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Volume 18, Issue 4, December 2019, Pages 402 - 415

Relations for Moments of Dual Generalized Order Statistics from Exponentiated Rayleigh Distribution and Associated Inference

Authors
M. A. R. Khan, R. U. Khan*, B. Singh
Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh-202 002, India
*Corresponding author. Email: aruke@rediffmail.com
Corresponding Author
R. U. Khan
Received 21 January 2018, Accepted 25 September 2019, Available Online 14 November 2019.
DOI
10.2991/jsta.d.191104.001How to use a DOI?
Keywords
Dual generalized order statistics; Order statistics; Lower records; Single moments; Product moments; Recurrence relations; Exponentiated Rayleigh distribution and characterization
Abstract

In this paper we obtain exact expressions and some recurrence relations satisfied by single and product moments of dual generalized order statistics from exponentiated Rayleigh distribution. These relations are deduced for moments of order statistics and lower record values. Further, conditional expectation, recurrence relations for single moments and truncated moment are used to characterize this distribution.

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

A random variable X is said to have exponentiated Rayleigh distribution [1] if its probability density function pdf is of the form

fx=2αβ x1−e−β  x2α−1e−β x,  x>0,  α,  β>0
with distribution function df
Fx=1−e−β x2α,  x>0,  α,  β>0.

The exponentiated Rayleigh distribution has many characteristics which are quite common to gamma, Weibull and exponentiated exponential distributions. The exponentiated Rayleigh distribution for the distribution function and the density function are found to have closed forms. Consequently, it can be applied very compatibly even on censored data.

The concept of lower generalized order statistics lg os was first introduced by Pawlas and Syznal [2] to enable a common approach to descending ordered random variables like reverse order statistics and lower record values. Further, the concept of lower (dual) generalized order statistics dgos was extensively studied by Burkschat et al. [3].

Let X*r, n, m, k, r=1, 2, …, n, be the r−th dgos and their joint pdf is of the from

k∏j=1n−1γj∏i=1n−1FximifxiFxnk−1fxn
for F−11>x1≥x2≥…≥xn>F−10.

For the case mi=m, i=1, 2, …, n−1, the pdf of r−th dgos X*r,n,m,k is given

fX*r,n,m,kx=Cr−1r−1!Fxγr−1fxgm r−1Fx
and the joint pdf of X*r,n,m,k and X*s,n,m,k, is
fX*r,n,m,k, X*s,n,m,kx,y=Cs−1r−1!s−r−1!Fxmfxgm r−1Fx  × hmFy−hmFxs−r−1fyFyγs−1, x>y,
where
hmx=− 1m+1xm+1,m≠−1− logx,m=−1
and
gmx=hmx−hm1,x∈0, 1.

Several authors utilized the concept of dgos in their work. References may be made to Pawlas and Szynal [2], Khan et al. [4], Ahsanullah [5,6], Mbah and Ahsanullah [7], Khan et al. [8], Khan and Kumar [9,10] and Khan and Khan [11] among others. In this paper, we mainly focus on the study of dgos arising from the exponentiated Rayleigh distribution.

2. RELATIONS FOR SINGLE MOMENTS

Note that for exponentiated Rayleigh distribution fx and Fx satisfy the relation

2αβ Fx=x−1eβ x2−1fx.

The relation in (6) will be used to derive some simple recurrence relations for the moments of dgos from the exponentiated Rayleigh distribution.

We shall first establish the exact expression for EX∗jr,n,m,k. Using (4), we have, when m≠−1

EX∗jr,n,m,k=Cr−1r−1!∫0∞x jFxγr−1fx gm r−1Fxdx =Cr−1r−1!Ijγr−1, r−1,
where
Ija,b=∫0∞x jFxafx gm bFx dx.

On expanding gm bFx=1m+11−Fxm+1b binomially in (8), we get

Ija,b=1m+1b∑u=0b−1ubu∫0∞x jFxa+um+1fxdx.

Making the substitution t=Fx1/α in (9), we find that

Ija,b=αβ j/2m+1b∑u=0b−1ubu∫01−ln1−t j/2t  αa+um+1+1−1dt.

