2.3.
ESTIMATION FOR PARAMETRIC MODELS
23
This formula is equivalent to assuming that every observation in a group is equal to the group’s
midpoint. For a limit of 125,000 the expected cost is
1
227
[3
,
750(99) + 12
,
500(42) + 25
,
000(29) + 50
,
000(28) + 96
,
250(17) + 125
,
000(12)] = 27
,
125
.
55
.
For a limit of 300,000, the expected cost is
1
227
[3
,
750(99) + 12
,
500(42) + 25
,
000(29) + 50
,
000(28) + 96
,
250(17) + 212
,
500(9) + 300
,
000(3)]
= 32
,
907
.
49
.
The ratio is
32
,
907
.
49
/
27
,
125
.
55 = 1
.
213
, or a 21.3% increase. Note that if the last group has a
boundary of in
fi
nity, the
fi
nal integral will be unde
fi
ned, but the contribution to the sum in the
last two lines is still correct.
¤
Exercise 16
Estimate the variance of the number of accidents using Data Set A assuming that all
seven drivers with
fi
ve or more accidents had exactly
fi
ve accidents.
Exercise 17
Estimate the expected cost per loss and per payment for a deductible of 25,000 and a
maximum payment of 275,000 using Data Set C.
Exercise 18
(*) You are studying the length of time attorneys are involved in settling bodily injury
lawsuits.
T
represents the number of months from the time an attorney is assigned such a case to
the time the case is settled. Nine cases were observed during the study period, two of which were
not settled at the conclusion of the study. For those two cases, the time spent up to the conclusion
of the study, 4 months and 6 months, was recorded instead. The observed values of
T
for the other
seven cases are as follows — 1, 3, 3, 5, 8, 8, 9. Estimate
Pr(3
≤
T
≤
5)
using the KaplanMeier
productlimit estimate.
2.3
Estimation for parametric models
2.3.1
Introduction
If a phenomenon is to be modeled using a parametric model, it is necessary to assign values to
the parameters. This could be done arbitrarily, but it would seem to be more reasonable to base
the assignment on observations from that phenomenon.
In particular, we will assume that
n
independent observations have been collected. For some of the techniques used in this Section it
will be further assumed that all the observations are from the same random variable. For others,
that restriction will be relaxed.
The methods introduced in the next Subsection are relatively easy to implement, but tend to
give poor results. The following Subsection covers maximum likelihood estimation. This method
is more di
ﬃ
cult to use, but has superior statistical properties and is considerably more
fl
exible.
2.3.2
Method of moments and percentile matching
For these methods we assume that all
n
observations are from the same parametric distribution.
In particular, let the distribution function be given by
F
(
x

θ
)
,
θ
T
= (
θ
1
,
θ
2
, . . . ,
θ
p
)
.
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24
CHAPTER 2.
MODEL ESTIMATION
That is,
θ
is a vector containing the
p
parameters to be estimated. Furthermore, let
μ
0
k
(
θ
)
=
E
(
X
k

θ
)
be the
k
th raw moment, and let
π
g
(
θ
)
be the 100
g
th percentile
of the random variable. That is,
F
[
π
g
(
θ
)

θ
] =
g
. For a sample of
n
independent observations from
this random variable, let
ˆ
μ
0
k
=
1
n
n
X
j
=1
x
k
j
be the empirical estimate of the
k
th moment, and let
ˆ
π
g
be the empirical estimate of the
100
g
th percentile.
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 Spring '12
 Arthuringham
 Normal Distribution, Probability theory, Maximum likelihood, maximum likelihood estimate

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