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Dependence structure of generalizedorder statisticsErhard Cramer aa Institute of Statistics, RWTH Aachen University, Aachen,D-52056, GermanyVersion of record first published: 01 Feb 2007.

To cite this article: Erhard Cramer (2006): Dependence structure of generalized order statistics,Statistics: A Journal of Theoretical and Applied Statistics, 40:5, 409-413

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Statistics, Vol. 40, No. 5, October 2006, 409–413

Dependence structure of generalized order statistics

ERHARD CRAMER*

Institute of Statistics, RWTH Aachen University, D-52056 Aachen, Germany

(Received 25 February 2005; in final form 14 May 2006)

In this article, simple expressions for marginal density functions of multiply censored generalized orderstatistics based on continuous distribution functions are obtained. Moreover, it is shown that generalizedorder statistics are multivariate totally positive and, thus, associated. This property is applied to showthat regressions of generalized order statistics are nondecreasing under weak conditions.

Keywords: Generalized order statistics; Marginal and conditional distributions; Multiple censoring;Association; MTP2

1. Marginal and conditional distributions of generalized order statistics

In ref. [1], it is shown that the joint density function (df) of generalized order statistics(gOSs) X

(1)∗ , . . . , X(n)∗ based on a cumulative distribution function (cdf) F and parameters

γ1, . . . , γn > 0 can be represented as

(X(1)∗ , . . . , X(n)

∗ ) ∼F

−1(B1), F

−1(B1B2), . . . , F

−1

n∏

j=1

Bj

, (1)

where B1, . . . , Bn are independent, power-function-distributed random variables with para-meters γ1, . . . , γn, i.e. Bj has the distribution function Gj(t) = tγj , t ∈ [0, 1], j = 1, . . . , n.F = 1 − F denotes the survival function of F .

Based on the following lemma [2], the joint df of a multiply censored sampleX

(j1)∗ , . . . , X(jr )∗ , 1 ≤ j1 < · · · < jr ≤ n, can be calculated.

LEMMA 1.1 Let B1, . . . , Bn be independently power-function-distributed random variableswith parameters γ1, . . . , γn, respectively, and 1 ≤ j1 < · · · < jr ≤ n.

*Email: erhard.cramer@rwth-aachen.de

StatisticsISSN 0233-1888 print/ISSN 1029-4910 online © 2006 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/02331880600822291

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410 E. Cramer

Then, the joint df of (V1, . . . , Vr), where Vρ = ∏jρ

ν=1 Bν, 1 ≤ ρ ≤ r, is given by

f V1,...,Vr (u1, . . . , ur) =(

jr∏ν=1

γν

) r−1∏

ρ=1

1

uρ

r∏

ρ=1

Gδρ ,0δρ ,δρ

[uρ

uρ−1

∣∣∣∣ γjρ−1+1, . . . , γjρ

γjρ−1+1 − 1, . . . , γjρ− 1

],

(2)where 1 = u0 ≥ u1 ≥ · · · ≥ ur > 0 and δρ = jρ − jρ−1, 1 ≤ ρ ≤ r, j0 = 0, and

Gδρ ,0δρ ,δρ

[·| γjρ−1+1, . . . , γjρ

γjρ−1+1 − 1, . . . , γjρ− 1

]is a particular Meijer’s G-function [3].

Let P F be the probability measure induced by the cdf F . The joint df of a selection of gOSsw.r.t. an appropriate product measure

⊗P F can now be easily deduced from equations (1)

and (2). In order to prove the result, it has to be taken into account that for a continuous cdfF, the following relation holds

1(−∞,F (t)] ◦ F = 1(−∞,t] P F a.e., t ∈ R,

where 1(−∞,t](x) = 1, x ≤ t, and 1(−∞,t](x) = 0, x > t .

THEOREM 1.2 Let 1 ≤ j1 < · · · < jr ≤ n and F be a continuous cdf with right endpoint ofsupport ω(F).

Then, a joint⊗r

j=1 P F -density of gOSs X(j1)∗ , . . . , X

(jr )∗ is given by

f X(j1)∗ ,...,X

(jr )∗ (x1, . . . , xr) =(

jr∏ν=1

γν

) r−1∏

ρ=1

1

F(xρ)

r∏

ρ=1

Gδρ ,0δρ ,δρ

[F(xρ)

F (xρ−1)

∣∣∣∣∣ γjρ−1+1, . . . , γjρ

],

(3)where −∞ = x0 < x1 ≤ · · · ≤ xr < ω(F) and δρ = jρ − jρ−1, 1 ≤ ρ ≤ r, j0 = 0.