On using the logarithmic expansion

−ln1−tj=∑p=1∞tppj=∑p=0∞zpjt j+p,  |t|<1,
where zpj is the coefficient of t j+p in the expansion of ∑p=1∞tppj (see Balakrishnan and Cohen [12], p. 44), we get
Ija,b=αβ j/2m+1b∑p=0∞∑u=0b−1ubu zpj/2 ∫01t αa+um+1+1+j/2+p−1dt =1β j/2m+1b∑p=0∞∑u=0b−1ubuzpj/2a+um+1+1+j/2+p/α.

When m=−1, we have

Ija,b=00, as ∑u=0b−1ubu=0.

Since (12) is of the form 00 at m≠1, therefore, we have

Ija,b=A∑u=0b−1ubuαa+um+1+1+j/2+p−1m+1b,
where
A=1β j/2∑p=0∞zpj/2.

Differentiating numerator and denominator of (13) b times with respect to m, we get

Ija,b=A∑u=0b−1u+bbuuba+um+1+1+j/2+p/αb+1,  b>0.

On applying L’ Hospital rule, we have

limm→−1Ija,b=A∑u=0b−1u+bbuuba+1+j/2+p/αb+1.

But for all integers n≥0 and for all real numbers x, we have Ruiz [13]

∑i=0n−1inix−in=n!.

Therefore,

∑u=0b−1u+bbuub=b!.

On substituting (16) in (14), we find that

Ija,b=b !β j/2∑p=0∞zpj/2a+1+j/2+p/αb+1,  m=−1.

Now substituting for Ijγr−1,r−1 from (12) in (7) and simplifying, we obtain when m≠−1

EX∗jr,n,m,k= Cr−1r−1! β j/2m+1r−1∑p=0∞∑u=0r−1−1ur−1u  ×  zpj/2γr−u+j/2+p/α
and when m=−1, in view of (17) and (7), we have
EX∗jr,n,−1,k=EZr kj=krβ j/2∑p=0∞zpj/2k+j/2+p/αr,
where Zr (k) denote the k−th lower record value.

Identity 2.1.

For γr≥1, k≥1, 1≤r≤n and m≠−1

∑u=0r−1−1ur−1u1γr−u=r−1!m+1r−1∏t=1rγt.

Proof.

At j=0 in (18), we have

1=Cr−1r−1!m+1r−1∑p=0∞∑u=0r−1−1ur−1uzp0γr−u+p/α.

Note that, if j=0, then

zp0=1, p=0 and zp0=0, p>0 (see Shawky and Bakoban [14]) and hence the result given in (20).

2.1. Special Cases

  1. Putting m=0, k=1 in (18), the explicit formula for single moments of order statistics of the exponentiated Rayleigh distribution can be obtained as

    EXn−r+1:n j=Cr:nβ j/2∑p=0∞∑u=0r−1−1ur−1uzpj/2n−r+1+u+j/2+p/α.

    That is

    EXr:n j=Cr:nβ j/2∑p=0∞∑u=0n−r−1un−ruzpj/2r+u+j/2+p/α,
    where
    Cr:n=n!r−1!n−r!.

  2. Putting k=1 in (19), we deduce the explicit expression for the moments of lower record values from the exponentiated Rayleigh distribution as

    EXL(r) j=1β j/2∑p=0∞zpj/21+j/2+p/αr.

    Now we obtain the recurrence relations for single moments of exponentiated Rayleigh distribution in the following theorem.

Theorem 2.1.

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

EX∗jr,n,m,k=EX∗jr−1,n,m,k  + j2 α β γrEX∗j−2r,n,m,k−EφX∗r,n,m,k
where
φx=x j−2 e β x2.

Proof.

In view of Khan et al. [15], note that

EX∗jr,n,m,k−EX∗jr−1,n,m,k       =−j Cr−1γrr−1!∫0∞x j−1Fxγrgm r−1Fx dx.

