Various parametric continuous probability models have been presented and discussed in this blog. The number of parameters in these models ranges from one to two, and in a small number of cases three. They are all potential candidates for models of severity in insurance applications and in other actuarial applications. This post highlights these models. The list presented here is not exhaustive; it is only a brief catalog. There are other models that are also suitable for actuarial applications but not accounted for here. However, the list is a good place to begin. This post also serves a navigation device (the table shown below contains links to the blog posts).
Many of the models highlighted here are related to gamma distribution either directly or indirectly. So the catalog starts with the gamma distribution at the top and then branches out to the other related models. Mathematically, the gamma distribution is a two-parameter continuous distribution defined using the gamma function. The gamma sub family includes the exponential distribution, Erlang distribution and chi-squared distribution. These are distributions that are gamma distributions with certain restrictions on the one or both of the gamma parameters. Other distributions are obtained by raising a distribution to a power. Others are obtained by mixing distributions.
Here’s a listing of the models. Click on the links to find out more about the distributions.
|Gamma sub families|
|Independent sum of gamma|
|Raising to a power||Raising exponential to a positive power|
|Raising exponential to a power|
|Raising gamma to a power|
|Raising Pareto to a power|
|Burr sub families|
The above table categorizes the distributions according to how they are mathematically derived. For example, the gamma distribution is derived from the gamma function. The Pareto distribution is mathematically an exponential-gamma mixture. The Burr distribution is a transformed Pareto distribution, i.e. obtained by raising a Pareto distribution to a positive power. Even though these distributions can be defined simply by giving the PDF and CDF, knowing how their mathematical origins informs us of the specific mathematical properties of the distributions. Organizing according to the mathematical origin gives us a concise summary of the models.
Further Comments on the Table
From a mathematical standpoint, the gamma distribution is defined using the gamma function.
In this above integral, the argument is a positive number. The expression in the integrand is always positive. The area in between the curve and the x-axis is . When this expression is normalized, i.e. divided by , it becomes a density function.
The above function is defined over all positive . The integral of over all positive is 1. Thus is a density function. It only has one parameter, the , which is the shape parameter. Adding the scale parameter making it a two-parameter distribution. The result is called the gamma distribution. The following is the density function.
Both parameters and are positive real numbers. The first parameter is the shape parameter and is the scale parameter.
As mentioned above, many of the distributions listed in the above table is related to the gamma distribution. Some of the distributions are sub families of gamma. For example, when are positive integers, the resulting distributions are called Erlang distribution (important in queuing theory). When , the results are the exponential distributions. When and where is a positive integer, the results are the chi-squared distributions (the parameter is referred to the degrees of freedom). The chi-squared distribution plays an important role in statistics.
Taking independent sum of independent and identically distributed exponential random variables produces the Erlang distribution, a sub gamma family of distribution. Taking independent sum of exponential random variables, with pairwise distinct means, produces the hypoexponential distributions. On the other hand, the mixture of independent exponential random variables produces the hyperexponential distribution.
The Pareto distribution (Pareto Type II Lomax) is the mixture of exponential distributions with gamma mixing weights. Despite the connection with the gamma distribution, the Pareto distribution is a heavy tailed distribution. Thus the Pareto distribution is suitable for modeling extreme losses, e.g. in modeling rare but potentially catastrophic losses.
As mentioned earlier, raising a Pareto distribution to a positive power generates the Burr distribution. Restricting the parameters in a Burr distribution in a certain way will produces the paralogistic distribution. The table indicates the relationships in a concise way. For details, go into the blog posts to get more information.
Another informative way to categorize the distributions listed in the table is through looking at the tail weight. At first glance, all the distributions may look similar. For example, the distributions in the table are right skewed distributions. Upon closer look, some of the distributions put more weights (probabilities) on the larger values. Hence some of the models are more suitable for models of phenomena with significantly higher probabilities of large or extreme values.
