4 Pdf:
, ,
1 ,
log log
] 2
[ ,
;
1 3
x x
x x
x f
7
Cdf:
2 ]
log 2
log 2
[ ,
; x
x x
x F
8
Moment:
,
2 ,
1 ,
1 ,
2 ,
, ]
2 [
] 2
[ ,
;
3 3
r r
r X
E
r
9
iv. for
2
p , we get new
,
2
LL distribution as
Pdf: ,
, 1
, log
log 2
6 ,
;
1 2
4
x x
x x
x f
10
Cdf:
2 6
}] log
3 log
2 6
{ log
2 6
[ ,
;
x
x x
x x
F 11
Moment:
,
2 ,
1 ,
1 ,
2 ,
, 3
2 ]
6 2
[ ,
;
4 4
r r
r X
E
r
12
2.1 Shape of
,
p
LL
distributions
Here we have drawn some illustrative pdf of
,
p
LL
distributions for different choices of parameters
, and index p to study their shapes. It is observed that the distribution
can be positively negatively skewed, symmetrical, and increasing deceasing type.
a b
5 c d
e f
Figure 1 . Pdf plots of
,
p
LL
when p = 0 red, 1 green, 3blue, 5 brown, 7 light green for a
001 .
, 20
b 1
. ,
5
c
5 ,
20
d
1 .
, 2
e 10
, 5
.
f
, 9
.
2.2 Moment
Here we derive general result for moments of
,
p
LL
distribution and obtain its
particular cases.
, 2
, 1
, 1
, 2
, ,
] 1
[ 1
2 1
, ;
2 2
r p
p r
p p
r X
E
p p
r
13
In particular,
p p
X E
p
1 1
1 1
, ;
2
. 14
6
2.3 Mode
The mode plays an important role in the usefulness of a distribution. For the
,
p
LL
distribution the mode occurs at ]
1 2
} 1
1 1
4 1
1 exp[{
2
r
r r
.
2.4 Survival and Hazard Rate functions
] 1
[ 1
log }]
log ,
1 {
1 [
, ,
;
1 1
1
p p
t t
t p
Ei p
p p
t S
r p
p
log }]
log ,
1 {
1 [
log ,
, ;
1 1
t t
t p
Ei p
p t
t p
t r
p
15
Some illustrative plots of the hazard function of
,
p
LL
distributions for different choices of parameters
, and index p are presented in Figure 2 reveals that it can
increasing and bath tub shaped.
Figure 2 . Hazard rate function plots of
,
p
LL
when p = 0 red, 1 green, 3blue, 5 brown, 7 light green for a
2 ,
9 .
b 2
, 2
c 2
, 5
2.5 Shannon Entropy of
,
p
LL
Here we derive the Shannon entropy of
,
p
LL
in terms of that of ,
LL .
] ,
, ;
log [
p X
f E
X H
p
1 2
log log
] 1
[ 1
log
X
X X
p p
E
p p
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
