A Cluster Truncated Pareto Distribution and Its Applications

The Pareto distribution is a heavy-tailed distribution with many applications in the real world. The tail of the distribution is important, but the threshold of the distribution is difficult to determine in some situations. In this paper we consider two real-world examples with heavy-tailed observations, which leads us to propose a mixture truncated Pareto distribution (MTPD) and study its properties. We construct a cluster truncated Pareto distribution (CTPD) by using a two-point slope technique to estimate the MTPD from a random sample. We apply the MTPD and CTPD to the two examples and compare the proposed method with existing estimation methods. The results of log-log plots and goodness-of-fit tests show that the MTPD and the cluster estimation method produce very good fitting distributions with real-world data. 1. Introduction There are many real-world problems modelled as heavy-tailed distributions, especially the Pareto distribution [1, 2]. However, there are some difficulties in estimation of Pareto distributions. First, the Pareto distribution has infinite moments in some heavy-tailed cases. Therefore the moment estimation method for the shape parameter cannot be used in these situations. It is a loss for the estimation process since the moment estimator is a robust estimator. Several authors suggest using a truncated Pareto distribution which always has finite moments (e.g., [3–5]). In some situations, data will behave differently within different thresholds. For example, losses from hurricane damage can be classified into small, medium, and large hurricane groups. The data in these classes may have different distributions, or by grouping, data with self-similarity may have the same kind of distribution but with different parameters. A cluster method for data is needed to determine these groups when dealing with real data sets. In this paper we study an example of 49 most damaging Atlantic hurricanes occurring between years 1900 and 2005 [6]. The costs are standardized to 2005 USD; see Figure 1. Figure 1: The 49 costliest Atlantic hurricanes between the years 1900 and 2005. Coia and Huang [7] applied Pareto and truncated Pareto models to fit the hurricane data set. The maximum likelihood estimator (MLE) and the moment estimator for the shape parameter were used. The results are shown in a log-log plot in Figure 2. Coia and Huang [7] also used Kolmogorov-Smirnov, Anderson-Darling, and Cramer-von-Mises goodness-of-fit tests. We note that the two estimated (by MLE and moment method) truncated Pareto curves fit the data set quite well; they fit