Προσεγγίσεις κατανομών Pareto από συνδυασμούς εκθετικών, κατανομών με εφαρμογές στην θεωρία κινδύνων
Approximations of Pareto distributions by combinations of exponential distributions, with applications in risk theory
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Keywords
Pareto ; Βαριά ουρά ; Μείξη εκθετικών κατανομών ; Κατανομή ισορροπίας ; Κλιμακωτά ύψη ; ΧρεοκοπίαAbstract
In actuarial science in order to assess risks in insurance and financial sectors, distributions with heavy tail are of particular interest. One of the widely known distributions with a heavy tail is the Pareto distribution that we will analyze in the first chapter and occupies a leading position in this diploma thesis.
In addition in the second chapter we will describe and analyze a very basic measure called failure rate or hazard rate that expresses the instantaneous number of system failures or the instantaneous number of deaths in the case of survival analysis and is a key pillar for designing secure systems in financial, trade and regulatory sectors. In the third chapter, through various methods, we will try to approach the Pareto distribution by bexponential distributions and through many criteria we will evaluate how good or not it is.
Approximating the Pareto distribution by mixed exponential distributions is a very useful tool because there are instances such as calculating the probability of ruin where it is not possible to calculate it for the Pareto distribution and in this way we may approach it through its approximation.
In the fourth chapter we will define the concept of equilibrium distribution and we will calculate it for the initial distribution of Pareto and the mixture of exponentials. We will also calculate the error rate using equilibrium distributions and compare the results with our initial distribution. We will also formulate and look at the concept of residual lifetime by calculating its mean value and its variance.
Last but not least in the fifth chapter we will introduce the notion of ruin for the collective model of risk theory. We study it both from a theoretical perceptive, but also compute the key quantities in mathematical terms and we discuss the basic link of the equilibrium distribution with ruin theory, showing that it is the distribution of the ladder heights.