Extreme Value Theory : a New Characterization of the Distribution Function for the Mixed Method

Consider the sample X1, X2, ... , XN of N independent and identically distributed (iid) random variables with common cumulative distribution function (cdf) F, and let Fu be their conditional excess distribution function. We define the ordered sample by X1 ≤ X2 ≤ ⋯ ≤ XN. Pickands (1975), Balkema and de Haan (1974) posed that for a large class of underlying distribution functions F, and large u, Fu is well approximated by the Generalized Pareto Distribution. The mixed method is a method for determining thresholds. This method consists in minimizing the variance of a convex combination of other thresholds. The objective of the mixed method is to determine by which probability distribution one can approach this conditional distribution. In this article, we propose a theorem which specifies the conditional distribution of excesses when the deterministic threshold tends to the end point.


Introduction
Pareto distribution is traditionally used by reinsurer's excess of loss mainly because of its good mathematical properties, particularly from the simplicity of the formulas resulting from its application.The new mixed method (MM) was proposed in [1,2,3,4] to determine a threshold  = ∑    =1 +  3  3 with 1 ≤  ≤ 2, at which a unit is declared atypical minimizing the variance of a convex combination of thresholds obtained by the mean excess function and generalized Pareto distribution (extreme quantile were estimated with a probability of 99.9% being an extreme value for the distribution of amounts of sinister with a confidence level of 95%).This method allows a compromi between the GPD method and FME method, between a minimum strategy GPD and maximum strategy FME (Mean Excess Function).It is more correlated with the GPD method and relatively smooth.

Method
This article focuses on two major paragraphs.The first paragraph (see paragraph 3.1) is based on determining a threshold  = ∑    =1 +  3  3 with 1 ≤  ≤ 2 by the mixed method (MM) and last paragraph (see paragraph 3.2) is to determine a distribution function of the laws of the mixed method.Let  3 : the threshold beyond which a unit is declared as extreme, obtained by the GPD function and : the threshold beyond which a unit is declared as extreme, obtained by the mixed method (MM).Let  1 ,  2 , . . .,    random variables (iid) common distribution function .We are looking from the distribution  of  to define a conditional distribution   3 compared to  3 threshold for random variables exceeding this threshold.It defines the excess over the threshold  3 as the set of random variables   defined by: Thus, for large threshold  3 , the law of excess is approximated by a generalized Pareto law: In this article, we will show that: where  ξ,σ(U)  () is the distribution function of the law of the mixed method and U is the threshold beyond which a unit is declared as extreme, obtained by the mixed method (MM).
Theorem Pickands (1975), Balkema and de Haan (1974) assures us that the law of the excess may be approaching a generalized Pareto law.In this article, we will use the theorem Pickands (1975), Balkema and de Haan (1974) to show that the law of the excess can be approached by a law of the mixed method.

Determination of Threshold U By the Mixed Method (MM)
The new mixed method (MM) was proposed in [1,2,3,4] to determine a threshold  = ∑    =1 +  3  3 with 1 ≤  ≤ 2, at which a unit is declared atypical minimizing the variance of a convex combination of thresholds obtained by the mean excess function and generalized Pareto distribution (extreme quantile were estimated with a probability of 99.9% being an extreme value for the distribution of amounts of sinister with a confidence level of 95%).
Let  1 be the threshold beyond which a unit is declared as extreme, obtained by the record values,  2 be the threshold beyond which a unit is declared as extreme, obtained by the mean excess function and  3 the threshold beyond which a unit is declared as extreme, obtained by the GPD function with  1 <  2 <  3 .Let  =   + (1 − )  with 0 <  < 1, minimizes the variance , ,  = 1, 2, 3 and  < .We get. ) Consider the sample  1 ,  2 , … ,   of  independent and identically distributed (iid) random variables.We define the ordered sample by  1 ≤  2 ≤ ⋯ ≤   .Let    ,  = 1,2,3 thresholds obtained by different methods.We consider a statistical series to a variable j U X ,taking the amount  1 ,  2 , … ,   and j U X ,which have been sorted in ascending order:  1 ≤  2 ≤ ⋯ ≤   ≤   ≤ ⋯ ≤   .We consider a statistical series 2 variables  and , taking the amount  1 ,  2 , … ,   and  1 ,  2 , … ,   .Which have been sorted in ascending order:  1 ≤  2 ≤ ⋯ ≤   and  1 ≤  2 ≤ ⋯ ≤   .We write:  The means of X and Y are :

