On joint distributions of order statistics for a discrete case

R E S E A R C H

Open Access

On joint distributions of order statistics for a

discrete case

M Güngör

_{, Y Bulut}

_{and B Yüzba¸sı}

*_{Correspondence:}

[email protected]

1_{Department of Econometrics,}

Inonu University, Malatya, 44280, Turkey

Full list of author information is available at the end of the article

Abstract

In this study, the joint distributions of order statistics ofinniddiscrete random variables are expressed. Also, the joint distributions are obtained in the form of an integral. Then, the results related topf anddf are given.

MSC: 62G30; 62E15

Keywords: order statistics; discrete random variable; probability function; distribution function

1 Introduction

The joint probability density function (pdf) and marginalpdf of order statistics of inde-pendent but not necessarily identically distributed (innid) random variables was derived by Vaughan and Venables [] by means of permanents. In addition, Balakrishnan [] and Bapat and Beg [] obtained the jointpdf and distribution function (df) of order statistics ofinnidrandom variables by means of permanents. Balasubramanianet al.[] obtained the distribution of single order statistics in terms of distribution functions of the mini-mum and maximini-mum order statistics of some subsets of{X,X, . . . ,Xn}whereXi’s areinnid random variables. Later, Balasubramanianet al.[] generalized their previous results [] to the case of the joint distribution function of several order statistics. Recurrence rela-tionships among the distribution functions of order statistics arising frominnidrandom variables were obtained by Cao and West []. Using multinomial arguments, thepdf of

Xr:n+(≤r≤n+ ) was obtained by Childs and Balakrishnan [] by adding another inde-pendent random variable to the originalnvariablesX,X, . . . ,Xn. Also, Balasubramanian

et al.[] established the identities satisfied by the distributions of order statistics from non-independent non-identical variables through operator methods based on the differ-ence and differential operators. In , Beg [] obtained several recurrdiffer-ence relations and identities for product moments of order statistics ofinnidrandom variables using perma-nents. Recently, Crameret al.[] derived the expressions for the distribution and density functions by Ryser’s method and the distributions of maxima and minima based on per-manents.

The notion of distribution theory has been applied in various branches of science for in-vestigations. Recently, the notion of a uniform distribution has been applied in sequence spaces and the notion of statistical convergent sequences has been studied and investi-gated from different aspects by Rath and Tripathy [], Tripathy [, ], Tripathy and Baruah [], Tripathy and Dutta [], Tripathy and Sarma [], Tripathy and Sen [], and others.

A multivariate generalization of classical order statistics for random samples from a con-tinuous multivariate distribution was defined by Corley []. Guilbaud [] expressed the probability of the functions of innidrandom vectors as a linear combination of prob-abilities of the functions of independent and identically distributed (iid) random vec-tors and thus also for order statistics of random variables. Expressions for generalized joint densities of order statistics of iid random variables in terms of Radon-Nikodym derivatives with respect to product measures based ondf were derived by Goldie and Maller [].

Several identities and recurrence relations forpdfanddfof order statistics ofiidrandom variables were established by numerous authors including Arnoldet al.[], Balasubrama-nian and Beg [], David [], and Reiss []. Furthermore, Arnoldet al.[], David [], Gan and Bain [], and Khatri [] obtained the probability function (pf) anddf of order statistics ofiidrandom variables from a discrete parent. Balakrishnan [] showed that several relations and identities that have been derived for order statistics from continuous distributions also hold for the discrete case. In , Nagaraja [] explored the behavior of higher order conditional probabilities of order statistics in an attempt to understand the structure of discrete order statistics. Later, Nagaraja [] considered some results on order statistics of a random sample taken from a discrete population.

In general, the distribution theory for order statistics is complex when the parent distri-bution is discrete. In this study, the joint distridistri-butions of order statistics ofinniddiscrete random variables are expressed in the form of an integral. As far as we know, these ap-proaches have not been considered in the framework of order statistics frominniddiscrete random variables.