On using (6) in (22), we get

EX∗jr,n,m,k−EX∗jr−1,n,m,k=j2αβγrCr−1r−1!∫0∞x j−2Fxγr−1fx gm r−1Fx dx  − Cr−1r−1!∫0∞x j−2eβ x2Fxγr−1fx gm r−1Fx dx
and hence the result given in (21).

Remark 2.1.

Putting m=0, k=1, in (21), we obtain a recurrence relation for single moments of order statistics of the exponentiated Rayleigh distribution in the form

EXn−r+1 : n j=EXn−r+2 : n j+j2αβ n−r+1EXn−r+1 : n j−2−Eφ Xn−r+1:n.

Replacing n−r+1 by r−1, we have

EXr:n j=EXr−1:n j+j2αβ r−1EXr : n j−2−EφXr:n.

Remark 2.2.

Setting m=−1 and k≥1 in (21), we get a recurrence relation for single moments of lower k record values from exponentiated Rayleigh distribution in the form

EZr kj=EZr−1 kj+j2αβ kEZr kj−2−EφZr k.

3. RELATIONS FOR PRODUCT MOMENTS

The explicit expressions for the product moments of dgos X∗ ir,n,m,k and X∗js,n,m,k, 1≤r<s≤n, can be obtained when m≠−1 as

EX∗ ir,n,m,kX∗js,n,m,k    =Cs−1r−1!s−r−1!∫0∞∫0xx iy jFxmfxgm r−1Fx × hmFy−hmFxs−r−1Fyγs−1fydydx.

On expanding gm r−1Fx=1m+11−Fxm+1r−1 binomially in (23), we get

EX∗ ir,n,m,kX∗js,n,m,k=Cs−1r−1!s−r−1!m+1r−1  × ∑u=0r−1−1ur−1u∫0∞∫0xx iy jFxm+um+1fx  × hmFy−hmFxs−r−1Fyγs−1fydydx =Cs−1r−1!s−r−1!m+1r−1∑u=0r−1−1ur−1u  × Ii,jm+um+1, s−r−1, γs−1,
where
Ii,ja, b, c=∫0∞∫0xx iy jFxafxhmFy−hmFxb  ×  Fycfydydx.

Expanding hmFy−hmFxb binomially in (25) after noting that hmFy−hmFx=gmFy−gmFx, we get

Ii,ja, b, c=1m+1b∑v=0b−1vbv ∫0∞x iFxa+b−vm+1fx Ix dx,
where
Ix=∫0xy jFyc+vm+1fy dy.

By setting t=[F (y)]1/α in (27) and simplifying on the lines of (12), we find that

Ix=1β j/2∑p=0∞zpj/2F xc+vm+1+1+j/2+p/αc+v m+1+1+j/2+p/α.

On substituting the expression of I x in (26), we have

Ii,ja, b, c=1β j/2m+1b∑p=0∞∑v=0b−1vbvzpj/2c+vm+1+1+j/2+p/α  ×  ∫0∞x iFxa+c+bm+1+1+j/2+p/αfx dx.

Again by setting w=Fx1/α in (28) and simplifying the resulting expression, we obtain

Ii,ja, b, c=1βi+j/2m+1b∑p=0∞∑q=0∞∑v=0b−1vbvzpj/2c+v m+1+1+j/2+p/α  ×  zqi/2a+c+bm+1+2+i/2+j/2+p+q/α
and when m=−1 that
Ii,ja,b,c=00,  as  ∑v=0b−1vbv=0.

Therefore, on applying L’ Hospital rule and using (16), we find that

limm→−1Ii,ja,b,c=b!βi+j∑p=0∞ ∑q=0∞zpj/2c+1+j/2+p/αb+1  ×  zqi/2a+c+2+i/2+j/2+p+q/α.

