This function gives the optimal rho involved in the moment and Pickands estimators of Daouia, Florens and Simar (2010).
rho_momt_pick(xtab, ytab, x, method="moment", lrho=1, urho=Inf)Returns a numeric vector with the same length as x.
a numeric vector containing the observed inputs \(x_1,\ldots,x_n\).
a numeric vector of the same length as xtab containing the observed outputs \(y_1,\ldots,y_n\).
a numeric vector of evaluation points in which the estimator is to be computed.
a character equal to "moment" or "pickands".
a scalar, minimum rho threshold value.
a scalar, maximum rho threshold value.
Abdelaati Daouia and Thibault Laurent (codes converted from Matlab's Leopold Simar code).
This function computes the moment and Pickands estimates of the extreme-value index
\(\rho_x\) involved in the frontier estimators \(\tilde\varphi_{momt}(x)\) [see dfs_momt] and
\(\hat\varphi_{pick}(x)\) [see dfs_pick].
In case method="moment", the estimator of \(\rho_x\) defined as
$$\tilde{\rho}_x = -\left(M^{(1)}_n + 1 -\frac{1}{2}\left[1-(M^{(1)}_n)^2/M^{(2)}_n\right]^{-1}\right)^{-1}$$
is based on the moments \(M^{(j)}_n = (1/k)\sum_{i=0}^{k-1}\left(\log z^x_{(n-i)}- \log z^x_{(n-k)}\right)^j\)
for \(j=1,2\), with \(z^{x}_{(1)}\leq \cdots\leq z^{x}_{(n)}\) are the ascending order statistics
corresponding to the transformed sample \(\{z^{x}_i := y_i\mathbf{1}_{\{x_i\le x\}}, \,i=1,\cdots,n\}\)
In case method="pickands", the estimator of \(\rho_x\) is given by
$$\hat{\rho}_x = - \log 2/\log\{(z^x_{(n-k+1)} - z^x_{(n-2k+1)})/(z^x_{(n-2k+1)} - z^x_{(n-4k+1)})\}.$$
To select the threshold \(k=k_n(x)\) in \(\tilde{\rho}_x\) and \(\hat{\rho}_x\), Daouia et al. (2010) have suggested to use the following data driven method for each
\(x\): They first select a grid of values for \(k=k_n(x)\).
For the Pickands estimator \(\hat{\rho}_x\), they choose \(k_n(x) = [N_x /4] - k + 1\), where \(k\) is an integer varying between 1
and the integer part \([N_x/4]\) of \(N_x/4\), with \(N_x=\sum_{i=1}^n1_{\{x_i\le x\}}\).
For the moment estimator \(\tilde{\rho}_x\), they choose \(k_n(x) = N_x - k\), where \(k\) is an integer varying between 1 and \(N_x -1\).
Then, they evaluate the estimator \(\hat{\rho}_x(k)\) (respectively, \(\tilde{\rho}_x(k)\)) and select the k where the variation of the results is the smallest.
They achieve this by computing the standard deviation of \(\hat{\rho}_x(k)\) (respectively, \(\tilde{\rho}_x(k)\)) over a ``window'' of
\(\max([\sqrt{N_x /4}],3)\) (respectively, \(\max([\sqrt{N_x-1}],3)\))
successive values of \(k\). The value of \(k\) where this standard deviation is minimal defines the value of \(k_n(x)\).
The user can also appreciably improve the estimation of \(\rho_x\) and \(\varphi(x)\) itself by tuning the choice of the lower limit (default option lrho=1)
and upper limit (default option urho=Inf).
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, The Annals of Statistics, 17(4), 1833-1855.
dfs_momt, dfs_pick
data("post")
x.post<- seq(post$xinput[100],max(post$xinput),
length.out=100)
if (FALSE) {
# a. Optimal rho for Pickands frontier estimator
rho_pick<-rho_momt_pick(post$xinput, post$yprod,
x.post, method="pickands")
# b. Optimal rho for moment frontier estimator
rho_momt<-rho_momt_pick(post$xinput, post$yprod,
x.post, method="moment")
}
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