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Volume 18, Issue 2, June 2019, Pages 129 - 135

Generalized Order Statistics from Marshall–Olkin Extended Exponential Distribution

Authors
Haseeb Athar1, *, Nayabuddin2, Saima Zarrin3
1Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah, Kingdom of Saudi Arabia
2Department of Epidemiology, Faculty of Public Health and Tropical Medicine, Jazan University, Kingdom of Saudi Arabia
3Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh, India
*Corresponding author. Email: haseebathar@hotmail.com
Corresponding Author
Haseeb Athar
Received 11 September 2017, Accepted 4 December 2018, Available Online 18 June 2019.
DOI
10.2991/jsta.d.190515.001How to use a DOI?
Keywords
Generalized order statistics; order statistics; record values; marginal moment generating function; joint moment generating function; Marshall–Olkin extended exponential distribution; characterization
Abstract

A.W. Marshall, I. Olkin, Biometrika. 84 (1997), 641–652, introduced an interesting method of adding a new parameter to an existing distribution. The resulting new distribution is called as Marshall–Olkin extended distribution. In this paper some recurrence relations for marginal and joint moment generating function of generalized order statistics from Marshall–Olkin extended exponential distribution are derived, and the characterization results are presented. Further, 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

The Marshall–Olkin extended exponential distribution is considered as a probability model for the lifetime of the product, if the lifetime shows a large variability.

A random variable X is said to have the Marshall–Olkin extended exponential distribution if its pdf is of the form (Marshall and Olkin [1])

fx=λe−x1−(1−λ)e−x2,   x>0,λ>0,
and corresponding survival function
F¯x=λe−x1−(1−λ)e−x,   x>0,λ>0.

Now in view of (1) and (2), we have

F¯x=1−1−λe−xfx.

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 (gos) 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−11−Fximifxi1−Fxnk−1fxn
on the cone F−10<x1≤x2≤…≤xn<F−11.

If mi=m=0; i=1…n−1, k=1, we obtain the joint pdf of the order statistics and for m=−1, k∈N, we get joint pdf of k−th record values. (Kamps [2]).

In view of (4), the marginal pdf of r−th gos Xr,n,m,k is

fXr,n,m,kx = Cr−1r−1! F¯xγr−1 fx gmr−1Fx
and the joint pdf of Xr,n,m,k and Xs,n,m,k, 1≤r<s≤n, is
fXr,n,m,k.Xs,n,m,kx,y=Cs−1r−1! s−r−1!F¯xmgmr−1Fx  ×hmFy−hmFxs−r−1F¯yγs−1fxfy, x<y,
where F¯x=1−Fx
Cr−1=∏i=1rγi
hmx=−1m+1 1−xm+1,  m≠−1log 11−x,  m=−1
and
gmx = hmx−hm0=∫0x1−tmdt,  x∈0,1.

Relations for marginal and joint moment generating functions of record values and gos for some specific distributions are investigated by several authors in literature. For more detailed survey one may refer to Ahsanullah and Raqab [3], Raqab and Ahsanullah [4,5], Saran and Pandey [6], Al-Hussaini et al. [7,8], and references therein. Here in this paper some recurrence relations for marginal and joint moment generating function of generalized order statistics from Marshall–Olkin extended exponential distribution are derived. Further the results are deduced for order statistics and k−th record values.

Let us denote the moment generating function of Xr,n,m,k by MXr,n,m,kt and it j−th derivative by MXr,n,m,kjt

MXr,n,m,kt=Cr−1r−1!∫−∞∞etxF¯xγi−1gmr−1Fxfx dx
and the joint moment generating function of Xr,n,m,k and Xs,n,m,k, 1≤r<s≤n is denoted by MXr,s,n,m,kt1,t2 and its i,j−th partial derivative by MXr,s,n,m,ki,jt1,t2
MXr,s,n,m,kt1,t2=Cs−1r−1!s−r−1! ∫−∞∞∫x∞et1x+t2yF¯xmfx  × gmr−1FxhmFy−hmFxs−r−1F¯yγs−1fydydx.

