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Parameter estimation
ActSCi 432/832
1 / 31
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View Full Document Distributions Summary
X is Binomial
with parameters
n
(positive integer) and
p
∈
(0
,
1)
p
.
f
.
f
(
x
) =
±
n
x
²
p
x
(1

p
)
n

x
for x
= 0
,
1
,
2
,
3
, ......
n
E
(
X
) =
np
,
Var
(
X
) =
np
(1

p
)
X is Bernoulli
with parameters
p
∈
(0
,
1)
→
X
is Binomial
with
n
= 1
X is Poisson
with parameter
λ >
0
p
.
f
.
f
(
x
) =
λ
x
e

λ
x
!
for x
= 0
,
1
,
2
,
3
, ......
E
(
X
) =
λ,
Var
(
X
) =
λ
2 / 31
Distributions Summary
X is Gamma
with parameters
α >
0
and
β >
0
p
.
f
.
f
(
x
) =
x
α

1
e

x
/β
Γ(
α
)
β
α
for x
>
0
E
(
X
k
) =
β
k
Γ(
α
+
k
)
Γ(
α
)
,
if
k
>

α
Note that Γ(
α
+ 1) =
α
Γ(
α
)
for
α >
0
and
Γ(
α
+ 1) =
α
!
for
α
= 0
,
1
,
2
, ....
X is Exponential
with parameters
β >
0
→
X is Gamma with
α
= 1
X is Pareto
with parameters
α >
0 and
β >
0
p
.
f
.
f
(
x
) =
αβ
α
(
x
+
β
)
α
+1
for x
>
0
E
(
X
) =
β
α

1
for
α >
1
,
Var
(
X
) =
αβ
2
(
α

1)
2
(
α

2)
for
α >
2
3 / 31
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View Full Document Introduction
Uncertainties in ﬁtting actuarial loss data
1
Model uncertainty
2
Parameter uncertainty
given
the model
3
Sampling error
This chapter is about the second item above.
Example: When you have
somehow
concluded that
Exponential distribution is suitable for the given data, how do
you obtain the parameter
β
?
Several popular methods available with diﬀerent implications
4 / 31
Method of moments estimation (MME)
Consider an iid sample
{
x
1
, ...,
x
n
}
of size
n
from the common
cdf
F
(
x

Θ)
Θ = (
θ
1
, ..., θ
p
) is the vector of unknown parameters. Eg, for
the normal distribution, Θ = (
μ, σ
)
Write
μ
0
k
(Θ) :=
E
(
X
k

Θ)
A method of moments estimate of Θ is any solution of the
p
eqns below:
μ
0
k
(Θ) = ˆ
μ
0
k
=
1
n
n
X
1
x
k
i
,
k
= 1
, ...,
p
.
(1)
Example (Gamma)
Determine the method of moments estimate of
Θ = (
α, β
)
from a
sample of size n generated from a Gamma distribution.
5 / 31
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View Full Document Percentile matching estimation (PME)
The
q
th percentile
π
q
of
F
(
x

Θ) is deﬁned from
F
(
π
q

Θ) =
q
,
0
<
q
<
1
(2)
Note that
π
q
is a function of Θ, so
π
q
(Θ) would be more
accurate expression
A percentil matching esimtate of Θ is any solution of the
p
eqns below:
π
q
k
(Θ) = ˆ
π
q
k
,
k
= 1
, ...,
p
,
(3)
where ˆ
π
q
k
are sample percentiles
Perhaps more readable version is obtained by applying
F
to
both sides:
q
k
=
F
(ˆ
π
q
k

Θ)
(4)
Idea is to match the theoretical percentiles to the empirical
(sample) ones, using
p
diﬀerent points
6 / 31
Sample quantiles ˆ
π
q
k
may not be uniquely deﬁned due to its
discreteness. Thus we use a midpoint approach.
The smoothed empirical estimate of a
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This note was uploaded on 02/08/2012 for the course ACTSC 432 taught by Professor Davidlandriault during the Fall '09 term at Waterloo.
 Fall '09
 davidlandriault

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