Abstract
We consider a problem from stock market modeling, precisely, choice of adequate distribution of modeling extremal behavior of stock market data. Generalized extreme value (GEV) distribution and generalized Pareto (GP) distribution are the classical distributions for this problem. However, from 2004, [1] and many other researchers have been empirically showing that generalized logistic (GL) distribution is a better model than GEV and GP distributions in modeling extreme movement of stock market data. In this paper, we show that these results are not accidental. We prove the theoretical importance of GL distribution in extreme value modeling. For proving this, we introduce a general multivariate limit theorem and deduce some important multivariate theorems in probability as special cases. By using the theorem, we derive a limit theorem in extreme value theory, where GL distribution plays central role instead of GEV distribution. The proof of this result is parallel to the proof of classical extremal types theorem, in the sense that, it possess important characteristic in classical extreme value theory, for e.g. distributional property, stability, convergence and multivariate extension etc.
Highlights
An important problem from the field of stock market modeling is the determination of adequate model for extreme stock movement
From 2004, [1] and many other researchers have been empirically showing that generalized logistic (GL) distribution is a better model than Generalized extreme value (GEV) and generalized Pareto (GP) distributions in modeling extreme movement of stock market data
From 2004, [1] and many other researchers have been empirically showing that generalized logistic (GL) distribution is a better model than GEV and GP in modeling extremal behavior of different stock market data, this includes US, UK, Germany, Japan, India, Athens, African stock markets etc
Summary
An important problem from the field of stock market modeling is the determination of adequate model for extreme stock movement. The asymptotic distribution for minima is the GEV(min) distribution defined in Definition 1.2 These theories have been extended to multivariate case, where multivariate generalized extreme value distributions play central role (see [13,14]). Suppose there exists a pair of sequences an and bn with an 0 for all n and a non-degenerate distribution function G x such that lim P n This theory extended to multivariate case, where multivariate generalized Pareto distribution plays central role (see [17]). Extremal types theorem (see [18]) and Peak over threshold theorem ([15,16]) facilitate a theoretical background to use generalized extreme value (GEV) distributions and GP distribution in modeling extreme movement of stock market indices. Definition 1.7 Two distributions F and G are equivalent in their right tail if they have the same right end point, i.e. xF xG , and lim x xF
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