From now on, the subscripts and superscripts are defined at first usage, and these defi-nitions will be valid unless they are redefined.

If a, a, . . . are column vectors, then [a m

a m

· · ·] will denote the matrix obtained by tak-ingmcopies of a,mcopies of aand so on.perA denotes the permanent of a square matrix A; the permanent is defined just like the determinant, except that all signs in the expansion are positive.

LetX,X, . . . ,Xnbeinniddiscrete random variables andX:n≤X:n≤ · · · ≤Xn:nbe the order statistics obtained by arranging then Xi’s in the increasing order of magnitude. Let

Fiandfibedf andpf ofXi(i= , , . . . ,n), respectively.

The df andpf ofXr:n,Xr:n, . . . ,Xrp:n, ≤r<r<· · ·<rn≤n(p= , , . . . ,n) will be

given. For notational convenience, we write_{z,z,...,zp}, _{mp,kp,...,m,k}, and_V instead ofx

z=

x

z=z

x

z=z· · ·

xp

zp=zp–,

n–rp

mp=

rp––rp–

kp= · · ·

r––r

m=

r––r

k=

r––r

m=

r–

k=,

F ς_()(x) F

ς_()(x–) F

ς_()(x) F

ς_()(x–)

· · ·

F ς_()_p(xp) F

ς_()_p(xp–) and

F ς_()(x) 

F ς_()(x) v

ς_() · · · F

ς_()_p(xp) v

ς_(()_p–) in the expressions below, respectively (xi= , , , . . .) (z= ).

2 Theorems for distribution and probability functions

In this section, the theorems related topf anddf ofXr:n,Xr:n, . . . ,Xrp:nwill be given. We

Theorem 

fr,r,...,rp:n(x,x, . . . ,xp)

=

mp,kp,...,m,k

ns,ns,...,nsp

_p+

w=

rw––kw–mw––rw–

iw–=

F

s(_i_ww_––)(xw–) –Fs(_i_ww_––)(xw–)

×

p

w= kw++mw

iw=

f

s(wiw)

(xw), ()

where x<x<· · ·<xp,

ns,ns,...,nsp denotes the sum over

p

=sfor which sυ∩sϑ=φ

forυ=ϑ,S=p+_= s,S={, , . . . ,n},r= ,rp+=n+ ,m= ,kp+= ,mw–+kw≤

rw–rw–– ,F_s(i_)



(x) = ,F

s_(i_p_+p+)(xp+–) = ,Fi(xw–) =P(Xi<xw) (w= , , . . . ,p+ ),nsis

the cardinality of sand

s= ⎧ ⎪ ⎨ ⎪ ⎩

{s() ,s () , . . . ,s

(k



++m



)

}, ifeven,

{s() ,s () , . . . ,s

(r+ 

––k+ 

–m– 

–r– 

)

}, ifodd.

Proof Consider the event{Xr:n=x,Xr:n=x, . . . ,Xrp:n=xp}.

The above event can be realized in mutually exclusive ways as follows:r–  –k ob-servations are less than x, kw+  +mw (w= , , . . . ,p) observations are equal to xw,

rξ–  –kξ–mξ––rξ–(ξ= , , . . . ,p) observations are in interval (xξ–,xξ) andn–mp–rp observations exceedxp. The probability function of the above event can be written as

fr,r,...,rp:n(x,x, . . . ,xp) =P{Xr:n=x,Xr:n=x, . . . ,Xrp:n=xp}. ()

Identity () can be expressed as

fr,r,...,rp:n(x,x, . . . ,xp)

=

mp,kp,...,m,k

CperF(x–) r––k

f(x) k++m

F(x–) – F(x) r––k–m–r

f(x) k++m

· · · f(xp) kp++mp

 – F(xp) n–mp–rp

=

mp,kp,...,m,k

C

ns,ns,...,nsp

perF(x–) r––k

[s/·)

×per f(x) k++m

[s/·)per

F(x–) – F(x) r––k–m–r

[s/·)

×per f(x) k++m

[s/·)· · ·per

f(xp) kp++mp

[sp/·)per

 – F(xp) n–mp–rp

[sp+/·),

where C = (p+_w=[(rw–  –kw–mw––rw–)!]–) p

w=[(kw+  +mw)!]–, F(xw) = (F(xw),

F(xw), . . . ,Fn(xw))and f(xw) = (f(xw),f(xw), . . . ,fn(xw))are column vectors. A[s/·) is the matrix obtained from A by taking rows whose indices are ins. Using the expansion of the

permanent in the above identity, we get ().

