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Volume 18, Issue 4, December 2019, Pages 367 - 374

On Moments Properties of Generalized Order Statistics from Marshall-Olkin-Extended General Class of Distribution

Authors
M. A. Khan1, *, Nayabuddin2
1Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh, India
2Department of Epidemiology, Jazan University, Jazan, Kingdom of Saudia Arabia
*Corresponding author. Email: khanazam2808@gmail.com
Corresponding Author
M. A. Khan
Received 21 February 2018, Accepted 11 August 2019, Available Online 22 November 2019.
DOI
10.2991/jsta.d.191112.004How to use a DOI?
Keywords
Marshall-Olkin extended general class of distribution; Generalized order statistics; Order statistics; Record values and recurrence relations
Abstract

Marshall and Olkin [Biometrika. 84 (1997), 641–652. https://doi.org/10.1093/biomet/84.3.641] introduced a new method of adding parameter to expand a family of distribution. In this paper the Marshall-Olkin extended general class of distribution is used. Further, some recurrence relations for single and product moments of generalized order statistics gos are studied. Also the results are deduced for order statistics and record values.

Copyright
© 2019 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Kamps [2] introduced the unifying concept of generalized order statistics gos, the use of such concept has been steadily growing along the years. This is due to the fact that such concept describes random variables arranged in ascending order of magnitude and includes important well known concept that have been separately treated in statistical literature. Examples of such concepts are the order statistics, sequential order statistics, progressive type II censored order statistics, record values and pfeifer's records. Application is multifarious in a variety of disciplines and particularly in reliability.

Let n≥2 be a given integer and m~=m1,m2,…,mn−1∈ℜn−1, k≥1 be the parameters such that

γi=k+n−i+∑j=in−1mj≥0   for   1≤i≤n−1.

The random variables X1,n,m~,k,X2,n,m~,k,…,Xn,n,m~,k are said to be generalized order statistics from an absolutely continuous distribution function F  with the probability density funtion pdf f , if their joint density function is of the form

k(∏j=1n−1γj)(∏i=1n−1[1−F(xi)]mif(xi))[1−F(xn)]k−1f(xn),
on the cone F−10<x1≤x2≤…≤xn<F−11.

If mi=0, i=1,2,…,n−1 and k=1, we obtain the joint pdf of the order statistics and for mi=−1, k∈N, we get the joint pdfk−th record values.

Let the Marshall-Olkin extended general form of distribution be

F¯x=λ e−hxc1−1−λ e−hxc,   α≤x≤β, λ>0,
where c is such that Fα=0, Fβ=1 and hx is a monotonic and differentiable function of x in the interval α,β.

Also we have,

F¯(x)=ch′(x)[1−(1−λ)e−h(x)c]f(x),
where, F¯x=1−Fx. The relation (3) will be utilized to establish recurrence relations for moments of gos.

2. RELATIONS FOR SINGLE MOMENTS

Case I: γi≠γj; i≠j=1,2,…,n−1.

In view of (1) the pdf of r−th generalized order statistic Xr,n,m~,k is

fX(r,n,m~,k)(x)=Cr−1f(x)∑i=1rai(r)F¯(x)γi−1,
where,
Cr−1=∏i=1rγi,γi=k+n−i+∑j=in−1mj>0,
and
ai(r)=∏j=1j≠ir1(γj−γi),1≤i≤r≤n.

Theorem 2.1.

For the Marshall-Olkin extended general class of distributions as given in (2) and n∈N, m~∈ℜ, k>0,1≤r≤n,λ>0

E[ξ{X(r,n,m~,k)}]=E[ξ{X(r−1,n,m~,k)}]+cγrE[ϕ{X(r,n,m~,k)}]−c(1−λ)γrE[ψ{X(r,n,m~,k)}],
where ϕ(x)=ξ′(x)h′(x) and ψ(x)=ξ′(x)h′(x)e−h(x)c.

Proof:

We have by Athar and Islam [3],

EξXr,n,m~,k−EξXr−1,n,m~,k          =Cr−2∫αβξ′x∑i=1rairF¯xγi dx.

Now on using (3) in (6), we get

E[ξ{X(r,n,m~,k)}]−E[ξ{X(r−1,n,m~,k)}]=cCr−1γr∫αβξ′(x)h′(x)∑i=1rai(r)[F¯(x)]γi−1[1−(1−λ)e−h(x)c]f(x)dx.
which after simplification yields (5).

