# mop: Mean of order p statistic for the extreme value index In evt0: Mean of order p, peaks over random threshold Hill and high quantile estimates

## Description

This function compute mean of order p (MOP) basic statistic for the extreme value index (EVI), which is indeed a simple generalisation of the Hill estimator.

## Usage

 1 mop(x, k, p, method = c("MOP", "RBMOP")) 

## Arguments

 x Data vector. k a vector of number of upper order statistics. p a vector of mean order. method Method used, ("MOP", default) and reduced-bias MOP ("RBMOP").

## Details

Basic statistics for the EVI estimation, the MOP of U_{ik}, where U_{ik}= \frac{X_{n-i+1:n}}{X_{n-k:n}} and X_{i:n} are order statistics, is

A(k)= ( \frac{1}{k} ∑^k_{i=1} U^p_{ik} )^{1/p},

for p \neq 0.

The new class of MOP EVI- estimators is

H_p(k)= (1 - A^{-p}(k))/p,

for p \neq 0. At p=0 the above MOP estimator is equal to classical Hill estimator.

Reduced bias MOP EVI-estimators is

RBA(k)=H_p(k) (1- \frac{β (1-p H_p(k) )}{1-ρ-p H_p(k)} (\frac{n}{k})^ρ ).

## Value

a matrix of EVI estimates, corresponds to k row and p columns. When Method = "RBMOP" shape and scale second order parameters estimates are also returned.

## Author(s)

B G Manjunath [email protected], Frederico Caeiro [email protected]

## References

Brilhante, M.F., Gomes, M.I. and Pestana, D. (2013). A simple generalisation of the Hill estimator. Computational Statistics and Data Analysis, 57, 518– 535.

Beran, J., Schell, D. and Stehlik, M. (2013). The harmonic moment tail index estimator: asymptotic distribution and robustness. Ann Inst Stat Math, Published Online.

Gomes, M.I., Brilhante, M.F. and Pestana, D. (2013). New reduced-bias estimators of a positive extreme value index. Submitted article.

## Examples

 1 2 3 4 5 # generate random samples x = rfrechet(50000, loc = 0, scale = 1,shape = 1/0.5) # estimate EVI mop(x,c(1,500,5000,49999), c(-1,0,1),"RBMOP") 

evt0 documentation built on May 30, 2017, 6:10 a.m.