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