# Pickands frontier estimator This function is an implementation of the Pickands type estimator developed by Daouia, Florens and Simar (2010). ```r dfs_pick(xtab, ytab, x, k, rho, ci=TRUE) ``` ## Arguments - `xtab`: a numeric vector containing the observed inputs $x1,...,xn$. - `ytab`: a numeric vector of the same length as `xtab` containing the observed outputs $y1,...,yn$. - `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 $$ \hat\varphi_{pick}(x) = \frac{z^x_{(n-k+1)} - z^x_{(n-2k+1)}}{2^{1/\rho_x} - 1} + z^x_{(n-k+1)}hat(varphi)[pick](x) = (z^x[(n-k+1)] - z^x[(n-2k+1)])/(2^(1/rho[x] - 1) + z^x_[(n-k+1)] $$ from the transformed data $z[i]^x, i=1,...,n$ 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 $$ [\hat\varphi_{pick}(x) \pm 1.96 \sqrt{v(\rho_x) / (2 k)} ( z^x_{(n-k+1)} - z^x_{(n-2k+1)})][hat(varphi)[pick](x) +- 1.96 sqrt(v(rho[x])/2k) (z^x[(n-k+1)] - z^x[(n-2k+1)])] $$ 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"`). ## Returns 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`. ## 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. ## Author(s) Abdelaati Daouia and Thibault Laurent (converted from Leopold Simar's Matlab code). ## See Also `dfs_momt`, `kopt_momt_pick`. ## Examples ```r 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) ```

dfs_pick.rd function