Two remarks on order statistics

71

Journal of Statistical Planning and Inference 11 (1985) 71-74 North-Holland

TWO REMARKS Ludger

ON ORDER

STATISTICS

RLJSCHENDORF

Institut fiir Mathematische Stochastik, Federal Republic of Germany

University of Freiburg, 07800 Freiburg im Breisgau,

Received 14 March 1983; revised manuscript received 16 February 1984 Recommended by S. Panchapakesan

Abstract: Let X,t,, . . . ,X,,, be the order statistics of n iid distributed random variables. We prove that (X(i)) have a certain Markov property for general distributions and secondly that the order statistics have monotone conditional regression dependence. Both properties are well known in the case of continuous distributions. AMS Subject Classification: 62630. Key words: Markov property; Monotone regression; Dependence; MTP2.

Let X=(X,,..., X,) be a random vector with iid components Xi and let Xc ) = Xcn,) denote the ordered vector, X(i,zz ... IX(,), so X, ) takes its values in (X(1,, ... , B={~ER”,~,Ix~I~-~Ix,}. For XEB with xi = . . . =X~IO. The following proposition is a somewhat more explicit version of Theorem 3.1 (b) of Arnold et al. (1984). It implies that the ordered values nearly are a Markov chain. Furthermore, the proof allows one to derive several explicit formulas for the distribution of Xc) (cf. Remark 1). For XEB,

kin-1,

let fk=fk(x)=

C,“=, 11,)(q),

and let Tk=?k(Xo).

Then:

Proposition 1. (Xckj,Tk)15k5n is a Markov chain. Proof. The distribution dPX( ’ --&x)= (Cf. COBOVer (1973),

of X,) is given by

n!

Tl! *-or,! Th.2.1).

(1)

le(x) For X=(X,,

. . . . Xk), xl 5 ... Ix,,

0378-3758/85/$3.30 0 1985, Elsevier Science Publishers B.V. (North-Holland)

let

12

L. Riischendorf / Remarks on order statistics

B,={~ER”-~;

(x,y)EB}=(y=(yl,...,y,-k);XklylI...~y,-k}.

Let sI, . . . . sI be the length of the ties in xl, . . . , xk. Then for y E B,, the length of ties in (x,y) are of the form s, ,..., s~_~,s,+~,s,+, ,..., s,, where OlrsnC:=, si, With d@%,>...,&,) fk

=

dp(x’,....xk)

’

(1) implies n!

fk(x1,..-,+)=

dp(xk+I>...,~n,(y)_

(2)

B,'1 *1.0-(s,+r)!***s,!

Since .s!= fk(x, y) we obtain n!

fk(x,,.**,xk)=

S,! “‘S/_

I!

g/&k, fk).

Similarly,

I q!

fk-l(%...,xk-l)=

n!

,!

“‘S,_

gk-

1(xk- 1, fk- 1)

fors/> 1,

n!

noting that for s,> 1, fk_ I = sI - 1, while for sI = 1, ?k_ 1= s/_ 1 (the functional form of &_ r, hk_ 1 is obvious from (2)). This observation implies the Markov property of (xc,,, Tk)*

0

Remark 1. If x = (x1, . . . ) xk) has ties sI, . . . , sI and fk(x) denotes (as in (2)) the density of the first k order statistics, then we obtain from (2) the following simplified formula: fk(x)=s,!.__s

_ I

,;;_k+s), hk(%Xk)~ I

1.

(3)

.

with

For Px’(xk)

= 0, (3) reduces

to

n! (1 -F(X/J)“? s,! . ..s.!(n - k)! For Px’(xk) > 0,

h,(l,Xk) =-

1

px'(xk)

[(l

-.(xk-))n-k+l

-(l

-F(Xk))“Pk+r],

(4)

13

L. Riischendorf / Remarks on order statistics

h&,x,)=

--&

[(I -F(_q-))“-k+2-(1

-F(X/J)“-k+2

-(n-k+2)PX1(Xk)(l Some further

formulas

The second remark

are given by Gupta concerns

dependence

XEB with ties r,, . . . , ~~ the functionf(x)= Lemma 1. Zf x, y E B, then f(xvy)f(xr\y) sup (inf)).

+x/J)“-k+‘].

and Gupta properties l/r,!...r,!.

(1981) and David of order statistics.

(V (A)

rf(x)f(y)

(5) (1980).

Define for

denoting componentwise

Proof. For a vector .Z= (Z,, . . . , Z,) let I(Z) denote the length of the last tie in Z, i.e. I(Z)= C;=, ll,l(Zi). The proof goes by induction in n E N. (a) n = 2. Consider the cases (xi
Letx=(x,,...,x,,x,+,)=(x’,~,+~),

&t

fw-(Y) =fWfW> f(xAY)f(x"Y)

Y=(Y~,...,Y,,Y,+~)=(Y’,Y,+~)

=f(x'Ay

(6)

ftxAy;f(xvy,.

‘>

')f(x"'r

Let x, y,, we obtain /(y) = 1, I(xVy) = 1 and, therefore, 1 /(xAy) /(xVy)

1 = /(x’Ay’)

1 +

11

Io+l

so this case follows from the assumption. IfY,+, = yn , then I(y) = /(y’) + 1, I(xvy) 1 i(xAy)@Vy)

1

=I(x)

= Qx’vy’)

+

1 and

1 = (,(x’Ay’)

+ l)(/(x’Vy’)

+ 1)

1

1 =p z (f(x) + 1)(&y’) + 1) f(x)I(y)’ Again (6) and the induction discussed similarly. 0

assumption

Lemma 1 implies that n!f(x)l,(x) Karlin and Rinott (1980), Def. 3.1).

settle this case. The other

is an MTP,

function

cases can be

(for the definition

cf.

74

L. Riischendorf / Remarks on order statistics

Proposition

2.

f,, = dPX’)/dPX

is MTP,.

Cl

Proposition 2 implies, that the distribution of XC1,, . . . , Xck, is stochastically increasing in (XCk+ ,), . . . , Xcn,) (cf. Th.4.1 of Karlin Rinott (1980)). Therefore, especially, X(i) is positively regression dependent on XU, for general underlying distributions. For absolutely continuous distributions Proposition 2 is due to Karlin and Rinott (1980), Prop. 3.11, while the likelihood ratio dependence of X~i,,X~, in the continuous case was observed by Lehmann (1966).

References Arnold,

B.C., A. Becker,

U. Gather

and H. Zahedi

(1984). On the Markov

property

of order statistics.

J. Statist. Plann. Inference 9, 147-154. Conover,

W.J.

(1973). Rank tests for one sample,

of a continuous

distribution

function.

(1980). Order Statistics. Wiley,

David,

H.A.

Gupta,

P.L. and R.C. Gupta

two samples

and k samples

without

the assumption

in discrete

order statistics.

Ann. Statist. 1, 1105-l 125.

(1981). Probability

New York. of ties and Markov

property

J. Statist. Plann. Inference 5, 213-279. Karlin,

S. and Y. Rinott

(1980). Classes

of orderings

of measures

and related

correlation

J. Multivariate Anal. 10, 467-498. Lehmann,

E.L.

(1966). Some concepts

of dependence.

Ann. Math. Statist. 37, 1137-1153.

inequalities.