Case II: mi=m, i=1,2,…,n−1.

The pdf of Xr,n,m,k is given as

fXr,n,m,kx=Cr−1r−1!F¯xγr−1fxgmr−1Fx,
where,
Cr−1=∏i=1rγi , γi=k+n−im+1,
hmx=−1m+11−xm+1,  m≠−1.−log1−x,  m=−1.
and
gmx=hmx−hm0, x∈0,1.

Theorem 2.2.

For the Marshall-Olkin extended general class of distributions as given in (2) and n∈N, m~∈ℜ, k>0,1≤r≤n,λ>0

E[ξ{X(r,n,m,k)}]=E[ξ{X(r−1,n,m,k)}]+cγrE[ϕ{X(r,n,m,k)}]−c(1−λ)γrE[ψ{X(r,n,m,k)}].

Proof:

It may be noted that for γi≠γj but at mi=m, i=1,2,…,n−1,

air=1m+1r−1−1r−i1i−1!r−i!.

Therefore the pdf of X(r,n,m~,k) given in (4) reduces to (7) (cf Khan et al. [4]).

Hence it can be seen that (8) is the partial case of (5) and is obtained by replacing m~ with m in (5).

Remark 2.1.

Recurrence relation for single moments of order statistics (at m=0,k=1) is

EξXr:n= EξXr−1:n+cn−r+1EϕXr:n−1−λ  EψXr:n,
at λ=1, we get
EξXr:n= EξXr−1:n+cn−r+1  EϕXr:n,
as obtained by Ali and Khan [5].

Remark 2.2.

Recurrence relation for single moments of k−th upper record (at m=−1) will be

EξXr,n,−1,k=EξXr−1,n,−1,k+ckEϕXr,n,−1,k  −1−λ EψXr,n,−1,k.

Remark 2.3.

Setting λ=1 in (8), we get

EξXr,n,m,k= EξXr−1,n,m,k+cγr EϕXr,n,m,k,
as obtained by Anwar et al. [6].

EXAMPLES

  1. Marshall-Olkin-Extended Exponential Distribution

    F¯x=λ e−θ x1+1−λ e−θ x,   0<x<∞,  λ >0,
    we have,
    c=1θ and hx=x,
    let ξx=xj+1, then
    ϕx=j+1 xj and ψx=j+1∑p=0∞−1pp!θp xj+p.

    Thus from relation (8), we have

    EXj+1r,n,m,k−EXj+1r−1,n,m,k=j+1θ γr ∑p=08(-1)pp!EXjr,n,m,k  −1−λ∑p=0∞−1pp!θp  EXj+pr,n,m,k.

  2. Marshall-Olkin-Extended Erlang Truncated Exponential Distribution

    F¯x=λ e−α1−e−βx1−1−λ e−α1−e−βx,   0<x<∞, λ>0, α, β>0,
    here we have
    c=1α1−e−β and hx=x,
    assuming ξx=xj+1, we get
    ϕx=j+1xj and ψx=j+1∑p=0∞ −1pp! α1−e−βpxj+p.

    Thus from relation (8),

    EXj+1r,n,m,k=EXj+1r−1,n,m,k+j+1γr α1−e−β∑p=0∞(−1)pp!EXjr,n,m,k  −1−λ∑p=0∞−1pp!α1−e−βp EXj+pr,n,m,k.

  3. Marshall-Olkin-Extended Rayleigh Distribution

    F¯x=λ e−x22θ21+1−λ e−x22θ2,   0<x<∞, λ >0, θ>0.
    we have,
    c=2θ2 and hx=x2,
    let ξx=xj+1, then
    ϕx=j+12xj−1 and ψx=j+1∑p=0∞−1pp! 12p+1 1θ2p x2p+j−1.

    Thus from relation (8), we have

    EXj+1r,n,m,k− EXj+1r−1,n,m,k=θ2j+1γr∑p=0∞(−1)pp!EXj−1r,n,m,k  −1−λ∑p=0∞−1pp!12θ2pEXj+2p−1r,n,m,k.

3. RELATIONS FOR PRODUCT MOMENTS

Case I: γi≠γj; i≠j=1,2,…,n−1.

