ProbStat Forum, Volume 05, July 2012, Pages 80–85
ISSN 0974-3235
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Conditional expectation of function of dual generalized order statistics
An alternative approach
M.J.S. Khana , M.I. Khanb
a Department
of Applied Mathematics and Statistics, Baba Saheb Bheem Rao Ambedkar University, Lucknow–226 025, India
of Statistics and Operations Research, Aligarh Muslim University, Aligarh–202 002, India
b Department
Abstract. A general form of continuous distribution has been characterized through the conditional
expectation of function of dual generalized order statistics and lower record values using Meijer’s Gfunction.
1. Introduction
Burkschat et al. (2003) introduced the concept of dual generalized order statistics (d1os) as: Let X1 ,
X2 , . . ., Xn is a sequence of independent and identically distributed random variables with distribution
function (d f ) F(x) and with probability density function pd f f (x). Further, let n ∈ N, n ≥ 2, k > 0,
n−1
P
m̃ = (m1 , m2 , . . . , mn−1 ) ∈ Rn−1 , Mr =
m j , such that γr = k + n − r + Mr > 0, ∀ r ∈ {1, 2, . . . , n − 1}. Then
j=r
X0 (r, n, m̃, k), r = 1, 2, . . . , n are called d1os if their joint pd f is given by



n−1
n−1
Y
 Y


 
mi
k 
γ j   [F(xi )] f (xi ) [F(xn )]k−1 f (xn )


j=1
(1)
i=1
for F−1 (1) > x1 ≥ x2 ≥ . . . ≥ xn > F−1 (0).
Conditional expectation of dual generalized order statistics are extensively used in characterizing the
probability distributions. Various approaches are available in the literature. For detailed survey and
discussion of characterization results one may refer to Ahasanullah (2004), Mbah and Ahsanullah (2007),
Khan et al. (2009), Khan et al. (2010a, b) and Faizan and Khan (2011). In this paper we have characterized
the distribution through conditional expectation of d1os conditioned on non-adjacent d1os using Meijer’s
G-function.
The pd f of X0 (r, n, m, k) with respect to a measure PF is given by
fr (x) = cr−1 Gr (F(x)|γ1 , . . . , γr )I(α,β) (x).
Here IA denotes the indicator function and Gr (x) =
(2)
Gr,0
r,r
x|γ1 , . . . , γr =
Gr,0
r,r
!
γ1 , . . . , γr
x
is the partiγ1 − 1, . . . , γr − 1
2010 Mathematics Subject Classification. Primary: 62E15; Secondary: 62E10, 62G30
Keywords. Characterization, Meijer’s G-function, dual generalized order statistics, lower records
Received: 03 December 2011; Accepted: 12 June 2012
Email addresses: [email protected] (M.J.S. Khan), [email protected] (Corresponding author) (M.I. Khan)
M.J.S. Khan, M.I. Khan / ProbStat Forum, Volume 05, July 2012, Pages 80–85
cular Meijer’s G-function defined by
!
Z
1
sz
γ1 , . . . , γr
r,0 Gr,r s
=
dz,
Qr
γ1 − 1, . . . , γr − 1
2πi L j=1 (γ j − 1 − z)
81
(3)
where L is an appropriate chosen contour of integration. See Mathai (1993, Chapter 3) for the definition of
Meijer’s G-function and its numerous properties and applications.
The joint PF ⊗ PF density of X0 (r, n, m̃, k) and X0 (s, n, m̃, k), 1 ≤ r < s ≤ n, is given by
!
F(y) 1
γr+1 , . . . , γs Gr (F(x)|γ1 , . . . , γr ),
fr,s (x, y) = cs−1
Gs−r
α ≤ y < x ≤ β,
(4)
F(x)
F(x)
where
cr−1 =
r
Y
γi .
i=1
Hence the conditional PF density function of X0 (s, n, m̃, k) given X0 (r, n, m̃, k) = x, 1 ≤ r < s ≤ n, is
!
F(y) 1
cs−1
γr+1 , . . . , γs
Gs−r
I(β,x) (y),
α ≤ y < x ≤ β,
fs|r (y|x) =
cr−1
F(x)
F(x)
and the conditional PF density function of X0 (r, n, m̃, k) given X0 (s, n, m̃, k) = y, 1 ≤ r < s ≤ n, is
F(y) Gs−r F(x) γr+1 , . . . , γs Gr (F(x)|γ1 , . . . , γr )
fr|s (x|y) =
I(y,α) (x).
F(x)Gs (F(y)|γ1 , . . . , γs )
(5)
(6)
Some auxiliary results
Here some results are given that are used in subsequent sections:
i) G1 (x|γ1 ) = xγ1 −1
ii) (γr − γ1 )Gr (x|γ1 , . . . , γr ) = Gr−1 (x|γ1 , . . . , γr−1 ) − Gr−1 (x|γ2 , . . . , γr )
iii) xa Gr (x|γ1 , . . . , γr ) = Gr(x|γ1 + a, . . . , γr + a), a ∈ R


