npbr (version 1.8)

kopt_momt_pick: Optimal \(k\) in moment and Pickands frontier estimators

Description

This function gives the optimal sample fraction k in the moment and Pickands type of estimators introduced by Daouia, Florens and Simar (2010).

Usage

kopt_momt_pick(xtab, ytab, x, rho, method="moment", wind.coef=0.1)

Value

Returns a numeric vector with the same length as x.

Arguments

xtab

a numeric vector containing the observed inputs \(x_1,\ldots,x_n\).

ytab

a numeric vector of the same length as xtab containing the observed outputs \(y_1,\ldots,y_n\).

x

a numeric vector of evaluation points in which the estimator is to be computed.

rho

a numeric vector of the same length as x or a scalar, which determines the values of rho.

method

a character equal to "moment" or "pickands".

wind.coef

a scalar coefficient to be selected in the interval (0,1].

Author

Abdelaati Daouia and Thibault Laurent (converted from Leopold Simar's Matlab code).

Details

This function is an implementation of an experimental method by Daouia et al. (2010) for the automated threshold selection (choice of \(k=k_n(x)\)) for the moment frontier estimator \(\tilde\varphi_{momt}(x)\) [see dfs_momt] in case method="moment" and for the Pickands frontier estimator \(\hat\varphi_{pick}(x)\) [see dfs_pick] in case method="pickands". The idea is to select first (for each \(x\)) a grid of values for the sample fraction \(k_n(x)\) given by \(k = 1, \cdots, [\sqrt{N_x}]\), where \([\sqrt{N_x}]\) stands for the integer part of \(\sqrt{N_x}\) with \(N_x=\sum_{i=1}^n1_{\{x_i\le x\}}\), and then select the \(k\) where the variation of the results is the smallest. To achieve this here, Daouia et al. (2010) compute the standard deviations of \(\tilde\varphi_{momt}(x)\) [option method="moment"] or \(\hat\varphi_{pick}(x)\) [option method="pickands"] over a ``window'' of size \(\max(3, [ wind.coef \times \sqrt{N_x} /2])\), where the coefficient wind.coef should be selected in the interval \((0,1]\) in such a way to avoid numerical instabilities. The default option wind.coef=0.1 corresponds to having a window large enough to cover around \(10\%\) of the possible values of \(k\) in the selected range of values for \(k_n(x)\). The value of \(k\) where the standard deviation is minimal defines the desired sample fraction \(k_n(x)\).

References

Daouia, A., Florens, J.P. and Simar, L. (2010). Frontier Estimation and Extreme Value Theory, Bernoulli, 16, 1039-1063.

Dekkers, A.L.M., Einmahl, J.H.J. and L. de Haan (1989), A moment estimator for the index of an extreme-value distribution, Annals of Statistics, 17, 1833-1855.

See Also

dfs_momt, dfs_pick.

Examples

data("post")
x.post<- seq(post$xinput[100],max(post$xinput), 
 length.out=100) 
# When rho[x] is known and equal to 2, we set:
rho<-2
# a. Optimal k in Pickands frontier estimators
best_kn.pick<-kopt_momt_pick(post$xinput, post$yprod, 
 x.post, method="pickands", rho=rho)
# b. Optimal k in moment frontier estimators
if (FALSE) {
best_kn.momt<-kopt_momt_pick(post$xinput, post$yprod, 
 x.post, rho=rho)
}