Abstract
Let {Xn,n⩾1}
be a sequence of independent random variables with common continuous
distribution function F having finite upper endpoint. A new tail index
estimator γ^
n
is defined based on only two numbers of near m-extremes
Kn(ai,m)=\# {j:X(n−m+:1)−ai<Xj⩽X(n−m+1:n)},m⩾1
where X (i:n) denotes the -th order statistic and a 2 > a 1 > 0. The weak and almost sure convergence of γ^ n to the tail index γ, as well as the asymptotic distribution is given. Moreover, the asymptotic distribution of K n (a n , m) for a n → 0 is derived.
Kn(ai,m)=\# {j:X(n−m+:1)−ai<Xj⩽X(n−m+1:n)},m⩾1
where X (i:n) denotes the -th order statistic and a 2 > a 1 > 0. The weak and almost sure convergence of γ^ n to the tail index γ, as well as the asymptotic distribution is given. Moreover, the asymptotic distribution of K n (a n , m) for a n → 0 is derived.
Original language | English |
---|---|
Pages (from-to) | 197-210 |
Number of pages | 14 |
Journal | Methodology and Computing in Applied Probability |
Volume | 5 |
Issue number | 2 |
DOIs | |
Publication status | Published - Jun 2003 |
Externally published | Yes |
Keywords
- extreme value theory
- near extremes
- tail index estimation