Example 1. Threshold Calculation
The data base provides a sample of 2020 observations for 4 wheel vehicle for personal use during the year 2013.The data come from a Malian insurance company and concern the amounts of claims caused by the insured of a risk class.This file contains only the amounts of claims during the insurance year.U 1 be the threshold beyond which a unit is declared as extreme, obtained by the record values.U 2 be the threshold beyond which a unit is declared as extreme, obtained by the mean excess function.U 3 the threshold beyond which a unit is declared as extreme, obtained by the GPD function and U the threshold beyond which a unit is declared as extreme, obtained by the method MM.
Let  be the number of claims and  1 ,  2 , … ,   the realizations of , which is the random variable representing the amounts of loss.As usual we assume mutual independence of random variables.with  = −0,0

Law (distribution) of The Mixed Method
In this section, we will give the main result of this paper is to write a new law of the mixed method (MM).Let U 3 : the threshold beyond which a unit is declared as extreme, obtained by the GPD function and U: the threshold beyond which a unit is declared as extreme, obtained by the mixed method (MM).Let  1 ,  2 , . . .,    random variables (iid) common distribution function .We are looking from the distribution  of  to define a conditional distribution   3 compared to  3 threshold for random variables exceeding this threshold.It defines the excess over the threshold  3 as the set of random variables   defined by:   =   −  3 for  ∈   3 = { ∈ *1,2, … , +/  >  3 }.It defines the excess over the threshold  3 as the set of random variables   defined by: The objective of the mixed method is to determine by which probability distribution one can approach this conditional distribution.In this article, we propose the following theorem (Theorem 2) which specifies the conditional distribution of excesses when the deterministic threshold tends to the end point   .
Theorem 1 (Pickands (1975), Balkema and de Haan (1974)): Let   3 be the conditional distribution of the excess over a threshold  3 , combined with unknown distribution function .This function F belongs to the domain of attraction of  ξ if and only if there exist a positive function  such Where  ξ,σ( 3 )  is the distribution function of GPD, define by: for  ∈ ,0, (  −  3 )if ξ ≥ 0 and  ∈ 00,  . is the distribution function of mixed method (MM), define by: for  ∈ ,0, (  − )if ξ ≥ 0 and  ∈ 00,  .

Proof:
The conditional distribution   of the excesses above the threshold  with is defined by: for 0 ≤  ≤   − .
This is equivalent to: for  ≥ .
U 1 : be the threshold beyond which a unit is declared as extreme, obtained by the record values, U 2 : be the threshold beyond which a unit is declared as extreme, obtained by the mean excess function, U 3 : the threshold beyond which a unit is declared as extreme, obtained by the GPD function and U: the threshold beyond which a unit is declared as extreme, obtained by the MM function.Let  be the number of claims and  1 ,  2 , … ,   the realizations of , which is the random variable representing the amounts of sinister.
Example 3: Threshold Calculation By the Graphical Method.
Knowledge of parameters (, ξ) allows to determine graphically the threshold  3 by the GPD method and  by MM method (mixed method).To do this, we will write a program on the MAPLE software to determine these thresholds.

Table 1 .
Determination of threshold U by the mixed method (MM)

Table 2 .
Knowing the parameters (, ξ) and the thresholds, we can write the distribution functions of GPD and MM.