| rho_momt_pick {npbr} | R Documentation |
Optimal rho for moment and Pickands frontier estimator
Description
This function gives the optimal rho involved in the moment and Pickands estimators of Daouia, Florens and Simar (2010).
Usage
rho_momt_pick(xtab, ytab, x, method="moment", lrho=1, urho=Inf)
Arguments
xtab |
a numeric vector containing the observed inputs |
ytab |
a numeric vector of the same length as |
x |
a numeric vector of evaluation points in which the estimator is to be computed. |
method |
a character equal to "moment" or "pickands". |
lrho |
a scalar, minimum rho threshold value. |
urho |
a scalar, maximum rho threshold value. |
Details
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).
Value
Returns a numeric vector with the same length as x.
Note
In order to choose a raisonable estimate \tilde\rho_x=\tilde\rho_x(k) and
\hat\rho_x=\hat\rho_x(k) of the extreme-value index \rho_x,
for each fixed x, one can construct the plot of the estimator of interest, consisting of the points \{(k,\tilde\rho_x(k))\}_k or
\{(k,\hat\rho_x(k))\}_k, and select a value of the estimate at which the obtained graph looks stable. This is this kind of idea
which guides the propoed automatic data-driven rule for a chosen grid of values of x. The main difficulty with such a method is that the plots of
\tilde\rho_x(k) or \hat\rho_x(k) as functions of k, for each x, may be so unstable that reasonable values of
k [which would correspond to the true value of \rho_x] may be hidden in the graphs. In results, the obtained extreme-value index estimator and the frontier estimator itself may
exhibits considerable volatility as functions of x. The user can appreciably improve the estimation of \rho_x and \varphi(x)
by tuning the choice of the lower limit (default option lrho=1) and upper limit (default option urho=Inf).
Author(s)
Abdelaati Daouia and Thibault Laurent (codes converted from Matlab's Leopold Simar code).
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, The Annals of Statistics, 17(4), 1833-1855.
See Also
Examples
data("post")
x.post<- seq(post$xinput[100],max(post$xinput),
length.out=100)
## Not run:
# 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")
## End(Not run)