rho_momt_pick: Optimal rho for moment and Pickands frontier estimator
In npbr: Nonparametric Boundary Regression
View source: R/rho_momt_pick.r
rho_momt_pick 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 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.
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
dfs_momt, dfs_pick
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)
npbr documentation built on March 31, 2023, 7:45 p.m.
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View source: R/rho_momt_pick.r
| rho_momt_pick | 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
dfs_momt, dfs_pick
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)
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