Theorem 

fr,r,...,rp:n(x,x, . . . ,xp) =

nς,nς,...,nςp

p+

w=

rw–rw––

iw=

v(w)

ς_(iw_w–) –v (w–) ς_(iw_w–)

p

w=

dv(w)

ς_w(), ()

where x<x<· · ·<xp,

nς,nς,...,nςp denotes the sum over

p

=ςfor whichςυ∩ςϑ=φ

forυ=ϑ,S=p+_= ς,v(t) ς_(iw_w–) = [v

(t) ς_()_t –Fς()t

(xt–)] f

ς_(iw_w–)(xt) f

ς_()_t(xt) +F_ς(iw)

w–

(xt–),v() ς_(i)= ,v

(p+)

ς_(ip_p++) = 

and

ς= ⎧ ⎨ ⎩

{ς()}, ifeven,

{ς(),ς(), . . . ,ς

(r+  –r– –)

}, ifodd.

Proof Consider the identity

nsw–=ϑ–kw––mw

_{ϑ–kw––mw}

j=

G

s(_j_w)_–

kw++mw

i=

f_s(i) w

= kw!mw! (kw+  +mw)!

nτw–=ϑ–kw––mw

nτw–=kw

nτw–=

_ϑ–k

w––mw

i=

G_τ(i) w–

kw

i=

f_τ(i) w– fτ

() w–

mw

i=

f_τ(i) w

()

and using () in (), it can be written as

fr,r,...,rp:n(x,x, . . . ,xp)

=

mp,kp,...,m,k

_p

w=

kw!mw! (kw+  +mw)!

nτ,nτ,...,nτp

_p+

w= _mw–

i(w–)=

f

τ_((i(_w–)w–))(xw–)

×

_{rw––kw–mw––rw–}

iw–=

F

τ_(i_ww_––)(xw–) –Fτ_(i_ww_––)(xw–)

×

kw

iw–=

f

τ_(i_w_–w–)(xw) _p

w=

f_τ() w–(xw),

where _nτ

,nτ,...,nτp denotes the sum over

p

l=τlfor which τυ∩τϑ =φ forυ=ϑ, S= p+

l= τl,

sw=τw–∪τw–∪τw, sw–=τw– and

τl= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

{τ_l(),τ_l(), . . . ,τ

(ml



)

l }, ifl≡ (mod),

{τ_l(),τ_l(), . . . ,τ

(rl+ 

––ml– 

–kl+ 

–rl– 

)

l }, ifl≡ (mod),

{τ_l(),τ_l(), . . . ,τ

(kl+ 

)

l }, ifl≡ (mod),

The above identity can be written as

fr,r,...,rp:n(x,x, . . . ,xp)

=

mp,kp,...,m,k

_p

w=

kw!mw! (kw+  +mw)!

×

nτ_,nτ_,...,nτ_{p p}

w=

(kw+  +mw)!

kw!mw!

× 

 



· · · 



yk

 ( –y)myk( –y)m· · ·y kp

p ( –yp)mpdydy· · ·dyp

×

_p+

w= _m

w–

i(w–)=

f

τ_((i(_ww_–)–))(xw–)

×

_{rw––kw–mw––rw–}

iw–=

F

τ_(i_ww_––)(xw–) –Fτ_(i_ww_––)(xw–)

×

kw

iw–=

f

τ_(i_w_–w–)(xw) _p

w=

f_τ() w–(xw).

