Abstract
Asymptotic results are obtained for several conditional measures of association. The chosen random variables are the first two order statistics and the total sum within a random sum. Many of the results have confirmed the “one-jump” property of the risk model. Non-trivial limits are obtained when the dependence among the first two order statistics is considered. Our results help in understanding the extreme behaviour of well-known reinsurance treaties that involve only few large claims. Interestingly, the Pearson product-moment correlation coefficient between the first two order statistics provides an alternative procedure to estimate tail index of the underlying distribution.
Highlights
Let X1, · · ·, Xn be independent and identically distributed random variables with distribution function F (·), tail function F = 1 − F and infinite right-end point.Extreme Value Theory (EVT) assumes that there are constants an > 0, bn ∈ R such that lim Pr n→∞ an max 1≤i≤n Xi − bn ≤x= G(x), for all x.G is called an Extreme Value Distribution and F is said to belong to the domain of attraction of G
Which is the same as Spearman’s rho if the marginals are uniform random variables. This third measure of association evaluates the linear correlation between two dependent rv’s, and it has been criticized for its lack of robustness, but is still a well-accepted measure in the presence of linear dependence, which is our case since we are interested only in extreme events that happen to be strongly correlated in the tail
The first step is to establish some preliminary results, which are given in Proposition 3.1
Summary
Kendall’s tau, τ := Pr (X1 − X2)(Y1 − Y2) > 0 − Pr (X1 − X2)(Y1 − Y2) < 0 , and Spearman’s rho rank correlation, ρR := 3 P (X1 − X2)(Y1 − Y3) > 0 − P (X1 − X2)(Y1 − Y3) < 0 , are based on the concordance and discordance probabilities, where (Xi, Yi), i = 1, 2, 3, are three iid copies from (X, Y ) It is well-known that both measures of association are scale-invariant, and robust, marginal-free whenever the marginal distributions are continuous. Which is the same as Spearman’s rho if the marginals are uniform random variables This third measure of association evaluates the linear correlation between two dependent rv’s, and it has been criticized for its lack of robustness, but is still a well-accepted measure in the presence of linear dependence, which is our case since we are interested only in extreme events that happen to be strongly correlated in the tail.
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