Now on substituting for Ii, jm+um+1, s−r−1, γs−1 from (29) in (24) and simplifying, we obtain when m≠−1

EX∗ir,n,m,kX∗js,n,m,k=Cs−1r−1!s−r−1! βi+j/2m+1s−2  ×  ∑p=0∞∑q=0∞∑u=0r−1∑v=0s−r−1−1u+vr−1us−r−1vzpj/2γs−v+j/2+p/α  ×  zqi/2γr−u+i/2+j/2+p+q/α.
and when m=−1, in view of (30) and (25), we have
EX∗ir,n,−1,kX∗js,n,−1,k=EZr kiZs kj =α ksβi+j/2∑p=0∞∑q=0∞zpjzqiα k+j/2+ps−rα k+(i/2)+(j/2)+p+qr.

Identity 3.1.

For γr, γs≥1, k≥1, 1≤r<s≤n and m≠−1

∑v=0s−r−1−1vs−r−1v1γs−v=s−r−1! m+1s−r−1∏t=r+1sγt.

Proof.

At i=j=0 in (31), we have

1=Cs−1r−1!s−r−1! m+1s−2∑p=0∞∑q=0∞∑u=0r−1∑v=0s−r−1−1u+vr−1us−r−1v  ×  zp0zq0γs−v+p/αγr−u+p+q/α.

In view of Shawky and Bakoban [14], for i=j=0, note that

αp0=1,  αq0=1,  p,  q=0 and αp0=0,  αq0=0,  p,  q>0.

Therefore,

∑v=0s−r−1−1vs−r−1v1γs−v=r−1!s−r−1! m+1s−2Cs−1∑u=0r−1−1ur−1u1γr−u.

Now on using (20), we get the result given in (33).

At r=0, (33) reduce to (20).

Remark 3.1.

At j=0 in (31), we have

EX∗ir,n,m,k=Cr−1r−1! βi/2m+1r−1∑p=0∞∑u=0r−1−1ur−1uzpi/2γr−u+i/2+p/α
which is the exact expression for single moment as given in (18).

3.1. Special Cases

  1. Putting m=0, k=1 in (31), the explicit formula for the product moments of order statistics of the exponentiated Rayleigh distribution is obtained as

    EXn−r+1:ni Xn−s+1:n j=Cr, s:nβi+j/2∑p=0∞∑q=0∞∑u=0r−1∑v=0s−r−1−1u+vr−1us−r−1  v  ×  zpj/2 zqi/2n−s+1+v+j/2+p/α n−r+1+u+i/2+j/2+p+q/α.

    That is

    EXr:ni Xs:n j=Cr, s:nβ(i+j)/2∑p=0∞∑q=0∞∑u=0n−s∑v=0s−r−1−1u+vn−sus−r−1v  ×  zpi/2 zqj/2r−1+v+j/2+p/α s−1+u+i/2+j/2+p+q/α,
    where
    Cr, s:n=n!r−1! s−r−1 ! n−s!.

  2. Putting k=1 in (32), the explicit formula for the product moments of lower record values for the exponentiated Rayleigh distribution can be obtained as

    EXLriXLs j=αsβi+j/2∑p=0∞∑q=0∞zpj zqiα+j/2+ps−rα+i/2+j/2+p+qr.

Theorem 3.1.

For the distribution as given in (2), for 1≤r<s≤n, n≥2 and k=1, 2, …

EX∗ ir,n,m,kX∗js,n,m,k−EX∗ ir,n,m,kX∗js−1,n,m,k      =j2αβγsEX∗ ir,n,m,kX∗j−2s,n,m,k − EφX∗r,n,m,kX∗s,n,m,k,
where
φx,y=x iyj−2eβ x2.

Proof.

In view of Khan et al. [15], note that

EX∗ir,n,m,kX∗js,n,m,k−EX∗ir,n,m,kX∗js−1,n,m,k    =− j Cs−1γsr−1!s−r−1!∫0∞∫0xx iyj−1Fxm fxgm r−1Fx × hmFy−hmFxs−r−1Fyγsdy dx.

On using relation (6) in (35), we get

EX∗ir,n,m,kX∗js,n,m,k−EX∗ir,n,m,kX∗js−1,n,m,k=− j Cs−12αβγsr−1!s−r−1!∫0∞∫0xx iyj−2eβx2Fxmfxgm r−1Fx × hmFy−hmFxs−r−1Fyγs−1fy dy dx −∫0∞∫0xx iy j−2Fxmfxgm r−1FxhmFy−hmFxs−r−1Fyγs−1fy dy dx
and hence the result given in (34).