1, r = 1
iv) lim Gr (x|γ1 , . . . , γr ) = 

0, r ≥ 2
x→1−


0,
if γ1:r > 1



Qr
1

γ j −γl , if γ1:r = 1 < γ2:r
v) lim Gr (x|γ1 , . . . , γr ) = 
 j=1

x→0+
j,1


∞,
if γ1:r = γ2:r = 1 or γ1:r < 1
d
1 vi) dx Gr (x|γ1 , . . . , γr ) = x (γr − 1)Gr (x|γ1 , . . . , γr ) − Gr−1 (x|γ1 , . . . , γr−1 )
d
vii) dx
Gr (x|γ1 , . . . , γr ) = 1x (γ1 − 1)Gr (x|γ1 , . . . , γr ) − Gr−1 (x|γ2 , . . . , γr−1 ) where γ1:r = min(γ1 , . . . , γr ) and
l = max{1 ≤ j ≤ r : γ j = γ1:r ).
For property (i), see Mathai (1993, p. 130), for property (ii), see Cramer and Kamps (2003), and property
(iii), see Mathai (1993, p. 69). Property (iv) can easily be deduced from Lemma 2.2 of Cramer et al. (2004 b).
Whereas (vi) and (vii) can be established from (3).
2. Characterization of distribution
Theorem 2.1. Let X0 (i, n, m̃, k), i = 1, . . . , n be the d1os from a continuous population with the d f F(x) and the pd f
f (x) over the support (α, β) and ξ(x) be a monotonic and differentiable function of x. If for two consecutive values r
M.J.S. Khan, M.I. Khan / ProbStat Forum, Volume 05, July 2012, Pages 80–85
82
and r + 1, 1 ≤ r < s − 1 < n
1s|l (x) = E [ξ{X0 (s, n, m̃, k)}|X0 (l, n, m̃, k) = x] ,
l = r, r + 1,
(7)
exists, then
F(x) = e−
A(t) =
Rβ
x
A(t)dt
10s|r (t)
γr+1 [1s|r+1 (t) − 1s|r (t)]
.
Proof. We have from (5)
!
Z
F(y) cs−1 x
γr+1 , . . . , γs dF(y)
ξ(y)Gs−r
1s|r (x)F(x) =
cr−1 α
F(x)
(8)
differentiate both the sides of (8) w.r.t. x to get
10s|r (x)F(x)+1s|r (x) f (x) = −
cs−1 f (x)
Cr−1 F(x)
x
Z
α
!
!#
"
F(y) F(y) γr+1 , . . . , γs −Gs−r−1
γr+2 , . . . , γs dF(y)
ξ(y) (γr+1 −1)Gs−r
F(x)
F(x)
or,
10s|r (x)F(x) + 1s|r (x) f (x) = −(γr+1 − 1)1s|r (x) f (x) + γr+1 1s|r+1 (x) f (x)
in view of property (vi) and equation (8).
Therefore,
10s|r (x)
f (x)
=
= A(x).
F(x) γr+1 [1s|r+1 (x) − 1s|r (x)]
Also for s = r + 1
10r+1|r (x)
f (x)
=
= A(x)
F(x) γr+1 [ξ(x) − 1s|r+1 (x)]
and hence the Theorem.
Remark 2.2. If
E[{X0 (s, n, m̃, k)}|X0 (r, n, m̃, k) = x] = a∗s|r x + b∗s|r = 1s|r (x)
then
F(x) = [ax + b]c
and
E[ξ{X0 (s, n, m̃, k)}|X0 (r, n, m̃, k) = x] = a∗s|r ξ(x) + b∗s|r
if and only if
F(x) = [aξ(x) + b]c
M.J.S. Khan, M.I. Khan / ProbStat Forum, Volume 05, July 2012, Pages 80–85
83
where
a∗s|r =
s
Y
cγ j
j=r+1
1 + cγ j
,
b
b∗s|r = − (1 − a∗s|r )
a
and ξ(x) be a monotonic and continuous function of x.
A number of distributions can be characterized by the proper choice of a, b, c and ξ(x) and the results can be
deduced for order statistics and lower records, Khan et al. (2010a).
Theorem 2.3. Let X0 (i, n, m̃, k), i = 1, . . . , n be the d1os from a continuous population with the d f F(x) and the pd f
f (x) over the support (α, β) and ξ(x) be a monotonic and differentiable function of x. If for two consecutive values
s − 1 and s, 1 ≤ r < s − 1 < n
1r|l (y) = E[ξ{X0 (r, n, m̃, k)}|X0 (l, n, m̃, k) = y],
l = s − 1, s
" Z
Gs (F(y)|γ1 − γs + 1, . . . , γs − γs + 1) = a(s) (s) exp −
y
(9)
then
α
#
D(t)dt ,
if γ1:s−1 > γs ,
(10)
and
y
" Z
Gs (F(y)|γ1 − γs + 1, . . . , γs − γs + 1) = exp −
#
D(t)dt ,
if γ1:s−1 ≤ γs .
(11)
p
To prove (11), we note that log F(y), α < y < β is a non-decreasing function in (−∞, 0), therefore exists a
p, α < p < β, such that
−lo1F(p) = 1
(12)
and
D(t) =
10r|s (t)
[1r|s (t) − 1r|s−1 (t)]
.
Proof. We have,
β
Z
1r|s (y)Gs (F(y)|γ1 , . . . , γs ) =
y
!
F(y) ξ(x)
γr+1 , . . . , γs Gr (F(x)|γ1 , . . . , γr )dF(x)
Gs−r
F(x)
F(x)
differentiate both the sides of (13) w.r .t. y, to get
f (y)
f (y)
Gs−1 (F(y)|γ1 , . . . , γs−1 )1r|s (y) −
Gs−1 (F(y)γ1 , . . . , γs−1 )1r|s−1 (y)
F(y)
F(y)
f (y)
10r|s (y)Gs (F(y)|γ1 , . . . , γs ) =
Gs−1 (F(y)|γ1 , . . . , γs−1 )[1r|s (y) − 1r|s−1 (y)]
F(y)
10r|s (y)
f (y) Gs−1 (F(y)|γ1 , . . . , γs−1 )
=
[1r|s (y) − 1r|s−1 (y)] F(y) Gs (F(y)|γ1 , . . . , γs )
10r|s (y)Gs (F(y)|γ1 , . . . , γs ) =
and s = r + 1
10r|r+1 (y)
f (y) Gr (F(y)|γ1 , . . . , γs−1 )
=
.
F(y) Gr+1 (F(y)|γ1 , . . . , γs ) [1r|r+1 (y) − ξ(y)]
(13)
M.J.S. Khan, M.I. Khan / ProbStat Forum, Volume 05, July 2012, Pages 80–85
Now using the property (vi), we have
d
10r|s (t)
f (y) dy Gs (F(y)|γ1 , . . . , γs )
(γs−1 − 1)
−
=
F(y)
Gs (F(y)|γ1 , . . . , γs )
[1r|s (t) − 1r|s−1 (t)]
integrating both the sides w.r.t. y over (α, y), to get
"
F(y)
F(α)
#1−γs
" Z y
#
Gs (F(y)|γ1 , . . . , γs )
= exp −
D(t)dt
Gs (F(α)|γ1 , . . . , γs )
α
or
" Z
Gs (F(y)|γ1 − γs + 1, . . . , γs − γs + 1) = a(s) exp −
y
α
#
D(t)dt ,
if γ1:s−1 > γs ,
or
Gs (F(α)|γ1 − γs + 1, . . . , γs − γs + 1) → a(s) (s)
when γ1:s−1 ≤ γs , then exists a p which satisfied (12). Hence
" Z y
#
Gs (F(y)|γ1 − γs + 1, . . . , γs − γs + 1) = exp −
D(t)dt ,
p
Corollary 2.4. For m1 = m2 = . . . = mn−1 = m
Gs (1 − x|γ1 − γs + 1, . . . , γs − γs + 1) =
1
[1m (x)]s−1
(s − 1)!
where


