7
}] log
[log{ 1
log log
] 1
[ 1
log
1 2
X X
E X
pE p
p
p
1 log
log ]
1 [
1 log
2
X pE
p p
p
1 log
log 1
log
2 1
2
X X
E
] 1
log [log
1 1
1 log
X H
X pE
p p
p
16
It can be checked that for
p ,
X H
p
reduces to X
H , the Shannon entropy of
,
LL
which is the Log-Lindley see proposition 2 in page 51 of Gomez et al., 2014. For integral values of index parameter p, exact expression for
] 1
log log
X E
can be obtained using Mathematica as follows.
2
log 2
3 }
1 {log
log ,
1 X
E p
2 6
log 3
2 3
11 }
1 {log
log ,
2
X E
p
6 24
log 4
6 11
50 }
1 {log
log ,
3
X E
p
24 120
log 5
24 50
274 }
1 {log
log ,
4
X E
p
,… For example, Shannon Entropy for
,
1
LL using 16 is given by
2 log
2 3
1 2
log
2 1
X H
X H
.
2.6 Stochastic Ordering
We will consider the likelihood ratio LR, hazard rate HR and stochastic ST orderings for
,
p
LL
random variables in this section.
8
Theorem1 . Let
1
X and
2
X be random variables following
,
1 1
1
p
LL
and
,
2 2
2
p
LL
distributions, respectively. If
2 1
2 1
,
and
1 2
p p
then
2 1
X X
LR
.
Proof: Consider the ratio
log ]
1 [
1 ]
1 [
1 ,
, ;
, ,
;
1 2
1 2
2 2
2 2
2 1
1 1
1 1
2 2
1 1
1 2
2 2
x h
x p
p p
p p
x f
p x
f
p p
p p
17
where
1 2
log log
1 2
x
x x
x h
. Gomez-Deniz et al. 2014 has shown that the function x
h is non-decreasing for
1 ,
x
if
2 1
2 1
,
. Here, if
1 2
p p
, it is clear that
1 2
log
p p
x
is non-decreasing for
1 ,
x
. This implies that if
2 1
2 1
,
and
1 2
p p
, then the ratio in 17 is non-decreasing for
1 ,
x
and hence,
2 1
X X
LR
.
The result on LR ordering leads to the following Gomez-Deniz et al., 2014:
2 1
2 1
2 1
X X
X X
X X
ST HR
LR
. Therefore, similar results as in Corollary 1 of
Gomez-Deniz et al. 2014 can be shown for the new log-Lindley distribution as follows.
Corollary 1 . Let
1
X and
2
X be random variables following
,
1 1
1
p
LL
and
,
2 2
2
p
LL
distributions, respectively. If
2 1
2 1
,
and
1 2
p p
, then
a. the moments,
] [
] [
2 1
k k
X E
X E
for all
k
; b.
the hazard rates,
2 1
x r
x r
for all
1 ,
x
.
2.7 Convexity and Concavity for Generalized Log-Lindley Distribution
For brevity, we suppress the cdf notation for
,
p
LL
distribution in 5 as x
F in this
section.
Theorem 2 : If
1
,
p and
, then
x F
is concave for 1
,
x . Hence,
for 1
,
x F
is also log-concave for 1
,
x since
x
F .
Proof: If 1
, then
p
x log
,
log x
and
1
x are decreasing in
1 ,
x
for
p and
. This implies that the pdf
9 ,
log log
] 1
[ 1
, ,
;
1 2
x
x x
p p
p x
f x
F
p p
is decreasing in 1
,
x . Thus,
x F
is concave for 1
,
x .
Since
x F
, concavity implies x
F is also log-concave for
1 ,
x
.
Theorem 3 . The function
x F
is not convex or concave for 1
,
x for any
1
,
p
and
.
Proof: For any
1
,
p
and
, consider the second order derivative of
x F
given by
1 log
1 1
log 1
log ]
1 [
1
2 1
1 2
p x
p x
x x
p p
x F
p p
In order for x
F to be convex,
x F
must be
or that the term
1
log 1
1 log
2
p x
p x
for all 1
,
x . However, when
1
x
,
1 1
log 1
1 log
2
p p
x p
x .
Thus, x
F is not convex for
1 ,
x
for any
1
,
p
and
.
When
x
,
1
log 1
1 log
2
p x
p x
. This implies that x
F
cannot be
for all 1
,
x . Therefore,
x F
is not concave for 1
,
x for any
1
,
p
and
.
Hence,
x F
is convex for 1
,
x only for
p
and 1
1
as shown in Gomez-
Deniz et al. 2014.
3. Application to insurance premium loading