The joint probability density function pdf of X(r,n,m~,k) and X(s,n,m~,k), 1≤r<s≤n is given as

fX(r,n,m~,k),X(s,n,m~,k)(x,y)=Cs−1(∑i=r+1sai(r)(s)F¯(y)F¯(x)γi)(∑i=1rai(r)F¯(x)γi)×(∑i=1rai(r)[F¯(x)]γi)f(x)F¯(x)f(y)F¯(y),α≤x<y≤β,
where,
airs=∏j=r+1  j≠is1γj−γi,  r+1≤i≤s≤n.

Theorem 3.1.

For the Marshall-Olkin extended general class of distributions as given in (2). Fix a positive integer k and for n∈N, m~∈ℜ,1≤r<s≤n,

E[ξ{X(r,n,m~,k),X(s,n,m~,k)}]=E[ξ{X(r,n,m~,k),X(s−1,n,m~,k)}]+cγsE[ϕ{X(r,n,m~,k),X(s,n,m~,k)}]− c(1−λ)γsE[ψ{X(r,n,m~,k),X(s,n,m~,k)}],
where,
ϕx,y=∂∂y ξx,yh′y,   ψx,y=e−hyc∂∂y ξx,yh′y,   ξx,y=ξ1x.ξ2y.

Proof:

We have by Athar and Islam [3],

E[ξ{X(r,n,m~,k),X(s,n,m~,k)}]−E[ξ{X(r,n,m~,k),X(s−1,n,m~,k)}]=Cs−2∫∫α≤x<y≤β∂∂yξ(x,y)∑i=r+1sai(r)(s)F¯(y)F¯(x)γi∑i=1rai(r)[F¯(x)]γif(x)F¯(x)dydx.

Now in view of (3) and (14), we have

E[ξ{X(r,n,m~,k),X(s,n,m~,k)}]−E[ξ{X(r,n,m~,k),X(s−1,n,m~,k)}]=cγsCs−1∫∫α≤x<y≤β∂∂yξ(x,y)h′(y)(∑i=r+1sai(r)(s)F¯(y)F¯(x)γi)(∑i=1rai(r)[F¯(x)]γi)×{[1−(1−λ)e−h(y)c]}f(x)F¯(x)f(y)F¯(y)dydx,
which leads to (13).

Case II: mi=m; i=1,2,…,n−1.

The joint pdf of Xr,n,m,k and Xs,n,m,k, 1≤r<s≤n is given as

fXr,n,m,k,Xs,n,m,kx,y=Cs−1r−1!s−r−1! F¯xmfx gmr−1Fx  × hmFy−hmFxs−r−1F¯yγs−1fy,α≤x<y≤β.

Theorem 3.2.

For distribution as given in (2) and condition stated as in Theorem 3.1.

EξXr,n,m,k,Xs,n,m,k=EξXr,n,m,k,Xs−1,n,m,k  + cγsEϕXr,n,m,k,Xs,n,m,k  − c 1−λγsEψXr,n,m,k,Xs,n,m,k.

Proof:

We have when γi≠γj but at mi=mj=m,

airs=1m+1s−r−1−1s−i1i−r−1!s−i!,
hence, joint pdf of X(r,n,m~,k) and X(s,n,m~,k) given in (12) reduces to (16). (cf Khan et al. [4]). Therefore, Theorem 3.2 can be established by replacing m~ with m in Theorem 3.1.

Remark 3.1.

Recurrence relation for product moments of order statistics (at m=0, k=1) is

EξXr,s:n= EξXr,s−1:n+cn−s+1 EϕXr,s:n−1−λ EψXr,s:n,
at λ=1, we get
EξXr,s:n=EξXr,s−1:n+cn−s+1  EϕXr,s:n,
as obtained by Ali and Khan [7].

Remark 3.2.

Recurrence relation for product moments of k−th record values will be

EξXr,n,−1,k,Xs,n,−1,k=EξXr,n,−1,k,Xs−1,n,−1,k  +ckEϕXr,n,−1,k,Xs,n,−1,k  −1−λEψXr,n,−1,k,Xs,n,−1,k.

Remark 3.3.

Set λ=1 in (17), we get

EξXr,n,m,k,Xs,n,m,k=EξXr,n,m,k,Xs−1,n,m,k  +cγs EϕXr,n,m,k,Xs,n,m,k,
as obtained by Anwer et al. [6].

