dfs_momt: Moment frontier estimator
In npbr: Nonparametric Boundary Regression
dfs_momt R Documentation
Moment frontier estimator
Description
This function is an implementation of the moment-type estimator developed by Daouia, Florens and Simar (2010).
Usage
dfs_momt(xtab, ytab, x, rho, k, 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.
rho
a numeric vector of the same length as x or a scalar, which determines the values of rho.
k
a numeric vector of the same length as x or a scalar, which determines the thresholds at which the moment estimator will be computed.
ci
a boolean, TRUE for computing the confidence interval.
Details
Combining ideas from Dekkers, Einmahl and de Haan (1989) with the dimensionless
transformation \{z^{x}_i := y_i\mathbf{1}_{\{x_i\le x\}}, \,i=1,\cdots,n\} of
the observed sample \{(x_i,y_i), \,i=1,\cdots,n\}, the authors
estimate the conditional endpoint \varphi(x) by
\tilde\varphi_{momt}(x) = z^{x}_{(n-k)} + z^{x}_{(n-k)} M^{(1)}_n \left\{1 + \rho_x \right\}
where M^{(1)}_n = (1/k)\sum_{i=0}^{k-1}\left(\log z^x_{(n-i)}- \log z^x_{(n-k)}\right),
z^{x}_{(1)}\leq \cdots\leq z^{x}_{(n)} are the ascending order statistics
corresponding to the transformed sample \{z^{x}_i, \,i=1,\cdots,n\}
and \rho_x>0 is referred to as the extreme-value index and has the following interpretation:
when \rho_x>2, the joint density of data decays smoothly to zero at a speed of power \rho_x -2 of the distance from the frontier;
when \rho_x=2, the density has sudden jumps at the frontier; when \rho_x<2, the density increases toward infinity at a speed of power \rho_x -2
of the distance from the frontier.
Most of the contributions to econometric literature on frontier analysis assume that the joint density is strictly positive at its support boundary, or equivalently, \rho_x=2 for all x.
When \rho_x is unknown, Daouia et al. (2010) suggest to use the following two-step estimator:
First, estimate \rho_x by the moment estimator \tilde\rho_x implemented in the function rho_momt_pick by utilizing the option method="moment",
or by the Pickands estimator \hat\rho_x by using the option method="pickands".
Second, use the estimator \tilde\varphi_{momt}(x), as if \rho_x were known, by substituting the estimated value \tilde\rho_x
or \hat\rho_x in place of \rho_x.
The 95\% confidence interval of \varphi(x) derived from the asymptotic normality of \tilde\varphi_{momt}(x) is given by
[\tilde\varphi_{momt}(x) \pm 1.96 \sqrt{V(\rho_x) / k} z^{x}_{(n-k)} M^{(1)}_n (1 + 1/\rho_x) ]
where V(\rho_x) = \rho^2_x (1+2/\rho_x)^{-1}.
The sample fraction k=k_n(x) plays here the role of the smoothing parameter and varies between 1 and N_x-1, with N_x=\sum_{i=1}^n\mathbf{1}_{\{x_i\le x\}}
being the number of observations (x_i,y_i) with x_i \leq x. See kopt_momt_pick for an automatic data-driven rule for selecting k.
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, nnals of Statistics, 17, 1833-1855.
See Also
dfs_pick, 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 tilde(varphi[momt])(x).
best_kn.1 <- kopt_momt_pick(post$xinput, post$yprod,
x.post, rho = rho)
# To compute the frontier estimates and confidence intervals:
res.momt.1 <- dfs_momt(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.momt.1[,1], lty = 1, col = "cyan")
lines(x.post, res.momt.1[,2], lty = 3, col = "magenta")
lines(x.post, res.momt.1[,3], lty = 3, col = "magenta")
## Not run:
# 2. rho[x] is unknown and estimated by
# the Pickands estimator tilde(rho[x])
rho_momt <- rho_momt_pick(post$xinput, post$yprod,
x.post)
best_kn.2 <- kopt_momt_pick(post$xinput, post$yprod,
x.post, rho = rho_momt)
res.momt.2 <- dfs_momt(post$xinput, post$yprod, x.post,
rho = rho_momt, k = best_kn.2)
# 3. rho[x] is unknown independent of x and estimated
# by the (trimmed) mean of tilde(rho[x])
rho_trimmean <- mean(rho_momt, trim=0.00)
best_kn.3 <- kopt_momt_pick(post$xinput, post$yprod,
x.post, rho = rho_trimmean)
res.momt.3 <- dfs_momt(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.momt.2[,1], lty = 1, lwd = 2, col = "cyan")
lines(x.post, res.momt.2[,2], lty = 3, lwd = 4, col = "magenta")
lines(x.post, res.momt.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.momt.3[,1], lty = 1, lwd = 2, col = "cyan")
lines(x.post, res.momt.3[,2], lty = 3, lwd = 4, col = "magenta")
lines(x.post, res.momt.3[,3], lty = 3, lwd = 4, col = "magenta")
## End(Not run)
npbr documentation built on March 31, 2023, 7:45 p.m.
