A Weighted Estimation for Risk Model

We propose a weighted estimation method for risk models. Two examples of natural disasters are studied: hurricane loss in the USA and forest fire loss in Canada. Risk data is often fitted by a heavy-tailed distribution, for example, a Pareto distribution, which has many applications in economics, actuarial science, survival analysis, networks, and other stochastic models. There is a difficulty in the inference of the Pareto distribution which has infinite moments in the heavy-tailed case. Firstly this paper applies the truncated Pareto distribution to overcome this difficulty. Secondly, we propose a weighted semiparametric method to estimate the truncated Pareto distribution. The idea of the new method is to place less weight on the extreme data values. This paper gives an exact efficiency function, -optimal weights and -optimal weights of the new estimator. Monte Carlo simulation results confirm the theoretical conclusions. The two above mentioned examples are analyzed by using the proposed method. This paper shows that the new estimation method is more efficient by mean square error relative to several existing methods and fits risk data well. 1. Introduction 1.1. Two Motivating Examples In the recent years, many extreme events have occurred in financial markets, natural disasters, disease control, and industrial quality control. Natural disasters, for example, earthquakes, hurricanes, forest fires, volcanoes, and floods affect human life. It is important to predict and prepare for the next disaster occurrence and to estimate losses to inhabitants, insurance companies, and governments. In this section, we study two examples. 1.1.1. A Hurricane Loss Example Strong winds, heavy rainfall, and storm surges caused by hurricanes cause death and destroy properties. They generate great losses to insurance companies as well. Figure 1 shows the 49 costliest Atlantic hurricane losses for the United States during 1900–2005 [1]. The measurement of this hurricane loss data is in US dollars; all dollars have been adjusted by using the inflation rates from 1900 to 2005. Figure 1: The 49 USA atlantic hurricane loss, 1900–2005. From the data in [1], we note that the most costly hurricane is the 1926 Great Miami Hurricane with cost of damage of 157 billion which is 1.58 times larger than the second worst hurricane, the 1900 Galveston Hurricane. After 1926, on August 28, 2005 Hurricane Katrina caused damage of billion. This is approximately 1.19 times larger than Hurricane Andrew in August 1992, which caused damage of billion. On the other hand, we note that 80% of the