1

 m+1 [1 − (1 − x)m+1 ] if m , −1
1m (x) = 

− log(1 − x)
if m = −1
and hence (10) and (11) can be rewritten as follows:
1
( Z y
)# m+1
F(y) = 1 − exp −
D(t)dt
, m > −1
α
"
( Z y
)#
F(x) = exp − exp −
D(t)dt , m = −1
"
p
as obtained by Khan et al. (2010a).
Remark 2.5.
1r|s (y) = E[{X0 (r, n, m̃, k)}|X0 (s, n, m̃, k) = y] = a∗r|s y + b∗r|s
if and only if
1 − [F(y)]m+1 = [ay + b]c ,
α < y < β, m > −1,
if γ1:s−1 ≤ γs .
84
M.J.S. Khan, M.I. Khan / ProbStat Forum, Volume 05, July 2012, Pages 80–85
85
where F(y) is d f over (α, β) and
a∗r|s
=
s−r
Y
j=1
c(s − j)
b
, b∗ = − (1 − a∗r|s )
1 + c(s − j) r|s
a
and − log F(y) = [ay + b]c at m = −1 with − log F(p) = 1.
For the proof and the related results for order statistics and lower records one may refer to Khan et al.
(2010a).
Acknowledgement
It is a pleasure to record our sincere thanks to Prof. A.H. Khan, Aligarh Muslim University, Aligarh
under whose supervision this work was done. The authors also thank the referees.
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