Using the relation (3), we shall derived the recurrence relation for moment generating functionmgf of generalized order statistics from Marshall–Olkin extended exponential distribution.

2. MARGINAL MOMENT GENERATING FUNCTION

Lemma 2.1.

For the distribution given in (1) and 1≤r≤n, MXr,n,m,kt exists.

Proof:

Since,

gmr−1Fx=1m+1r−1∑i=0r−1−1ir−1iF¯xm+1i.

Therefore, (7) may also be written as

MXr,n,m,kt=Cr−1r−1!m+1r−1∑i=0r−1−1ir−1i∫0∞etxF¯xγr−i−1fxdx.

Now in view of (2) and (3), we have

MXr,n,m,kt=Cr−1r−1!m+1r−1∑i=0r−1−1ir−1iλγr−i  ×∫0∞e−γr−i−tx1−1−λe−xγr−i+1 dx.

Since,

∫0∞e−μx1−βe−xρdx=Bμ,1 2F1ρ,μ;μ+1;β,
where Bm,n is complete beta function and
 2F1a,b;c;z=∑n=0∞an bn zncn n!
xn=xx+1…x+n−1,  n>0.

(Gradshteyn and Ryzhik [9])

Therefore,

MXr,n,m,kt=Cr−1r−1!m+1r−1∑i=0r−1−1ir−1iλγr−i  ×Bγr−i−t,1 2F1γr−i+1,γr−i−t;γr−i−t+1;1−λ.

Hence the lemma.

Lemma 2.2.

For 2≤r≤n,n≥2, k=1,2,…

  1. MXr,n,m,kt−MXr−1,n,m,kt=Cr−1γr r−1!  t∫−∞∞etxF¯xγrgmr−1Fx dx

  2. MXr−1,n,m,kt−MXr−1,n−1,m,kt =−m+1Cr−2nγ1 r−2!  t∫−∞∞etxF¯xγrgmr−1Fx dx

  3. MXr,n,m,kt−MXr−1,n−1,m,kt=Cr−1γ1 r−1!  t∫−∞∞etxF¯xγrgmr−1Fx dx.

Proof:

Relations (9–11) can be seen in view of Athar and Islam [10].

Theorem 2.1.

Fix a positive integer k. For n∈ℕ, m∈ℝ, 2≤r≤n, n≥2 and j=1,2,…,

MX(r,n,m,k)(j+1)(t)=γrγr−tMX(r−1,n,m,k)(j+1)(t)−t(1−λ)γr−tMX(r,n,m,k)(j+1)(t−1)  +j+1γr−t{MX(r,n,m,k)(j)(t)−(1−λ) MX(r,n,m,k)(j)(t−1)},
where MXr,n,m,kjt is the j−th derivative of MXr,n,m,kt.

Proof:

On application of (3) in (9), we get

MXr,n,m,kt−MXr−1,n,m,kt  =tγrCr−1r−1!∫0∞etx1−1−λe−xF¯xγr−1gmr−1Fxfx dx
MXr,n,m,kt−MXr−1,n,m,kt=tγrMXr,n,m,kt−1−λMXr,n,m,kt−1.

Differentiating both the sides of (13) j+1 times w.r.t. t and rearranging the terms, we get the required result.

Remark 2.1.

For m=0,k=1, the recurrence relation for marginal moment generating function of order statistics from Marshall–Olkin extended exponential distribution is

MXr:nj+1t=n−r+1n−r+1−tMXr−1:nj+1t−t1−λn−r+1−tMXr:nj+1t−1  +j+1n−r+1−tMXr:njt−1−λ MXr:njt−1.

At λ=1 in (14), we get the result for standard exponential distribution.

Remark 2.2.

Letting m→−1 in (12), we get the recurrence relation for marginal moment generating function of k−th upper record values as

MXUrkj+1t=kk−tMXUr−1kj+1t−t1−λk−tMXUrkj+1t−1  +j+1k−tMXUrkjt−1−λ MXUrkjt−1.

