.
For instance, let
Xj
n
1
= -
that
when the random variables are not identically
Pr(Uk*
~ u),
(11)
verIfIed
that
~
thIs
(l.ot~erw.ise),
dIstrIbutIon
between the bounds of Proposition
neither one nor the other.
It m.ay b,e
functIon
IS
3 but reaches
nk=1
we easily see that the random variables Uj are
distributed
identically.
Moreover,
it follows from
the definition
(8) and from the right-hand
side of
(11) that for any U.,
J
Pu(u)
= u
Vu E [0,1].
(12)
J
Then the random variables Uj are identically dist ' b t d .th
'J'
d ' t 'b t .
f t"
n u e
WI
a unllorm
IS n u Ion
,
unc Ion m
the range [0, 1]. By anamorphosIs let
(Uj,)
Xj = P -I
(13)
Arnold, B.C. (1988), Bounds on the expectedmaximum,
Comm.Statist.Theoryand Methods17(7),2135-2150.
David, H.A. (1970), Order Statistics (Wiley, New York).
David,.H:A.(1988),Gen~ralboundsand inequalitiesin order
statIstIcs,Comm. StatISt.TheoryMethods 17(7),2119-2134.
..
Hoover, D.R. (1989), Bounds on expectations of order statls-
tics for dependentsamples,Statist.Probab.Lett. 8, 261265.
Lai, T.L. and H. Robbins(1976),Maximallydependentrandom variables, Proc. Nat. Acad. Sci. U.S.A. 73(2), 286-288.
Maurer, W. and B.H. Margolin (1976), The multivariate inclu-
where
(u ) -'
p-l
References
m
f { x I P( x )
}
~ u .
sion-exclusion formula and order statistics from dependent variates, Ann. Statist. 4(6), 1190-1199.
105