evt0 (version 1.1.5)

mop: Mean of order p statistic for the extreme value index

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

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

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.

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").

Author

B G Manjunath , Frederico Caeiro

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 $$A(k)= ( \frac{1}{k} \sum^k_{i=1} U^p_{ik} )^{1/p},$$ 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
$$RBA(k)=H_p(k) (1- \frac{\beta (1-p H_p(k) )}{1-\rho-p H_p(k)} (\frac{n}{k})^\rho ).$$

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

# 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")