Theorem 2.2.

For distribution as given in (1) and 2≤r≤n, n≥2, j=1,2,…

  1. MXr,n,m,kj+1t=1−λMXr,n,m,kj+1t−1  −j+1tMXr,n,m,kjt−1−λMXr,n,m,kjt−1  −γ1 γrm+1r−1tMXr−1,n,m,kj+1t−MXr−1,n−1,m,kj+1t.

  2. MXr,n,m,kj+1t=j+1γ1−tMXr,n,m,kjt−1−λMXr,n,m,kjt−1  +γ1γ1−tMXr−1,n−1,m,kj+1t−1−λMXr,n,m,kj+1t−1.

Proof:

Results can be established on the lines of Theorem 2.1 in view of (10) and (11).

3. JOINT MOMENT GENERATING FUNCTION

Lemma 3.1.

For 1≤r<s<n−1, n≥2 and k=1,2,…

MXr,s,n,m,kt1,t2−MXr,s−1,n,m,kt1,t2  =t2γsCs−1r−1!s−r−1!∫−∞∞∫x∞et1x+t2yF¯xmfxgmr−1Fx  ×hmFy−hmFxs−r−1F¯yγsdydx.

Proof:

Relation (15) can be established in view of Athar and Islam [10].

Theorem 3.1.

For the distribution as given in (1) and n∈ℕ, m∈ℝ, 1≤r<s−1≤n, k≥1,

MXr,s,n,m,ki,j+1t1,t2=γsγs−t2MXr,s−1,n,m,ki,j+1t1,t2  −1−λt2γs−t2MXr,s−1,n,m,ki,j+1t1,t2−1  +j+1γs−t2MXr,s,n,m,ki,jt1,t2−1−λMXr,s−1,n,m,ki,jt1,t2−1.

Proof:

In view of (3) and (15), we have

MXr,s,n,m,kt1,t2−MXr,s−1,n,m,kt1,t2  =t2γsCs−1r−1!s−r−1!∫0∞∫x∞et1x+t2yF¯xm fx gmr−1Fx  ×hmFy−hmFxs−r−1F¯yγs−11−1−λe−yfydydx.

After simplification, we get

MXr,s,n,m,kt1,t2−MXr,s−1,n,m,kt1,t2   =t2γsMXr,s,n,m,kt1,t2−1−λMXr,s−1,n,m,kt1,t2−1.

Differentiating (17) i times w.r.t. t1 and j+1 times w.r.t. t2, we get the required result.

Remark 3.1.

Putting m=0 and k=1 in Theorem 3.1, we get the recurrence relations for joint moment generating function of order statistics and at m→−1, we get the result for k−th upper record values.

Remark 3.2.

Theorem 2.1 can be deduced from Theorem 3.1 by letting t1→0.

4. CHARACTERIZATIONS

This section contains characterization results for the given distribution using the recurrence relations for the marginal as well as joint moment generating functions using Müntz-Szász theorem.

Theorem 4.1.

The necessary and sufficient condition for a random variable X to be distributed with pdf given by (1) is that

MXr,n,m,kt−MXr−1,n,m,kt=tγrMXr,n,m,kt−1−λMXr,n,m,kt−1.

Proof:

The necessary part follows immediately from (13). On the other hand if the recurrence relation (18) is satisfied, then

Cr−1r−1!∫−∞∞etxF¯xγr−1gmr−1Fxfx dx
−Cr−2r−2!∫−∞∞etxF¯xγr−1−1gmr−2Fxfx dx
 =tγrCr−1r−1!∫−∞∞etxF¯xγr−1gmr−1Fxfx dx −1−λCr−1r−1!∫−∞∞e(t−1)xF¯xγr−1gmr−1Fxfx dx.