When a distribution significantly puts more probabilities on larger values, the distribution is said to be a heavy tailed distribution (or said to have a larger tail weight). In general tail weight is a relative concept. For example, we say model A has a larger tail weight than model B (or model A has a heavier tail than model B). However, there are several ways to check for tail weight of a given distribution. Here are the four criteria.
|Tail Weight Measure||What to Look for|
|1||Existence of moments||The existence of more positive moments indicates a lighter tailed distribution.|
|2||Hazard rate function||An increasing hazard rate function indicates a lighter tailed distribution.|
|3||Mean excess loss function||An increasing mean excess loss function indicates a heavier tailed distribution.|
|4||Speed of decay of survival function||A survival function that decays rapidly to zero (as compared to another distribution) indicates a lighter tailed distribution.|
Existence of moments
For a positive real number , the moment is defined by the integral where is the density function of the distribution in question. If the distribution puts significantly more probabilities in the larger values in the right tail, this integral may not exist (may not converge) for some . Thus the existence of moments for all positive is an indication that the distribution is a light tailed distribution.
In the above table, the only distributions for which all positive moments exist are gamma (including all gamma sub families such as exponential), Weibull, lognormal, hyperexponential, hypoexponential and beta. Such distributions are considered light tailed distributions.
The existence of positive moments exists only up to a certain value of a positive integer is an indication that the distribution has a heavy right tail. All the other distributions in the table are considered heavy tailed distribution as compared to gamma, Weibull and lognormal. Consider a Pareto distribution with shape parameter and scale parameter . Note that the existence of the Pareto higher moments is capped by the shape parameter . If the Pareto distribution is to model a random loss, and if the mean is infinite (when ), the risk is uninsurable! On the other hand, when , the Pareto variance does not exist. This shows that for a heavy tailed distribution, the variance may not be a good measure of risk.
Hazard rate function
The hazard rate function of a random variable is defined as the ratio of the density function and the survival function.
The hazard rate is called the force of mortality in a life contingency context and can be interpreted as the rate that a person aged will die in the next instant. The hazard rate is called the failure rate in reliability theory and can be interpreted as the rate that a machine will fail at the next instant given that it has been functioning for units of time.
Another indication of heavy tail weight is that the distribution has a decreasing hazard rate function. On the other hand, a distribution with an increasing hazard rate function has a light tailed distribution. If the hazard rate function is decreasing (over time if the random variable is a time variable), then the population die off at a decreasing rate, hence a heavier tail for the distribution in question.
The Pareto distribution is a heavy tailed distribution since the hazard rate is (Pareto Type I) and (Pareto Type II Lomax). Both hazard rates are decreasing function.
The Weibull distribution is a flexible model in that when its shape parameter is , the Weibull hazard rate is decreasing and when , the hazard rate is increasing. When , Weibull is the exponential distribution, which has a constant hazard rate.
The point about decreasing hazard rate as an indication of a heavy tailed distribution has a connection with the fourth criterion. The idea is that a decreasing hazard rate means that the survival function decays to zero slowly. This point is due to the fact that the hazard rate function generates the survival function through the following.
Thus if the hazard rate function is decreasing in , then the survival function will decay more slowly to zero. To see this, let , which is called the cumulative hazard rate function. As indicated above, . If is decreasing in , has a lower rate of increase and consequently has a slower rate of decrease to zero.
In contrast, the exponential distribution has a constant hazard rate function, making it a medium tailed distribution. As explained above, any distribution having an increasing hazard rate function is a light tailed distribution.
The mean excess loss function
The mean excess loss is the conditional expectation . If the random variable represents insurance losses, mean excess loss is the expected loss in excess of a threshold conditional on the event that the threshold has been exceeded. Suppose that the threshold is an ordinary deductible that is part of an insurance coverage. Then is the expected payment made by the insurer in the event that the loss exceeds the deductible.
Whenever is an increasing function of the deductible , the loss is a heavy tailed distribution. If the mean excess loss function is a decreasing function of , then the loss is a lighter tailed distribution.