The following expression can be written from the above identity

fr,r,...,rp:n(x,x, . . . ,xp)

=

mp,kp,...,m,k

nτ,nτ,...,nτp



 



· · ·



 _p+

w= _mw–

i(w–)=

( –yw–)f

τ_((i(_ww_–)–))(xw–)

×

_{rw––kw–mw––rw–}

iw–=

F

τ_(i_ww_––)(xw–) –Fτ_(i_ww_––)(xw–)

×

kw

iw–=

ywf_τ(i_w–)

w–

(xw) _p

w=

f_τ()

w–(xw)dyw

and here, ifv(w)

τ_l(ij) =ywf

τ_l(ij)(xw) +Fτ_l(ij)(xw–), the following identity is obtained

fr,r,...,rp:n(x,x, . . . ,xp)

=

mp,kp,...,m,k

nτ,nτ,...,nτp

F τ_()(x) F

τ_()(x–) F

τ_()(x) F

τ_()(x–)

· · ·

F τ_()_p–(xp) F

τ_()_p–(xp–) _p+

w= _mw–

i(w–)=

F

τ_((i(_ww_–)–))(xw–)

–v(w–)

τ_((i(_ww_–)–))

rw––kw–mw––rw–

iw–=

F

τ_(i_ww_––)(xw–) –Fτ_(i_ww_––)(xw–)

×

kw

iw–=

v(w)

τ_(i_ww_––)–Fτ_(i_ww_––)(xw–) p

w=

dv(w)

By considering

l

ξ= l

ζ=

nτ_(w–)=ζ nτw–=l–ξ–ζ

_ζ

i(w–)=

g

τ(i(w–))

(w–)

l–ξ–ζ

iw–=

f_τ(iw–) w–

ξ

iw–=

h_τ(iw–) w–

= l

iw= [g_ς(iw)

w–+fς (iw) w–+hς

(iw)

w–], ()

whereξ+ζ ≤l, and using () for eachmw–andkwin (), we get

fr,r,...,rp:n(x,x, . . . ,xp)

=

nς,nς,...,nςp

p+

w=

rw–rw––

iw=

F_ς(iw)

w–(xw–) –v

(w–)

ς_(iw_w–) +Fς(iww–)(xw–)

–F_ς(iw)

w–(xw–) +v

(w)

ς_(iw_w–) –Fς(iww–)(xw–)

p

w=

dv(w)

ς_w(),

whereςw–=τ(w–)∪τw–∪τw–andςw=τw–. This completes the proof of the

the-orem.

We have the following special cases obtained from (). Consider by takingp= ,n=

,r= ,r=  andv() ς_() = [v

()

ς_()–Fς()(x–)]

f ς()_ (x) f

ς()_ (x) +F_ς()

 (x–), the following identity is

obtained

f,:(x,x) =

nς,nς

F ς_()(x) F

ς_()(x–)

F

ς_()(x) F

ς_()(x–)

 –v()

ς_()

dv()

ς_()

dv()

ς_()

=

nς,nς

f_ς()  (x)

f_ς()  (x) +



fς_()(x)Fς_()(x–)

–

fς_()(x)Fς_()(x) –fς_()(x)Fς_()(x–)

=f(x)

f(x) + 

f(x)F(x–) – 

f(x)F(x) –f(x)F(x–)

+f(x)

f(x) + 

f(x)F(x–) – 

f(x)F(x) –f(x)F(x–)

+f(x)

f(x) + 

f(x)F(x–) – 

f(x)F(x) –f(x)F(x–)

+f(x)

f(x) + 

f(x)F(x–) – 

f(x)F(x) –f(x)F(x–)

+f(x)

f(x) + 

f(x)F(x–) – 

f(x)F(x) –f(x)F(x–)

+f(x)

f(x) + 

f(x)F(x–) – 

f(x)F(x) –f(x)F(x–)

Moreover, the above identity in theiidcase can be expressed by

f,:(x,x) = f(x)f(x) – f(x)f(x)F(x) + f(x)f(x).