Remark 1.3

(i) If jρ = ρ, 1 ≤ ρ ≤ r, the joint df (w.r.t.⊗r

j=1 P F ) of the first r gOSs results. Thisassumption implies δρ = 1, 1 ≤ ρ ≤ r, and, thus,

G1,01,1

[F(xρ)

F (xρ−1)

∣∣∣∣∣ γρ

]=

(F(xρ)

F (xρ−1)

)γρ−1

holds. Combining these arguments, evaluation of equation (3) leads to

f X(1)∗ ,...,X

(r)∗ (x1, . . . , xr) =(

r∏ν=1

γν

) r−1∏

ρ=1

1

F(xρ)

r∏

ρ=1

(F(xρ)

F (xρ−1)

)γρ

1x1≤···≤xr(4)

=(

r∏ν=1

γν

) r−1∏

ρ=1

Fγρ−γρ+1−1

(xρ)

F

γr−1(xr)1x1≤···≤xr

. (5)

For an absolutely continuous cdf F , this expression can be found in ref. [4].(ii) In the particular case of order statistics, a

⊗rj=1 P F -density of the multiply censored

sample Xj1:n, . . . , Xjr :n [5, p. 319] is given by

f Xj1 :n,...,Xjr :n (x1, . . . , xr) = n!(n − jr)!

r∏ρ=1

1

δρ !r∏

ρ=1

(F(xρ) − F(xρ−1)

)δρ−1F

n−jr(xr),

where −∞ = x0 < x1 ≤ · · · ≤ xr < ω(F) and δρ = jρ − jρ−1, 1 ≤ ρ ≤ r, j0 = 0.

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Dependence structure of gOSs 411

Further explicit representations can be deduced in the model of progressive censoring using theresult of ref. [6] for one-dimensional marginal distributions of progressively censored-orderstatistics. More information on progressive censoring can be found in ref. [7].

It is well known that gOSs based on a continuous cdf obey the Markov chain property[4]: A P F -density of the conditional distribution P X

(r+1)∗ |X(r)∗ =x can be directly deduced fromequation (3)

f X(r+1)∗ |X(r)∗ (t |x) = γr+1

F(t)

(F(t)

F (x)

)γr+1

1(x,ω(F ))(t), x, t ∈ R.

Using the representation of the joint df, expressions for arbitrary conditional distributions canbe obtained.

2. Dependence structure of gOSs

In this section, we are focusing on the notion of multivariate total positivity. This property ofa df is important in, e.g., reliability theory, as it implies association of the components of thecorresponding random vector. This notion of dependence was introduced by Esary et al. [8].

DEFINITION 2.1 An n-dimensional real-valued random vector X is associated if

Cov(g(X), h(X)) ≥ 0

for every pair of increasing functions g, h: Rn → R.

This definition has some interesting implications [9, chapter 3]. An important feature dueto Esary et al. [8] is that any subset of associated random variables is associated as well.For instance, this implies that the covariance of two random variables, which are componentsof an associated random vector is nonnegative. Furthermore, association of a random vector(X1, . . . , Xn)

′ implies the following inequality

P(X1 > x1, . . . , Xn > xn) ≥n∏

j=1

P(Xi > xi) for all x1, . . . , xn ∈ R. (6)

Although association is a desirable feature of random variables, it is often difficult tocheck. A more restrictive property, which implies association but is often easy to check,is multidimensional total positivity.

DEFINITION 2.2 [10] A df f : Rn → R is multidimensional totally positive ( MTP2), if

f (x)f (y) ≤ f (x ∧ y)f (x ∨ y) for all x, y ∈ Rn.

x ∧ y and x ∨ y are defined via x ∧ y = (min{x1, y1}, . . . , min{xn, yn})′ and x ∨ y =(max{x1, y1}, . . . , max{xn, yn})′.

A random vector X is said to be MTP2 if its df is MTP2.

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412 E. Cramer

The basic properties of MTP2 are derived in ref. [10]. In our setup, it is important that theindicator function

1x1≤···≤xn

is MTP2, and that a product of the form(n∏

i=1

fi(xi)

)g(x1, . . . , xn)

has this property provided that fi are nonnegative and that g is MTP2. Recalling the repre-sentation of the joint

⊗nj=1 P F -df of gOSs based on a continuous cdf given in equation (5),

this yields directly the following result well known for order statistics from absolutely con-tinuous distributions [10]. The MTP2 property for discrete order statistics was established byRüschendorf [11]. Although it is near at hand to investigate this property for record values,it seems not to be known in this model. By choosing appropriate values for γ1, . . . , γn [1], itfollows that the progressively censored order statistics have this property as well.