The Pareto distribution can also be classified as a heavy tailed distribution based on an increasing mean excess loss function. For a Pareto distribution (Type I) with shape parameter and scale parameter , the mean excess loss is , which is increasing. The mean excess loss for Pareto Type II Lomax is , which is also decreasing. They are both increasing functions of the deductible ! This means that the larger the deductible, the larger the expected claim if such a large loss occurs! If the underlying distribution for a random loss is Pareto, it is a catastrophic risk situation.
In general, an increasing mean excess loss function is an indication of a heavy tailed distribution. On the other hand, a decreasing mean excess loss function indicates a light tailed distribution. The exponential distribution has a constant mean excess loss function and is considered a medium tailed distribution.
Speed of decay of the survival function to zero
The survival function captures the probability of the tail of a distribution. If a distribution whose survival function decays slowly to zero (equivalently the cdf goes slowly to one), it is another indication that the distribution is heavy tailed. This point is touched on when discussing hazard rate function.
The following is a comparison of a Pareto Type II survival function and an exponential survival function. The Pareto survival function has parameters ( and ). The two survival functions are set to have the same 75th percentile, which is . The following table is a comparison of the two survival functions.
Note that at the large values, the Pareto right tails retain much more probabilities. This is also confirmed by the ratio of the two survival functions, with the ratio approaching infinity. Using an exponential distribution to model a Pareto random phenomenon would be a severe modeling error even though the exponential distribution may be a good model for describing the loss up to the 75th percentile (in the above comparison). It is the large right tail that is problematic (and catastrophic)!
Since the Pareto survival function and the exponential survival function have closed forms, We can also look at their ratio.
In the above ratio, the numerator has an exponential function with a positive quantity in the exponent, while the denominator has a polynomial in . This ratio goes to infinity as .
In general, whenever the ratio of two survival functions diverges to infinity, it is an indication that the distribution in the numerator of the ratio has a heavier tail. When the ratio goes to infinity, the survival function in the numerator is said to decay slowly to zero as compared to the denominator.
It is important to examine the tail behavior of a distribution when considering it as a candidate for a model. The four criteria discussed here provide a crucial way to classify parametric models according to the tail weight.
The notion of mixtures is discussed in this previous post. Many probability distributions useful for actuarial modeling are mixture distributions. The previous post touches on some examples – negative binomial distribution (a Poisson-Gamma mixture), Pareto distribution (an exponential-gamma mixture) and the normal-normal mixture. In this post we present additional examples. We discuss the following examples.
- Poisson-Gamma mixture = Negative Binomial.
- Normal-Normal mixture = Normal.
- Exponential-Gamma mixture = Pareto.
- Exponential-Inverse Gamma mixture = Pareto.
- Gamma-Gamma mixture = Generalized Pareto.
- Weibull-Exponential mixture = Loglogistic.
- Gamma-Geometric mixture = Exponential.
- Normal-Gamma mixture = Student t.
The first three examples are discussed in the previous post. We discuss the remaining examples in this post.
The Pareto Family
Examples 3 and 4 show that Pareto distributions are mixtures of exponential distributions with either gamma or inverse gamma mixing weights. In Example 3, is an exponential distribution with being a rate parameter. When follows a gamma distribution, the resulting mixture is a (Type I Lomax) Pareto distribution. In Example 4, is an exponential distribution with being a scale parameter. When follows an inverse gamma distribution, the resulting mixture is also a (Type I Lomax) Pareto distribution.
As a mixture, Example 5 is like Example 3, except that it is a gamma-gamma mixture resulting in a generalized Pareto distribution. Example 3 has been discussed in the previous post. We now discuss Example 4 and Example 5.
The following gives the cumulative distribution function (CDF) and survival function of the conditional random variable .
The random parameter follows an inverse gamma distribution with parameters and . The following is the pdf of :
We show that the unconditional survival function for is the survival function for the Pareto distribution with parameters (shape parameter) and (scale parameter).
Note that the the integrand in the last integral is a density function for an inverse gamma distribution. Thus the integral is 1 and can be eliminated. The result that remains is the survival function for a Pareto distribution with parameters and . The following gives the CDF and density function of this Pareto distribution.
See here for further information on Pareto Type I Lomax distribution.
Conditional on , the following is the density function of .
The following is the density function of the random parameter .