Remark 3.2.

Putting m=0, k=1 in (34), we obtain recurrence relations for product moments of order statistics of the exponentiated Rayleigh distribution in the form

EXn−r+1 : niXn−s+1 : n j−EXn−r+1 : niXn−s+2:n j=j2αβn−s+1  × EXn−r+1 : niXn−s+1 : n j−2−EφXn−r+1 : nXn−s+1 : n.

That is

EXr : ni Xs : n j−EXr−1 : niXs:n j=i2αβ r−1EXr : ni−2Xs : n j−Eφ Xr : nXs : n.

Remark 3.3.

Setting m=−1 and k≥1, in (34), we obtain the recurrence relations for product moments of lower k record values from exponentiated Rayleigh distribution in the form

EZr kiZs kj−EZr kiZs−1 kj    =j2αβ kEZr kiZs kj−2−EφZr kZs k.

Remark 3.4.

At i=0, Theorem 3.1 reduces to Theorem 2.1.

4. CHARACTERIZATION BY CONDITIONAL EXPECTATION AND RECURRENCE RELATION

Let X∗r,n,m,k, r=1, 2, …, n be dgos from a continuous population with df F x and pdf fx, then the conditional pdf of X∗s, n, m, k given X∗r, n, m, k=x, 1≤r<s≤n, in view of (4) and (5), is

fX∗s,n,m,k | X∗r,n,m,ky|x=Cs−1(s−r−1)! Cr−1Fxm−γr+1  × hmFy−hmFxs−r−1Fyγs−1fy,  y<x,  m≠−1
fZs k | Zr ky|x=ks−rs−r−1!lnFx−lnFys−r−1  × FyFxk−1fyFxdy,  y<x,  m=−1.

Theorem 4.1.

Let X be a non-negative random variable having an absolutely continuous df Fx with F0=0 and 0<Fx<1 for all x>0, then

EξX∗s,n,m,k|X∗l,n,m,k=x=1β∑p=1∞1−e−β x2pp∏j=1s−lγr+jγr+j+p/α,  l=r,  r+1,  m≠−1.
EξZs k|Zl k=x=1β∑p=1∞kk+p/αs−l1−e−β x2p/p,  l=r,  r+1,  m=−1,
where
ξy=y2
if and only if
Fx=1−e−β x2α,  x>0,   α,β>0.

Proof.

When m≠−1, we have from (36) for s > r+1

EξXs,n,m,k|Xr,n,m,k=x=Cs−1s−r−1! Cr−1m+1s−r−1  × ∫0xy21−F yF xm+1s−r−1FyFxγs−1fyFx dy.

By setting u=FyFx=1−e−β  y21−e− β x2α from (2) in (40), we obtain

EξXs,n,m,k|Xr,n,m,k=x=Cs−1s−r−1! Cr−1m+1s−r−1  × 1β∫01−ln1−1−e−β x2 u1/α uγs−11−um+1s−r−1du =Cs−1s−r−1! Cr−1m+1s−r−1β∑p=1∞1−e−β x2pp  × ∫01up/α+γs−11−um+1s−r−1du.

Again by setting t=um+1 in (41), we get

EξXs,n,m,k|Xr,n,m,k=x=Cs−1s−r−1! Cr−1m+1s−rβ  × ∑p=1∞1−e−β x2pp∫01t p+αkαm+1+n−s−11−ts−r−1dt =Cs−1Cr−1m+1s−rβ∑p=1∞1−e−β x2ppΓp+α kαm+1+n−sΓp+α kαm+1+n−r =Cs−1β Cr−1∑p=1∞1−e−β x2pp∏j=1s−rγr+j+p/α,
where
Cs−1Cr−1=∏j=1s−rγr+j
and hence the result given in (38).

