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
Theorem 4. If
, ;
x F
of
,
p
LL
given by 5 then ,
; 1
1
x
F
is a convex
function from 0, 1 to 0, 1 for
1
and
,
p .
10 Proof: It has been shown in Theorem 2 that the cdf
, ;
x F
of
,
p
LL
is concave for
1
.
Hence, ,
; 1
1
x
F
is a convex function from 0, 1 to 0, 1.
Remark: ,
;
x
F can be used as a distortion function to distort survival function sf of
a given random variable as stated in the corollary below.
Corollary 2.
If X is the risk with sf x
G and let Z be a distorted random variable with
survival function
] ,
; [
x G
F x
H
for
1
and
,
p .
Then
] ,
; [
] [
x P
dx x
G F
Z E
is a premium principle such that
i.
max ,
X P
X P
X P
ii.
b x
aP b
ax P
iii.
if
1
X precedes
2
X under first stochastic dominance that is if
2 2
1 1
x G
x G
then
2 ,
1 ,
X P
X P
iv.
if
1
X precedes
2
X under second stochastic dominance that is if
2 2
2 1
1 1
dx x
G dx
x G
then
2 ,
1 ,
X P
X P
Where X
E X
P
is the net premium average loss, }]
, ,
[max{
1 max
n
X X
E X
P
is
the maximum premium of the insurance product
1 1
x G
and
1 1
x G
are sf of two non negative risk r.v. respectively.
Proof: Since from theorem 2 above we know that ,
;
x
F is concave for
1 ,
x
when 1
,
p
and
and is an increasing function of x with
, ;
F
and 1
, ;
1
F . Therefore by definition 6 of distortion premium principle
and subsequent properties thereof in Wang 1996 the results follow immediately.
4. Parameter Estimation: Maximum Likelihood Estimation
The likelihood function for a random sample of size n from the
,
p
LL
is
11
n i
i p
n i
i n
p n
x x
x p
p L
1 1
1 2
log log
1 1
.
The log-likelihood function is then given by
n i
i
x p
p n
p n
p n
L l
1
log log
1 log
1 log
log 2
log
n i
i n
i i
x x
1 1
log 1
log log
. The first and second order derivatives of the log-likelihood function are:
n i
i
x p
n p
n l
1
log 1
2
n i
i
x p
n l
1
log 1
1
n i
i
x p
n p
p n
n p
l
1
]] 1
log[log[ 1
1 1
log
2 2
2 2
2 2
2 2
2 2
1 ]
1 2
[ 1
2
p p
p n
p n
p n
l
n i
i
x p
n l
1 2
2 2
2 2
log 1
1
1 1
1 1
2 2
2
p p
n p
p n
p l
p n
p n
l 1
1
2 2
2 2
1
p
n p
l
2 2
1
p n
n p
l
12 For information matrix we obtain the following result
, 1
] 1
[ 1
log 1
2 1
2
r r
p p
p e
x E
p r
p r
p n
i i
. For
p
, this reduces to the result given in Gomez et al. 2014. The information matrix is given by
This matrix can be inverted to get asymptotic variance-covariance matrix for the maximum likelihood estimates.
5. Numerical Applications