Integrating the first part on the left-hand side of the equation by parts, treating −ddxF¯xγr for integration and the rest of the terms for differentiation, we get after simplification that

tγrCr−1r−1!∫−∞∞etxF¯xγr−1gmr−1Fx−F¯x+fx−1−λe−xfx dx=0.

Using Müntz − Sza'sz theorem (See, Hwang and Lin [11]), we get

fx=λe−x1−1−λe−x2,
which proves that fx has the form (1).

Theorem 4.2.

The necessary and sufficient condition for a random variable X to be distributed with pdf given by (1) is that

MXr,s,n,m,kt1,t2−MXr,s−1,n,m,kt1,t2   =t2γsMXr,s,n,m,kt1,t2−1−λMXr,s−1,n,m,kt1,t2−1.

Proof:

The necessary part follows immediately from (17). On the other hand if the recurrence relation (19) is satisfied, then

Cs−1r−1!s−r−1!∫0∞∫x∞et1x+t2yF¯xmfxgmr−1FxhmFy−hmFxs−r−1
× F¯yγs−1fydydx−Cs−2r−1!s−r−2!∫0∞∫x∞et1x+t2yF¯xmfxgmr−1Fx× F¯yγs−1−1hmFy−hmFxs−r−2fydydx
=t2γsCs−1r−1!s−r−1!∫0∞∫x∞et1x+t2yF¯xmfxgmr−1Fx × hmFy−hmFxs−r−1F¯yγs−1fydydx−1−λt2γsCs−1r−1!s−r−1! × ∫0∞∫x∞et1x+t2yF¯xmfxgmr−1FxhmFy−hmFxs−r−1F¯yγs−1fydydx.

Integrating the first part on the left-hand side of the equation by parts, treating −ddyF¯(y)γs for integration and the rest of the terms for differentiation, we get after simplification that

t2γsCr−1r−1!s−r−1!∫−∞∞∫x∞et1x+t2yF¯xmgmr−1Fx  ×hmFy−hmFxs−r−1F¯yγs−1−F¯y+fy−1−λe−yfy dydx=0.

Now on application of Müntz − Sza'sz theorem (See, Hwang and Lin [11]), we get

fx=λe−x1−1−λe−x2
which proves that fx has the form (1).

ACKNOWLEDGMENT

Authors are thankful to Professor M. Ahsanullah, Editor in Chief, JSTA and learned referee who spent their valuable time to review this manuscript.

REFERENCES

3.M. Ahsanullah and M.Z. Raqab, Stoch. Model. Appl., Vol. 2, No. 2, 1999, pp. 35-48.
4.M.Z. Raqab and M. Ahsanullah, J. Appl. Stat. Sci., Vol. 10, 2000, pp. 27-36.
5.M.Z. Raqab and M. Ahsanullah, Pak. J. Stat., Vol. 19, No. 1, 2003, pp. 1-13.
6.J. Saran and A. Pandey, Metron, Vol. 61, 2003, pp. 27-33. https://ideas.repec.org/a/mtn/ancoec/030104.html
8.E.K. Al-Hussaini, A.A. Ahmad, and M.A. Al-Kashif, J. Stat. Theory Appl., Vol. 6, 2007, pp. 134-155.
10.H. Athar and H.M. Islam, Metron, 2004, pp. 327-337. https://econpapers.repec.org/RePEc:mtn:ancoec:040303
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Journal
Journal of Statistical Theory and Applications
Volume-Issue
18 - 2
Pages
129 - 135
Publication Date
2019/06/18
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
10.2991/jsta.d.190515.001How 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

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TY  - JOUR
AU  - Haseeb Athar
AU  - Nayabuddin
AU  - Saima Zarrin
PY  - 2019
DA  - 2019/06/18
TI  - Generalized Order Statistics from Marshall–Olkin Extended Exponential Distribution
JO  - Journal of Statistical Theory and Applications
SP  - 129
EP  - 135
VL  - 18
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.190515.001
DO  - 10.2991/jsta.d.190515.001
ID  - Athar2019
ER  -