To prove sufficient part, we have from (36) and (38)

Cs−1s−r−1!Cr−1m+1s−r−1∫0xyFxm+1−Fym+1s−r−1    ×Fyγs−1fydy=Fxγr+1Hrx,
where
Hrx=1β∑p=1∞1−e−β x2pp∏j=1s−rγr+jγr+j+p/α.

Differentiating (42) both sides with respect to x, we get

Cs−1Fxmfxs−r−2!Cr−1m+1s−r−2∫0xyFxm+1−Fym+1s−r−2Fyγs−1fydy    =H′rxFxγr+1+γr+1HrxFxγr+1−1fx
or
γr+1Hr+1xFxγr+2+mfx   =H′rxFxγr+1+γr+1HrxFxγr+1−1fx.

Therefore,

fxFx=H′rxγr+1Hr+1x−Hrx =2αβ x e−β x21−e−β x2,
where
H′rx=2x e−β x2∑p=1∞1−e−β x2p−1∏j=1s−rγr+jγr+j+p/α,
Hr+1x−Hrx=1αβ γr+1∑p=1∞1−e−β x2p∏j=1s−rγr+jγr+j+p/α.

Integrating both the sides of (43) with respect to x between 0,y, the sufficiency part is proved.

For the case when m=−1, from (37) on using the transformation u=FyFx=1−e−β  y21−e− β x2α, we find that

EξZs k|Zl k=x=A*∫01−ln us−r−1 uk+p/α−1du,
where
A*=ks−rs−r−1 !β∑p=1∞1−e−β x2p/p.

We have Gradshteyn and Ryzhik ([16], p. 551)

∫01−lnxμ−1xυ−1dx=Γμυμ,  μ>0,  υ>0.

On using (45) in (44), we have the result given in (39).

Sufficiency part can be proved on the lines of case m≠−1.

Theorem 4.2.

Let X be a non-negative random variable having an absolutely continuous df Fx with F0=0 and 0<Fx<1 for all x>0, then

EX∗jr,n,m,k=EX∗jr−1,n,m,k−j2αβ γrEφX*r,n,m,k  + j2αβ γrEX∗j−2r,n,m,k
if and only if
Fx=1−e−β x2α,  x>0,   α>0,  β>0.

Proof.

The necessary part follows immediately from (21). On the other hand if the recurrence relation in (46) is satisfied, then on using (4), we have

Cr−1r−1! ∫0∞x jFxγr−1fxgm r−1Fx dx  =r−1Cr−1γrr−1! ∫0∞x jFxγr+mfx gmr−2Fx dx − j Cr−12αβγrr−1!∫0∞x j−2eβ x2Fxγr−1fxgm r−1Fxdx + j Cr−12αβγrr−1!∫0∞x j−2Fxγr−1fx gm r−1Fx dx.

Integrating the first integral on the right hand side in (47) by parts and simplifying the resulting expression, we get

j Cr−1γrr−1! ∫0∞xj−1Fxγr−1gm r−1Fx dx  ×Fx−12αβ xeβ x2fx+12αβ xfxdx=0.

Now applying a generalization of the Müntz-Szász Theorem [17] to (48), we get

fxFx=2αβ xeβ x2−1,
which proves that
Fx=1−e−β x2α,  x>0,   α,  β>0.

5. CHARACTERIZATION BY TRUNCATED MOMENT

Theorem 5.1.

Suppose an absolutely continuous (with respect to Lebesgue measure) random variable X has the df Fx and pdf fx for 0<x<∞, such that f′x and EX|X≤x exist for all x, 0<x<∞, then

EX|X≤x=gxηx,
where
ηx=fxFx
and
gx=1−e−β  x22αβ x e−β x2−∫0x1−e−β  u2αdu2αβ x1−e−β  x2α−1e−β x2,
if and only if
fx=2αβ x 1−e−β  x2α−1e−β x2,  x>0,  α,  β>0.

Proof.

In view of Ahsanullah et al. [18] and (1), we have

EX|X≤x=2αβFx∫0xu21−e−β  u2α−1e−β x2du.

