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 previous post opens up a discussion on generating distributions by raising an existing distribution to a power. The previous post focuses on the example of raising an exponential distribution to a power. This post focuses on the distributions generated by raising a gamma distribution to a power.
Raising to a Power
Let be a random variable. Let be a positive constant. The random variables , and are called transformed, inverse and inverse transformed, respectively.
Let , and be the probability density function (PDF), the cumulative distribution function (CDF) and the survival function of the random variable (the base distribution). The following derivation seeks to express the CDFs of the “transformed” variables in terms of the base CDF .
Unlike the exponential distribution, the CDF of the gamma distribution does not have a closed form. Suppose that is a random variable that has a gamma distribution with shape parameter and scale parameter . The following is the expression of the gamma CDF.
By a change of variable, the CDF can be expressed as the following integral.
Note that and are the gamma function and incomplete gamma function, respectively, defined as follows.
The CDF can be evaluated numerically using software. When the gamma distribution is raised to a power, the resulting CDF will be defined as a function of .
“Transformed” Gamma CDFs
The CDF of the “transformed” gamma distributions does not have a closed form. Thus the CDFs are to be defined based on an integral or the incomplete gamma function, shown in the preceding section. We still use the two-step approach – first deriving the CDF without the scale parameter and then add it at the end. Based on the two preceding sections, the following shows the CDFs of the three different cases.
The transformed gamma distribution and the inverse transformed gamma distribution are three-parameter distributions with being the shape parameter, being the scale parameter and being in the power to which the base gamma distribution is raised. The inverse gamma distribution has two parameters with being the scale parameter and being shape parameter (the same two parameters in the base gamma distribution). Computation of these CDFs would require the use of software that can evaluate incomplete gamma function.
Another Way to Work with “Transformed” Gamma
The CDFs derived in the preceding section is a two-step approach – first raising a gamma distribution with scale parameter equals to 1 to a power and then adding a scale parameter. The end result gives CDFs that are a function of the incomplete gamma function. Calculating a CDF would require using a software that has the capability of evaluating incomplete gamma function (or evaluating an equivalent integral). If the software that is used does not have the incomplete gamma function but has gamma CDF (e.g. Excel), then there is another way of generating the “transformed” gamma CDF.
Observe that the CDFs in the last section are the results of raising a base gamma distribution with shape parameter and (transformed), shape parameter and (inverse) and shape parameter and (inverse transformed). Based on this observation, we can evaluate the CDFs and the moments. This section shows how to evaluate CDFs using this approach. The next section shows how to evaluate the moments.
Transformed Gamma Distribution
Inverse Gamma Distribution
Inverse Transformed Gamma Distribution
Example 1 below uses Excel to compute the transformed gamma CDF.
“Transformed” Gamma Moments
Following the idea in the preceding section, the moments for the “transformed” gamma moments can be derived using gamma moments with the appropriate parameters (see here for the gamma moments). The following table shows the results.
|Base Gamma Parameters||Moments|
Note that the moments for the transformed gamma distribution exists for all positive . The moments for inverse gamma are limited (capped by the shape parameter ). The moments for inverse transformed gamma are also limited, this time limited by both parameters. The computation for these moments is usually done by software, except when the argument of the gamma function is a positive integer.
“Transformed” Gamma PDFs
Once the CDFs are known, the PDFs are derived by taking derivative. The following table gives the three PDFs.
Each PDF is obtained by taking derivative of the integral for the corresponding CDF.
The post is concluded with one example demonstrating the calculation for CDF and percentiles using the gamma distribution function in Excel.
The size of a collision claim from a large pool of auto insurance policies has a transformed gamma distribution with parameters , and . Determine the following.
- The probability that a randomly selected claim is greater than than 6.75.
- The probability that a randomly selected claim is between 4.25 and 8.25.
- The median size of a claim from this pool of insurance policies.
- The mean and variance of a claim from this pool of insurance policies.
- The probability that a randomly selected claim is within one standard deviation of the mean claim size.
- The probability that a randomly selected claim is within two standard deviations of the mean claim size.
Since , this is a transformed gamma distribution. There are two ways to get a handle on this distribution. One way is to plug in the three parameters , and into the transformed gamma CDF (in the table Step 2. “Transformed” Gamma CDF (with scale parameter)). This would require using a software that can evaluate the incomplete gamma function or its equivalent integrals. The other way is to know that this distribution is the result of raising the gamma distribution with shape parameter and scale parameter to the power of 1/2. We take the latter approach.
In Excel, =GAMMA.DIST(A1,B1,C1,TRUE) is the function that produces the gamma CDF , assuming that the value is in cell A1, the shape parameter is in cell B1 and the scale parameter is in cell C1. When the last parameter is TRUE, it gives the CDF. If it is FALSE, it gives the PDF. For example, =GAMMA.DIST(2.5, 2, 1, TRUE) gives the value of 0.712702505.
The transformed gamma distribution in question (random variable ) is the result of raising gamma and scale parameter 16 (random variable ) to 1/2. Thus the CDF is obtained from evaluating the gamma CDF , which is =GAMMA.DIST(, 2.5, 16, TRUE). As a result, the following gives the answers for the first two bullet points.
=GAMMA.DIST(, 2.5, 16, TRUE)
The median or other percentiles of transformed gamma distribution are obtained by a trial and error approach, i.e. by plugging in values of into the gamma CDF =GAMMA.DIST(, 2.5, 16, TRUE). After performing the trial and error process, we see that and . Thus we take the median to be 5.9. So the median size of a claim is around 5.9.
Computation of the moments of a gamma distribution requires the evaluation of the gamma function . Excel does not have an explicit function for gamma function. Since is in the gamma PDF, we can derive the gamma function from the gamma PDF in Excel =GAMMA.DIST(1, a, 1, FALSE). The value of this gamma PDF is . This the value of is divided by the value of this gamma PDF value. For example, the following Excel formulas give and .
=EXP(-1)/GAMMA.DIST(1, 5, 1, FALSE) = 24
=EXP(-1)/GAMMA.DIST(1, 2.5, 1, FALSE) = 1.329340388
The moments is . Based on the results in the section on “Transformed” gamma moments, the following gives the mean and variance.
The following gives the probability of a claim within one or two standard deviations of the mean.