EXAMPLES

  1. Marshall-Olkin-Extended Exponential Distribution

    F¯x=λ e−θ x1+1−λ e−θ x,   0<x<∞,  λ >0,
    we have,
    c=1θ and hx=x,
    let ξx,y=xiyj+1, then
    ϕx,y=j+1 xiyj  and  ψx=j+1∑p=0∞−1pp!θp xiyj+p.

    Thus from relation (8), we have

    EXir,n,m,k,Xj+1s,n,m,k−EXir,n,m,k,Xj+1s−1,n,m,k=j+1θ γsEXir,n,m,k,Xjs,n,m,k−1−λ∑p=0∞−1pp!θpEXir,n,m,k,Xj+ps,n,m,k.

  2. Marshall-Olkin-Extended Erlang Truncated Exponential Distribution

    F¯x=λ e−α1−e−βx1−1−λ e−α(1−e−β)x,   0<x<∞,  λ>0, α, β>0.

    Here we have

    c=1α1−e−β and hx=x,
    assuming ξx,y=xiyj+1, we get
    ϕx,y=j+1xiyj  and  ψx,y=j+1∑p=0∞−1pp! α1−e−βpxiyj+p.

    Thus from relation (8),

    EXir,n,m,k,Xjs,n,m,k=EXir,n,m,k,Xjs−1,n,m,k  + j+1γs α1−e−β∑p=0∞−1pp!EXir,n,m,k,Xjs,n,m,k  − 1−λ∑p=0∞−1pp! α1−e−βp EXir,n,m,k,Xj+ps,n,m,k.

  3. Marshall-Olkin-Extended Rayleigh Distribution

    F¯x=λ e−x22θ21+1−λ e−x22θ2,   0<x<∞,  λ >0,  θ>0.

    We have,

    c=2θ2 and hx=x2,
    let ξx,y=xiyj+1, then
    ϕx,y=j+12xiyj−1 and ψx,y=j+1 ∑p=0∞−1pp! 12p+1 1θ2p xiy2p+j−1.

    Thus from relation (8), we have

    EXir,n,m,k,Xj+1s,n,m,k− EXir,n,m,k,Xj+1s−1,n,m,k        =θ2j+1γr ∑p=0∞(−1)pp!EXir,n,m,k,Xj−1s,n,m,k  −1−λ∑p=0∞ −1pp!12θ2pEXir,n,m,k,Xj+2p−1s,n,m,k.

CONFLICT OF INTEREST

The authors declare that they have no competing interests.

AUTHORS' CONTRIBUTIONS

All authors equally contributed in the manuscript. All authors read and approved the final manuscript.

ACKNOWLEDGMENTS

The authors acknowledge with thanks to the referees and the Editor-in-Chief for their fruitful suggestions and comments which led to the overall improvement in the manuscript.

REFERENCES

3.H. Athar and H.M. Islam, Metron, Vol. LXII, 2004, pp. 327-337.
4.A.H. Khan, R.U. Khan, and M. Yaqub, J. Appl. Prob. Statist., Vol. 1, 2006, pp. 115-131.
5.M.A. Ali and A.H. Khan, J. Statist. Assoc., Vol. 35, 1997, pp. 1-9.
6.Z. Anwar, H. Athar, and R.U. Khan, J. Statist. Res., Vol. 40, 2007, pp. 93-102.
7.M.A. Ali and A.H. Khan, Metron, Vol. LVII, 1998, pp. 107-119.
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Journal
Journal of Statistical Theory and Applications
Volume-Issue
18 - 4
Pages
367 - 374
Publication Date
2019/11/22
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
10.2991/jsta.d.191112.004How to use a DOI?
Copyright
© 2019 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

risenwbib
TY  - JOUR
AU  - M. A. Khan
AU  - Nayabuddin
PY  - 2019
DA  - 2019/11/22
TI  - On Moments Properties of Generalized Order Statistics from Marshall-Olkin-Extended General Class of Distribution
JO  - Journal of Statistical Theory and Applications
SP  - 367
EP  - 374
VL  - 18
IS  - 4
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.191112.004
DO  - 10.2991/jsta.d.191112.004
ID  - Khan2019
ER  -