This result is obtained ifi= ,j= , andn=  in equation () in Khatri []. If x=x=· · ·=xp=x, it should be written as · · ·

instead of in (), where

· · · is to be carried out over the region: F_ς()

 (x–)≤v

() ς_() ≤v

()

ς_() ≤ · · · ≤v (p) ς_p() ≤

F_ς()

p(xp–),Fς ()

 (x–)≤v

()

ς_() ≤Fς()(x),Fς ()

 (x–)≤v

()

ς_() ≤Fς()(x), . . . ,Fς ()

p(xp–)≤v

(p) ς_p() ≤

F_ς() p(xp).

Further on consideringp= ,n= ,r= ,r= , andv() ς_() = [v

()

ς_()–Fς_()(x–)] f

ς_()(x) f

ς_()(x) +

F_ς() 

(x–) in (), ifX,X,Xareinniddiscrete random variables and forx=x=x, then

f,:(x,x)

=

nς,nς

F ς_()(x) F

ς_()(x–)

F

ς_()(x) v

ς_()

 –v()

ς_()

dv()

ς_()

dv()

ς_()

=

nς,nς

F_ς()  (x)fς

()  (x) –

 

F_ς()  (x) +Fς

()  (x–)

f_ς()  (x) –

 F



ς_()(x)fς_()(x)

f_ς()  (x)

f_ς() 

(x)

+  

F

ς_()(x) –F  ς_()(x–)

f_ς() 

(x)

f_ς()  (x)

+F_ς()  (x)Fς

()  (x–)

f_ς()  (x)fς

() 

(x)

f_ς()  (x)

–  

F_ς()  (x) +Fς

()  (x–)

f_ς()  (x)Fς

()  (x–)

f_ς()  (x)

f_ς()  (x)

–F_ς()  (x)Fς

() 

(x–)f_ς()  (x)

+ 

F_ς() 

(x) +F_ς() 

(x–)f_ς() 

(x)F_ς() 

(x–)

=

F(x)f(x) –  

F(x) +F(x–)

f(x) –  F

 (x)f(x)

f(x)

f(x) +



F_(x) –F_(x–)f(x)

f(x)

+F(x)F(x–)

f(x)f(x)

f(x) –



F(x) +F(x–)

f(x)F(x–)

f(x)

f(x)

–F(x)F(x–)f(x)

+ 

F(x) +F(x–)

f(x)F(x–)

+

F(x)f(x) –  

F(x) +F(x–)

f(x) –  F

 (x)f(x)

f(x)

f(x)

+  

F_(x) –F_(x–)f(x)

f(x)

+F(x)F(x–)

f(x)f(x)

f(x)

–  

F(x) +F(x–)

f(x)F(x–)

f(x)

f(x)

–F(x)F(x–)f(x)

+ 

F(x) +F(x–)

f(x)F(x–)

+

F(x)f(x) –  

F(x) +F(x–)

f(x) –  F

 (x)f(x)

f(x)

+  

F_(x) –F_(x–)f(x)

f(x)

+F(x)F(x–)

f(x)f(x)

f(x)

–  

F(x) +F(x–)

f(x)F(x–)

f(x)

f(x)

–F(x)F(x–)f(x)

+ 

F(x) +F(x–)

f(x)F(x–)

+

F(x)f(x) –  

F(x) +F(x–)

f(x) –  F

 (x)f(x)

f(x)

f(x)

+  

F_(x) –F_(x–)f(x)

f(x)

+F(x)F(x–)

f(x)f(x)

f(x)

–  

F(x) +F(x–)

f(x)F(x–)

f(x)

f(x)

–F(x)F(x–)f(x)

+ 

F(x) +F(x–)

f(x)F(x–)

+

F(x)f(x) –  

F(x) +F(x–)

f(x) –  F

 (x)f(x)

f(x)

f(x)

+  

F_(x) –F_(x–)f(x)

f(x)

+F(x)F(x–)

f(x)f(x)

f(x)

–  

F(x) +F(x–)

f(x)F(x–)

f(x)

f(x)

–F(x)F(x–)f(x)

+ 

F(x) +F(x–)

f(x)F(x–)

+

F(x)f(x) –  

F(x) +F(x–)

f(x) –  F

 (x)f(x)

f(x)

f(x)

+  

F_(x) –F_(x–)f(x)

f(x)

+F(x)F(x–)

f(x)f(x)

f(x)

–  

F(x) +F(x–)

f(x)F(x–)

f(x)

f(x)

–F(x)F(x–)f(x)

+ 

F(x) +F(x–)

f(x)F(x–)

.

