Extreme value distributions of mixing two sequences with the same MDA

Area of Science:

• Probability Theory
• Statistics
• Extreme Value Theory

Background:

• Independent random variables X(i) and Y(i) with distribution functions F(X)(x) and F(Y)(x) are considered.
• A combined sequence Z(i,n) is formed by mixing X(i) and Y(i) with probability p(n).

Purpose of the Study:

• To investigate the extreme value distribution G(Z)(x) of a triangular array of these combined random variables Z(1,n), ..., Z(n,n).
• To identify novel forms of extreme value distributions and their properties, specifically focusing on max-stability.

Main Methods:

• Analysis of a triangular array of independent and identically distributed (i.i.d.) random variables Z(i,n).
• Derivation and characterization of the limiting extreme value distributions for the sequence Z(n).

Main Results:

• Discovery of new extreme value distribution forms: Phi(alpha(1))(A)(x)Phi(alpha(2))(x) and Psi(alpha(1))(A)(x)Psi(alpha(2))(x), where alpha(1) < alpha(2).
• These newly found distributions are demonstrated to be not max-stable.
• The occurrence of these distributions is contingent upon F(X) and F(Y) belonging to the same class of the domain of attraction of the Gumbel (MDA(Phi)) or Fisher-Tippett (MDA(Psi)) extreme value distributions.

Conclusions:

• The study contributes new non-max-stable extreme value distributions arising from mixtures of independent random variables.
• The findings highlight the importance of the domain of attraction of the underlying distributions in determining the properties of the extreme value distributions of combined sequences.