It is often desirable to find an explicit analytical expression for a loss distribution. This is particularly the case if the claim statistics are too sparse to use the empirical approach. It should be stressed, however, that many standard models in statistics - like the Gaussian distribution - are unsuitable for fitting the claim size distribution. The main reason for this is the strongly skewed nature of loss distributions. The log-normal, Pareto, Burr, Weibull, and gamma distributions are typical candidates for claim size distributions to be considered in applications.
Consider a random variable which has the normal distribution with density
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(13.2) |
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(13.3) |
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(13.4) |
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(13.5) |
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(13.6) |
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(13.7) |
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(13.8) |
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(13.9) |
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(13.10) |
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(13.11) |
The log-normal distribution is very useful in modeling of claim sizes. It is right-skewed, has a thick tail and fits many situations well. For
small it resembles a normal distribution (see the left panel in Figure 13.2) although this is not always desirable. It is
infinitely divisible and closed under scale and power transformations. However, it also suffers from some drawbacks. Most notably, the Laplace
transform does not have a closed form representation and the moment generating function does not exist.
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Consider the random variable with the following density and distribution functions, respectively:
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(13.15) |
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(13.16) |
To generate an exponential random variable with intensity
we can use the inverse transform method (Ross; 2002; L'Ecuyer; 2004). The method consists of taking a random number
distributed uniformly on the interval
and setting
, where
is the inverse of the exponential cdf (13.13). In fact we can set
since
has the same distribution as
.
The exponential distribution has many interesting features. For example, it has the memoryless property, i.e.
. It also arises as the inter-occurrence times of the events in a Poisson process, see Chapter 14. The
-th root of the Laplace transform (13.14) is
The exponential distribution is often used in developing models of insurance risks. This usefulness stems in a large part from its many and varied tractable mathematical properties. However, a disadvantage of the exponential distribution is that its density is monotone decreasing (see the right panel in Figure 13.2), a situation which may not be appropriate in some practical situations.
Suppose that a variate has (conditional on
) an exponential distribution with mean
. Further, suppose that
itself
has a gamma distribution (see Section 13.3.6). The unconditional distribution of
is a mixture and is called the Pareto distribution.
Moreover, it can be shown that if
is an exponential random variable and
is a gamma random variable, then
is a Pareto random variable.
The density and distribution functions of a Pareto variate are given by:
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(13.21) |
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(13.25) |
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(13.26) |
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Like for many other distributions the simulation of a Pareto variate can be conducted via the inverse transform method. The inverse of the cdf
(13.20) has a simple analytical form
. Hence, we can set
, where
is distributed uniformly on the unit interval. We have to be cautious, however, when
is
larger but very close to one. The theoretical mean exists, but the right tail is very heavy. The sample mean will, in general, be significantly
lower than
.
The Pareto law is very useful in modeling claim sizes in insurance, due in large part to its extremely thick tail. Its main drawback lies in its lack of mathematical tractability in some situations. Like for the log-normal distribution, the Laplace transform does not have a closed form representation and the moment generating function does not exist. Moreover, like the exponential pdf the Pareto density (13.19) is monotone decreasing, which may not be adequate in some practical situations.
Experience has shown that the Pareto formula is often an appropriate model for the claim size distribution, particularly where exceptionally large claims may occur. However, there is sometimes a need to find heavy tailed distributions which offer greater flexibility than the Pareto law, including a non-monotone pdf. Such flexibility is provided by the Burr distribution and its additional shape parameter . If
has the Pareto distribution, then the distribution of
is known as the Burr distribution, see the left panel in Figure 13.4. Its density and distribution functions are given by:
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(13.29) |
The maximum likelihood and method of moments estimators for the Burr distribution can only be evaluated numerically. A Burr variate can be generated using the inverse transform method. The inverse of the cdf (13.28) has a simple analytical form
. Hence, we can set
, where
is distributed uniformly on the unit interval. Like in the Pareto case, we have to be cautious when
is larger but very close to one. The theoretical mean exists, but the right tail is very heavy. The sample mean will, in general, be significantly lower than
.
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If is an exponential variate, then the distribution of
,
, is called the Weibull (or Frechet) distribution. Its density and distribution functions are given by:
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(13.30) |
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(13.31) |
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(13.32) |
The probability law with density and distribution functions given by:
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The Laplace transform of the gamma distribution is given by:
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(13.36) |
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(13.37) |
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(13.38) |
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(13.39) |
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(13.40) |
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(13.41) |
The gamma distribution is closed under convolution, i.e. a sum of independent gamma variates with the same parameter is again gamma distributed with this
. Hence, it is infinitely divisible.
Moreover, it is right-skewed and approaches a normal distribution in the limit as
goes to infinity.
The gamma law is one of the most important distributions for modeling because it has very tractable mathematical properties. As we have seen above it is also very useful in creating other distributions, but by itself is rarely a reasonable model for insurance claim sizes.
Let
denote a series of non-negative weights satisfying
. Let
denote an
arbitrary sequence of exponential distribution functions given by the parameters
, respectively. Then, the
distribution function:
The Laplace transform of (13.43) is
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(13.44) |
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(13.45) |
Simulation of variates defined by (13.42) can be performed using the composition approach (Ross; 2002). First generate a random variable , equal to
with probability
,
. Then simulate an exponential variate with intensity
. Note, that the method is general in the sense that it can be used for any set of distributions
's.