Moreover, the above identity in theiidcase can be expressed by

= F(x)f(x) – F(x) +F(x–)f(x) – F(x)f(x) +F(x) –F(x–)+ F(x)F(x–)f(x)

– F(x) +F(x–)F(x–)f(x) – F(x)F(x–)f(x) + F(x) +F(x–)f(x)F(x–)

= F(x)f(x) – F(x)f(x) – F(x–)f(x) – F(x)f(x) +F(x) –F(x–)

= f(x) – F(x)f(x) +f(x)F(x) – F(x)f(x) +f(x)

=f(x) + f(x) –F(x).

This result is obtained ifr= ,s= , andn=  in equation (..) in David []. Furthermore, ifx≤x≤ · · · ≤xp, it should be written · · ·

instead ofin (), where

· · · is to be carried out over the region:v()

ς_() ≤v ()

ς_() ≤ · · · ≤v (p)

ς_p(),Fς()(x–)≤v

() ς_() ≤

F_ς()  (x),Fς

()

 (x–)≤v

()

ς_()≤Fς()(x), . . . ,Fς ()

p(xp–)≤v

(p)

We now express the following theorem to obtain the jointdf of order statistics ofinnid

discrete random variables.

Theorem 

Fr,r,...,rp:n(x,x, . . . ,xp)

=

z,z,...,zp

mp,kp,...,m,k

ns_,ns_,...,ns_p

_p+

w=

rw––kw–mw––rw–

iw–=

F

s(_i_ww_––)(zw–) –Fs(_i_ww_––)(zw–)

×

p

w= kw++mw

iw=

f

s(_wiw)(zw). ()

Proof We have

Fr,r,...,rp:n(x,x, . . . ,xp) =

z,z,...,zp

fr,r,...,rp:n(z,z, . . . ,zp) ()

and using () in (), () is obtained.

The identity () can also be written in the form of an integral as follows.

Theorem 

Fr,r,...,rp:n(x,x, . . . ,xp) =

nς,nς,...,nςp

V p+

w= rw–rw––

iw=

v(w)

ς_(iw_w–) –v (w–) ς_(iw_w–)

p

w=

dv(w)

ς_w(). ()

Proof Using () in (), () is obtained.

3 Results for distribution and probability functions

In this section, the results related topf anddf ofXr:n,Xr:n, . . . ,Xrp:nwill be determined. We express the following result forpf of therth order statistic ofinniddiscrete random variables.

Result 

fr:n(x) =

n–r

m=

r–

k=

ns_,ns_

_r

––k

i=

F

s(_i)(x–)

k++m

i=

f

s(_i)(x)

n–m–r

i=

 –F

s(_i)(x)

=

nς,nς

F ς_()(x) F

ς_()(x–) _r

–

i=

v()

ς_(i) n–r

i=

 –v()

ς_(i)

dv()

ς_(). ()

Proof In () and (), ifp= , () is obtained.

In Result  and Result , thepf’s of minimum and maximum order statistics ofinnid

Result 

f:n(x) = n–

m=

ns

_+m

i=

f

s(i) 

(x) n–m–

i=

 –F

s(i) 

(x)

=

nς_ F

ς_()(x) F

ς_()(x–) _n–

i=

 –v()

ς_(i)

dv()

ς_(). ()

Proof In (), ifr= , () is obtained.