Integrating (50) by parts treating 'u 1−e−β  u2α−1e−β x2' for integration and rest of the integrant for differentiation, we get

EX|X≤x=1Fxx 1−e−β  x2−∫0x1−e−β  u2αdu.

After multiplying and dividing by fx in (51), we have the result given (49).

To prove sufficient part, we have from (49)

 1Fx∫0xu fu du=g(x)fxFxor∫0xu fu du=gxfx.

Differentiating (52) on both the sides with respect to x, we find that

x fx=g′xfx+gxf′x.

Therefore,

f′(x)f(x)=x−g′(x)g(x)=2(α−1)βxe−β x21−e−βx2+1x−2βx,
where
g′(x)=x+g(x)(2(α−1)βxe−β x21−e−βx2+1x−2βx).

Integrating both the sides in (53) with respect to x, we get

f(x)=cx1−e−βx2α−1e−β x2.

It is known that

∫0∞fx dx=1.

Thus,

1c=∫0∞x1−e−βx2α−1e−β x2dx=12αβ,
which proved that
f(x)=2αβx1−e−βx2α−1e−β x2,x>0,α,β>0.

CONFLICT OF INTEREST

The authors are declare no competing interests.

AUTHORS' CONTRIBUTIONS

The author carried out the proof of the main results and approved the final manuscript.

ACKNOWLEDGMENT

The authors acknowledge with thanks to both the referee and the Editor-in-Chief Prof. M. Ahsanullah for their fruitful suggestions and comments which led the overall improvement in the manuscript. Authors are also thankful to Prof. A. H. Khan, Aligarh Muslim University, Aligarh, who helped in preparation of this manuscript.

REFERENCES

3.M. Burkschat, E. Cramer, and U. Kamps, Metron, Vol. LXI, 2003, pp. 13-26. ftp://metron.sta.uniroma1.it/RePEc/articoli/2003-1-13-26.pdf
4.A.H. Khan, R.U. Khan, and M. Yaqub, J. Appl. Probab. Stat., Vol. 1, 2006, pp. 115-131.
6.M. Ahsanullah, Stat. Methods, Vol. 7, 2005, pp. 16-28.
7.A.K. Mbah and M. Ahsanullah, Pak. J. Stat., Vol. 23, 2007, pp. 139-146.
8.A.H. Khan, Z. Anwar, and S. Chisti, Pak. J. Stat., Vol. 26, 2010, pp. 615-628.
9.R.U. Khan and D. Kumar, Appl. Math. Sci. (Ruse)., Vol. 4, 2010, pp. 2711-2722. http://www.m-hikari.com/ams/ams-2010/ams-53-56-2010/khanAMS53-56-2010.pdf
12.N. Balakrishnan and A.C. Cohen, Order Statistics and Inference: Estimation Methods, Academic Press, San Diego, 1991.
14.A.I. Shawky and R.A. Bakoban, J. Stat. Theory Appl., Vol. 7, 2008, pp. 263-277. https://www.kau.edu.sa/files/0008606/researches/24613_33404.pdf
15.R.U. Khan, Z. Anwar, and H. Athar, Aligarh J. Stat., Vol. 28, 2008, pp. 37-45.
16.I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 2007. https://www.sciencedirect.com/science/article/pii/B9780080471112500030?via%3Dihu
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Journal of Statistical Theory and Applications
Volume-Issue
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Pages
402 - 415
Publication Date
2019/11/14
ISSN (Online)
2214-1766
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1538-7887
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10.2991/jsta.d.191104.001How to use a DOI?
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risenwbib
TY  - JOUR
AU  - M. A. R. Khan
AU  - R. U. Khan
AU  - B. Singh
PY  - 2019
DA  - 2019/11/14
TI  - Relations for Moments of Dual Generalized Order Statistics from Exponentiated Rayleigh Distribution and Associated Inference
JO  - Journal of Statistical Theory and Applications
SP  - 402
EP  - 415
VL  - 18
IS  - 4
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.191104.001
DO  - 10.2991/jsta.d.191104.001
ID  - Khan2019
ER  -