The following gives the unconditional density function for .
Any distribution that has a density function described above is said to be a generalized Pareto distribution with the parameters , and . Its CDF cannot be written in closed form but can be expressed using the incomplete beta function.
The moments can be easily derived for the generalized Pareto distribution but on a limited basis. Since it is a mixture distribution, the unconditional mean is the weighted average of the conditional means.
Note that has a simple expression when .
When the parameter , the conditional distribution for is an exponential distribution. Then the situation reverts back to Example 3, leading to a Pareto distribution. Thus the Pareto distribution is a special case of the generalized Pareto distribution. Both the Pareto distribution and the generalized Pareto distribution have thicker and longer tails than the original conditional gamma distribution.
It turns out that the F distribution is also a special case of the generalized Pareto distribution. The F distribution with and degrees of freedom is the generalized Pareto distribution with parameters , and . As a result, the following is the density function.
Another way to generate the F distribution is from taking a ratio of two chi-squared distributions (see Theorem 9 in this previous post). Of course, there is no need to use the explicit form of the density function of the F distribution. In a statistical application, the F distribution is accessed using tables or software.
The Loglogistic Distribution
The loglogistic distribution can be derived as a mixture of Weillbull distribution with exponential mixing weights.
The following gives the conditional survival function for and the exponential mixing weight.
The following gives the unconditional survival function and CDF of as well as the PDF.
Any distribution that has any one of the above three distributional quantities is said to be a loglogistic distribution with shape parameter and scale parameter .
One interesting point about loglogistic distribution that an inverse loglogistic distribution is another loglogistic distribution. Suppose that has a loglogistic distribution with shape parameter and scale parameter . Let . Then has a loglogistic distribution with shape parameter and scale parameter .
The above is a survival function for the loglogistic distribution with the desired parameters. Thus there is no need to specially call out the inverse loglogistic distribution.
In order to find the mean and higher moments of the loglogistic distribution, we take the approach of identifying the conditional Weibull means and the weight these means by the exponential mixing weights. Note that the parameter in the conditional CDF is not a scale parameter. The Weibull distribution in this conditional CDF is equivalent to a Weibull distribution with shape parameter and scale parameter . According to formula (4) in this previous post, the th moment of this Weillbull distribution is
The following gives the unconditional th moment of the Weibull-exponential mixure.
The range follows from the fact that the arguments of the gamma function must be positive. Thus the th moments of the loglogistic distribution are limited by its shape parameter . If , then does not exist. For a larger , more moments exist but always a finite number of moments. This is an indication that the loglogistic distribution has a thick (right) tail. This is not surprising since mixture distributions (loglogistic in this case) tend to have thicker tails than the conditional distributions (Weibull in this case). The thicker tail is a result of the uncertainty in the random parameter in the conditional distribution (the Weibull in this case).
Another Way to Obtain Exponential Distribution
We now consider Example 7. The following is a precise statement of the gamma-geometric mixture.
The conditional gamma distribution has an uncertain shape parameter that can take on positive integers. The parameter follows a geometric distribution. Here’s the ingredients that go into the mixture.
The following is the unconditional probability density function of .
The above density function is that of an exponential distribution with rate parameter .
Student t Distribution
Example 3 (discussed in the previous post) involves a normal distribution with a random mean. Example 8 involves a normal distribution with mean 0 and an uncertain variance, which follows a gamma distribution such that the two gamma parameters are related to a common parameter , which will be the degrees of freedom of the student t distribution. The following is a precise description of the normal-gamma mixture.
The following gives the ingredients of the normal-gamma mixture. The first item is the conditional density function of given . The second is the density function of the mixing weight .
The following calculation derives the unconditional density function of .
The above density function is in terms of the two parameters and . In the assumptions, the two parameters are related to a common parameter such that and . The following derivation converts to the common .
The above density function is that of a student t distribution with degrees of freedom. Of course, in performing test of significance, the t distribution is accessed by using tables or software. A usual textbook definition of the student t distribution is the ratio of a normal distribution and a chi-squared distribution (see Theorem 6 in this previous post.