Result 

fn:n(x) = n–

k=

ns,ns

_{n––k}

i=

F

s(i) 

(x–) k+

i=

f

s(i) 

(x)

=

nς,nς

F ς_()(x) F

ς_()(x–) _n–

i=

v()

ς_(i) dv ()

ς_(). ()

Proof In (), ifr=n, () is obtained.

In the following result, we determine the jointpf ofX:n,X:n, . . . ,Xp:n.

Result  If x≤x≤ · · · ≤xp,

f,,...,p:n(x,x, . . . ,xp) = n–p

mp=

ns_,ns_,...,ns_p

n–mp–p

ip+=

 –F

s(_i_p+p+)(xp)

p

w= +mw

iw= f

s(_i_ww)(xw)

=

nς,nς,...,nςp

· · ·

n–p

ip+=

 –v(p)

ς_(ip_p++)

p

w=

dv(w)

ς_w(), ()

where · · · is to be carried out over the region:v()

ς_() ≤v ()

ς_() ≤ · · · ≤v (p)

ς_p(),Fς_()(x–)≤

v()

ς_()≤Fς_()(x),Fς_()(x–)≤v ()

ς_()≤Fς_()(x), . . . ,Fς_p()(xp–)≤v (p)

ς_p()≤Fς_p()(xp).

Proof In () and (), ifr= ,r= , . . . ,rp=pand · · ·

instead of, () is obtained.

Specially, in (), by takingp= ,n=  andv()

ς_() = [v () ς_() –Fς()

(x–)] f

ς_()(x) f

ς_()(x) +F_ς()



(x–),

the following identity is obtained

f,:(x,x) =

ns,ns

 –F

s()_ (x)

f

s()_ (x)fs()_ (x) +

ns

f

s()_ (x)fs()_(x)fs()_ (x) = –F(x)

f(x)f(x) +

 –F(x)

f(x)f(x) +

 –F(x)

f(x)f(x)

+ –F(x)

f(x)f(x) +

 –F(x)

f(x)f(x)

+ –F(x)

f(x)f(x) +f(x)f(x)f(x)

Moreover, the above identity in theiidcase can be expressed as

f,:(x,x) = f(x)f(x) – f(x)f(x)F(x) + f(x)f(x).

We now establish three results for thedfof single order statistic ofinniddiscrete random variables.

Result 

Fr:n(x) =

x

z=

n–r

m=

r–

k=

ns,ns

_{r––k}

i=

F

s(i) 

(z–)

k++m 

i=

f

s(i) 

(z)

n–m–r

i=

 –F

s(i) 

(z)

=

nς,nς

F ς_()(x) 

_r–

i=

v()

ς_(i) n–r

i=

 –v()

ς_(i)

dv()

ς_(). ()

Proof In () and (), ifp= , () is obtained.

Result 

F:n(x) = x

z=

n–

m=

ns

_+m

i=

f

s(i)

(z) n–m–

i=

 –F

s(i)

(z)

=

nς

F ς_()(x) 

_n–

i=

 –v()

ς_(i)

dv()

ς_(). ()

Proof In (), ifr= , () is obtained.

Result 

Fn:n(x) = x

z=

n–

k=

ns_,ns_

_n––k



i=

F

s(_i)(z–) k+

i=

f

s(_i)(z)

=

nς,nς

F ς_()(x) 

_n–

i=

v()

ς_(i) dv ()

ς_(). ()

Proof In (), ifr=n, () is obtained.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

MG, YB and BY have contributed to all parts of the article. All authors read and approved the final manuscript.

Author details

1_{Department of Econometrics, Inonu University, Malatya, 44280, Turkey.}2_{Department of Mathematics, Bingol University,}

Bingol, 12000, Turkey.