THEOREM 2.3 gOSs X(1)∗ , . . . , X

(n)∗ based on a continuous cdf are MTP2. In particular, thisyields Cov(X

(j)∗ , X

(r)∗ ) ≥ 0 for all 1 ≤ j, r ≤ n.

Obviously, the MTP2 property implies nonnegative covariances. For order statistics,nonnegative correlation was first claimed by Bickel [12]. Arnold et al. [13, p. 33] stated thatrecord values are always positively correlated. Nagaraja and Nevzorov [14] considered, e.g.,covariances of functions of record values. They point out that the correlation of transformedadjacent record values based on continuous distributions is always nonnegative. However,they present an example with negative covariance if g is nonmonotone and the underlyingdistribution is discrete. This could not happen for discrete order statistics.

From Theorem 2.3, we deduce that any marginal distribution of at least two (different) gOSshas the MTP2-property; see [10, Proposition 3.2] for the general result on associated randomvariables. This feature of gOSs has many interesting implications concerning the dependencestructure of gOSs. As mentioned above, it implies association of gOSs, which means thatall the covariances are nonnegative. It implies inequality (6) giving a lower bound for themultivariate survival function in terms of the univariate survival functions.

Moreover, it yields the following lower bound for expectations of products of gOSs

E

s∏

j=1

�j(X(1)∗ , . . . , X(n)

∗ )

≥

s∏j=1

E�j(X(1)∗ , . . . , X(n)

∗ ),

where �j : Rn → R are nonnegative functions. In particular, we obtain for functions

�j(X(1)∗ , . . . , X

(n)∗ ) = ψj(X(j)∗ ), 1 ≤ j ≤ n, i.e. �j depends on the j th component X

(j)∗ only,

E

s∏

j=1

ψνj(X

(νj )∗ )

≥

s∏j=1

Eψνj(X

(νj )∗ ), 1 ≤ ν1 < · · · < νs ≤ n.

Another interesting conclusion is deduced from a result originally due to Sarkar [10,Theorem 4.1]. It deals with monotonicity of regressions of random variables having theMTP2-property. We state the result in terms of gOSs.

THEOREM 2.4 LetX(1)∗ , . . . , X(n)∗ be gOSs based on a continuous cdf with joint

⊗nj=1 P F -df f.

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Dependence structure of gOSs 413

Then, for any increasing function h: Rν → R, 1 ≤ ν ≤ n − 1, the version of the conditionalexpectation

E(h(X(1)∗ , . . . , X(ν)

∗ )|X(ν+1)∗ = xν+1, . . . , X

(n)∗ = xn)

calculated w.r.t. the conditional density is a nondecreasing function of xν+1, . . . , xn.

This property, called conditional monotone regression endowment by Lehmann [15], isa reversed version of the monotonicity of conditional expectation established in ref. [16,Lemma 3.1]. In particular, we have for 1 ≤ r ≤ n − 1 and 1 ≤ l ≤ n − r that

E(h(X(r)∗ )|X(r+l)

∗ = x)

is an increasing function of x. Therefore, this result provides an extension of the second part ofLemma 2.2 in ref. [17], which establishes this property for m-gOSs only. The continuity of theconditional expectation E(h(X

(r)∗ )|X(r+l)∗ = ·) calculated w.r.t. the conditional density maybe proved similar to that of E(h(X

(r+l)∗ )|X(r)∗ = ·) (see the proof of Lemma 3.1 in ref. [16]),using the representation of the joint df of X

(r)∗ , X(r+l)∗ . Therefore, the conditional expectation

is a nondecreasing and continuous function.This result has some implications regarding characterization problems

E(h(X(r)∗ )|X(r+l)

∗ = ·) = g(·),with some function g and an increasing function h. A cdf F with this property does only existif g is continuous and nondecreasing.

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58, 293–310.[2] Cramer, E., 2002, Contributions to generalized order statistics. Habilitationsschrift, University of Oldenburg.[3] Mathai, A.M., 1993, A Handbook of Generalized Special Functions for Statistical and Physical Sciences

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Order Statistics: Applications (Amsterdam: North-Holland), pp. 315–335.[6] Kamps, U. and Cramer, E., 2001, On distributions of generalized order statistics. Statistics, 35, 269–280.[7] Balakrishnan, N. and Aggarwala, R., 2000, Progressive Censoring (Boston: Birkhäuser).[8] Esary, J.D., Proschan, F. and Walkup, D.W., 1967, Association of random variables with applications. Annals

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[13] Arnold, B.C., Balakrishnan, N. and Nagaraja, H.N., 1998, Records (New York: Wiley).[14] Nagaraja, H.N. and Nevzorov, V.B., 1996, Correlations between functions of records can be negative. Statistics

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