Pingback: Introducing the gamma distribution | Topics in Actuarial Modeling, Pingback: Pareto Distribution | Topics in Actuarial Modeling. If the Pareto model is used as a model of income or wealth of individuals, then a higher would mean a smaller proportion of the people are in the higher income brackets (or more people in the lower income ranges). The parameter is a scale parameter and is a shape parameter. Change ), Integrating survival function to calculate the mean « Practice Problems in Actuarial Modeling, Mathematical models for insurance payments – part 1 – policy limit « Practice Problems in Actuarial Modeling, Transformed Pareto distribution | Topics in Actuarial Modeling, Mixing probability distributions | Topics in Actuarial Modeling, Examples of mixtures | Topics in Actuarial Modeling, Pareto Type I versus Pareto Type II « Practice Problems in Actuarial Modeling, More on Pareto distribution | Applied Probability and Statistics, The Pareto distribution | Applied Probability and Statistics, A catalog of parametric severity models | Topics in Actuarial Modeling, Value-at risk and tail-value-at-risk | Topics in Actuarial Modeling. The following is the pdf of . In general, an increasing mean excess loss function is an indication of a heavy tailed distribution. According to [1], there are four ways to look for indication that a distribution is heavy tailed. There is uncertainty in the parameter, which can be viewed as a random variable . The survival function captures the probability of the tail of a distribution. The link to the catalog is found in that blog post. The larger the shape parameter , the more moments that can be calculated. The following is the mean excess loss function of the Pareto distribution: Note that the Pareto mean excess loss function is a linear increasing function of the deductible . Change ), You are commenting using your Facebook account. So within the Pareto family, a lower means a distribution with a heavier tail and a larger means a lighter tail. The Pareto survival function has parameters ( and ). This blog post introduces a catalog of many other parametric severity models in addition to Pareto distribution. The gamma distribution has been written extensively in this blog. On the other hand, shifting by a constant does not change the variance. Then is the variance. In general, the th moment exists only when the shape parameter is greater than . 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. For a given random variable , the existence of all moments , for all positive integers , indicates with a light (right) tail for the distribution of . However it is conditional one since the parameter is uncertain. Thus the shape parameter is called the Pareto index, which is a measure of the breath of income/wealth. On the other hand, when , the Pareto variance does not exist. Let’s examine the graphs of Pareto survival functions and CDFs. Since it is a heavy tailed distribution, it is a good candidate for modeling income above a theoretical value and the distribution of insurance claims above a threshold value. Pareto Distributions. All the density curves in Figure 1 are skewed to the right and have a long tail. Change ), You are commenting using your Google account. Letting the first relation subtract the second gives the following: Raising the natural log constant to each side gives the following: Note that the left hand side of the last equation is the proportion of the people having income greater than , which of course is the survival function described at the beginning. For an actuarial perspective, refer to the text Loss Models. For example, Lomax is Type I shifted to the left by the amount . In the following discussion, will denote the Pareto distribution as defined above. The following gives the definition for these distributional quantities. For a continuous random variable, it is computed by. In contrast, the exponential distribution has a constant hazard rate function, making it a medium tailed distribution. In comparing the tail weight, it is equivalent to consider the ratio of density functions (due to the L’Hopital’s rule).