References

1. Vaughan, RJ, Venables, WN: Permanent expressions for order statistics densities. J. R. Stat. Soc. B34, 308-310 (1972) 2. Balakrishnan, N: Permanents order statistics, outliers and robustness. Rev. Mat. Complut.20, 7-107 (2007) 3. Bapat, RB, Beg, MI: Order statistics for nonidentically distributed variables and permanents. Sankhya, Ser. A51, 79-93

(1989)

4. Balasubramanian, K, Beg, MI, Bapat, RB: On families of distributions closed under extrema. Sankhya, Ser. A53, 375-388 (1991)

5. Balasubramanian, K, Beg, MI, Bapat, RB: An identity for the joint distribution of order statistics and its applications. J. Stat. Plan. Inference55, 13-21 (1996)

6. Cao, G, West, M: Computing distributions of order statistics. Commun. Stat., Theory Methods26, 755-764 (1997) 7. Childs, A, Balakrishnan, N: Relations for order statistics from non-identical logistic random variables and assessment

of the effect of multiple outliers on bias of linear estimators. J. Stat. Plan. Inference136, 2227-2253 (2006) 8. Balasubramanian, K, Balakrishnan, N, Malik, HJ: Identities for order statistics from non-independent non-identical

variables. Sankhya, Ser. B56, 67-75 (1994)

9. Beg, MI: Recurrence relations and identities for product moments of order statistics corresponding to nonidentically distributed variables. Sankhya, Ser. A53, 365-374 (1991)

10. Cramer, E, Herle, K, Balakrishnan, N: Permanent expansions and distributions of order statistics in the INID case. Commun. Stat., Theory Methods38, 2078-2088 (2009)

11. Rath, D, Tripathy, BC: Matrix maps on sequence spaces associated with sets of integers. Indian J. Pure Appl. Math.

27(2), 197-206 (1996)

12. Tripathy, BC: Matrix transformations between some classes of sequences. J. Math. Anal. Appl.206, 448-450 (2001) 13. Tripathy, BC: On generalized difference paranormed statistically convergent sequences. Indian J. Pure Appl. Math.

35(5), 655-663 (2004)

14. Tripathy, BC, Baruah, A: Lacunary statistical convergent and lacunary strongly convergent generalized difference sequences of fuzzy real numbers. Kyungpook Math. J.50(4), 565-574 (2010)

15. Tripathy, BC, Dutta, H: On some lacunary difference sequence spaces defined by a sequence of Orlicz functions and q-lacunaryn

m-statistical convergence. An. Stiint. Univ. Ovidius, Ser. Mat.20(1), 417-430 (2012)

16. Tripathy, BC, Sarma, B: Statistically convergent difference double sequence spaces. Acta Math. Sin.24(5), 737-742 (2008)

17. Tripathy, BC, Sen, M: On generalized statistically convergent sequences. Indian J. Pure Appl. Math.32(11), 1689-1964 (2001)

18. Corley, HW: Multivariate order statistics. Commun. Stat., Theory Methods13, 1299-1304 (1984)

19. Guilbaud, O: Functions of non-i.i.d. random vectors expressed as functions of i.i.d. random vectors. Scand. J. Stat.9, 229-233 (1982)

20. Goldie, CM, Maller, RA: Generalized densities of order statistics. Stat. Neerl.53, 222-246 (1999) 21. Arnold, BC, Balakrishnan, N, Nagaraja, HN: A First Course in Order Statistics. Wiley, New York (1992) 22. Balasubramanian, K, Beg, MI: On special linear identities for order statistics. Statistics37, 335-339 (2003) 23. David, HA: Order Statistics. Wiley, New York (1970)

24. Reiss, R-D: Approximate Distributions of Order Statistics. Springer, New York (1989)

25. Gan, G, Bain, LJ: Distribution of order statistics for discrete parents with applications to censored sampling. J. Stat. Plan. Inference44, 37-46 (1995)

26. Khatri, CG: Distributions of order statistics for discrete case. Ann. Inst. Stat. Math.14, 167-171 (1962)

27. Balakrishnan, N: Order statistics from discrete distributions. Commun. Stat., Theory Methods15, 657-675 (1986) 28. Nagaraja, HN: Structure of discrete order statistics. J. Stat. Plan. Inference13, 165-177 (1986)

29. Nagaraja, HN: Order statistics from discrete distributions. Statistics23, 189-216 (1992)

doi:10.1186/1029-242X-2012-264