dfs_pick {npbr}

Pickands frontier estimator

Description

This function is an implementation of the Pickands type estimator developed by Daouia, Florens and Simar (2010).

Usage

dfs_pick(xtab, ytab, x, k, rho, ci=TRUE)

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.

k

a numeric vector of the same length as x or a scalar, which determines the thresholds at which the Pickands estimator will be computed.

rho

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

ci

a boolean, TRUE for computing the confidence interval.

Details

Built on the ideas of Dekkers and de Haan (1989), Daouia et al. (2010) propose to estimate the frontier point \varphi(x) by

\hat\varphi_{pick}(x) = \frac{z^x_{(n-k+1)} - z^x_{(n-2k+1)}}{2^{1/\rho_x} - 1} + z^x_{(n-k+1)}

from the transformed data \{z^{x}_i, \,i=1,\cdots,n\} described in dfs_momt, where \rho_x>0 is the same tail-index as in dfs_momt. If \rho_x is known (typically equal to 2 if the joint density of data is believed to have sudden jumps at the frontier), then one can use the estimator \hat\varphi_{pick}(x) in conjunction with the function kopt_momt_pick which implements an automatic data-driven method for selecting the threshold k. In contrast, if \rho_x is unknown, one could consider using the following two-step estimator: First, estimate \rho_x by the Pickands estimator \hat\rho_x implemented in the function rho_momt_pick by using the option method="pickands", or by the moment estimator \tilde\rho_x by utilizing the option method="moment". Second, use the estimator \hat\varphi_{pick}(x), as if \rho_x were known, by substituting the estimated value \hat\rho_x or \tilde\rho_x in place of \rho_x. The pointwise 95\% confidence interval of the frontier function obtained from the asymptotic normality of \hat\varphi_{pick}(x) is given by

[\hat\varphi_{pick}(x) \pm 1.96 \sqrt{v(\rho_x) / (2 k)} ( z^x_{(n-k+1)} - z^x_{(n-2k+1)})]

where v(\rho_x) =\rho^{-2}_x 2^{-2/\rho_x}/(2^{-1/\rho_x} -1)^4. Finally, to select the threshold k=k_n(x), one could use the automatic data-driven method of Daouia et al. (2010) implemented in the function kopt_momt_pick (option method="pickands").

Value

Returns a numeric vector with the same length as x.

Note

As it is common in extreme-value theory, good results require a large sample size N_x at each evaluation point x. See also the note in kopt_momt_pick.

Author(s)

Abdelaati Daouia and Thibault Laurent (converted from Leopold Simar's Matlab 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, Annals of Statistics, 17, 1833-1855.

See Also

dfs_momt, kopt_momt_pick.

Examples

data("post")
x.post<- seq(post$xinput[100],max(post$xinput), 
 length.out=100) 
# 1. When rho[x] is known and equal to 2, we set:
rho<-2
# To determine the sample fraction k=k[n](x) 
# in hat(varphi[pick])(x).
best_kn.1<-kopt_momt_pick(post$xinput, post$yprod, 
 x.post, method="pickands", rho=rho)
# To compute the frontier estimates and confidence intervals:  
res.pick.1<-dfs_pick(post$xinput, post$yprod, x.post, 
 rho=rho, k=best_kn.1)
# Representation
plot(yprod~xinput, data=post, xlab="Quantity of labor", 
 ylab="Volume of delivered mail")
lines(x.post, res.pick.1[,1], lty=1, col="cyan")  
lines(x.post, res.pick.1[,2], lty=3, col="magenta")  
lines(x.post, res.pick.1[,3], lty=3, col="magenta")  

## Not run: 
# 2. rho[x] is unknown and estimated by 
# the Pickands estimator hat(rho[x])
rho_pick<-rho_momt_pick(post$xinput, post$yprod, 
 x.post, method="pickands")
best_kn.2<-kopt_momt_pick(post$xinput, post$yprod,
  x.post, method="pickands", rho=rho_pick)
res.pick.2<-dfs_pick(post$xinput, post$yprod, x.post, 
 rho=rho_pick, k=best_kn.2)  
# 3. rho[x] is unknown independent of x and estimated
# by the (trimmed) mean of hat(rho[x])
rho_trimmean<-mean(rho_pick, trim=0.00)
best_kn.3<-kopt_momt_pick(post$xinput, post$yprod,
  x.post, rho=rho_trimmean, method="pickands")   
res.pick.3<-dfs_pick(post$xinput, post$yprod, x.post, 
 rho=rho_trimmean, k=best_kn.3)  

# Representation 
plot(yprod~xinput, data=post, col="grey", xlab="Quantity of labor", 
 ylab="Volume of delivered mail")
lines(x.post, res.pick.2[,1], lty=1, lwd=2, col="cyan")  
lines(x.post, res.pick.2[,2], lty=3, lwd=4, col="magenta")  
lines(x.post, res.pick.2[,3], lty=3, lwd=4, col="magenta")  
plot(yprod~xinput, data=post, col="grey", xlab="Quantity of labor", 
 ylab="Volume of delivered mail")
lines(x.post, res.pick.3[,1], lty=1, lwd=2, col="cyan")  
lines(x.post, res.pick.3[,2], lty=3, lwd=4, col="magenta")  
lines(x.post, res.pick.3[,3], lty=3, lwd=4, col="magenta") 

## End(Not run)