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| dfs_momt | R Documentation |
Moment frontier estimator
Description
This function is an implementation of the moment-type estimator developed by Daouia, Florens and Simar (2010).
Usage
dfs_momt(xtab, ytab, x, rho, k, ci=TRUE)
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. |
rho |
a numeric vector of the same length as |
k |
a numeric vector of the same length as |
ci |
a boolean, TRUE for computing the confidence interval. |
Details
Combining ideas from Dekkers, Einmahl and de Haan (1989) with the dimensionless
transformation \{z^{x}_i := y_i\mathbf{1}_{\{x_i\le x\}}, \,i=1,\cdots,n\} of
the observed sample \{(x_i,y_i), \,i=1,\cdots,n\}, the authors
estimate the conditional endpoint \varphi(x) by
\tilde\varphi_{momt}(x) = z^{x}_{(n-k)} + z^{x}_{(n-k)} M^{(1)}_n \left\{1 + \rho_x \right\}
where M^{(1)}_n = (1/k)\sum_{i=0}^{k-1}\left(\log z^x_{(n-i)}- \log z^x_{(n-k)}\right),
z^{x}_{(1)}\leq \cdots\leq z^{x}_{(n)} are the ascending order statistics
corresponding to the transformed sample \{z^{x}_i, \,i=1,\cdots,n\}
and \rho_x>0 is referred to as the extreme-value index and has the following interpretation:
when \rho_x>2, the joint density of data decays smoothly to zero at a speed of power \rho_x -2 of the distance from the frontier;
when \rho_x=2, the density has sudden jumps at the frontier; when \rho_x<2, the density increases toward infinity at a speed of power \rho_x -2
of the distance from the frontier.
Most of the contributions to econometric literature on frontier analysis assume that the joint density is strictly positive at its support boundary, or equivalently, \rho_x=2 for all x.
When \rho_x is unknown, Daouia et al. (2010) suggest to use the following two-step estimator:
First, estimate \rho_x by the moment estimator \tilde\rho_x implemented in the function rho_momt_pick by utilizing the option method="moment",
or by the Pickands estimator \hat\rho_x by using the option method="pickands".
Second, use the estimator \tilde\varphi_{momt}(x), as if \rho_x were known, by substituting the estimated value \tilde\rho_x
or \hat\rho_x in place of \rho_x.
The 95\% confidence interval of \varphi(x) derived from the asymptotic normality of \tilde\varphi_{momt}(x) is given by
[\tilde\varphi_{momt}(x) \pm 1.96 \sqrt{V(\rho_x) / k} z^{x}_{(n-k)} M^{(1)}_n (1 + 1/\rho_x) ]
where V(\rho_x) = \rho^2_x (1+2/\rho_x)^{-1}.
The sample fraction k=k_n(x) plays here the role of the smoothing parameter and varies between 1 and N_x-1, with N_x=\sum_{i=1}^n\mathbf{1}_{\{x_i\le x\}}
being the number of observations (x_i,y_i) with x_i \leq x. See kopt_momt_pick for an automatic data-driven rule for selecting k.
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, nnals of Statistics, 17, 1833-1855.
See Also
dfs_pick, 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 tilde(varphi[momt])(x).
best_kn.1 <- kopt_momt_pick(post$xinput, post$yprod,
x.post, rho = rho)
# To compute the frontier estimates and confidence intervals:
res.momt.1 <- dfs_momt(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.momt.1[,1], lty = 1, col = "cyan")
lines(x.post, res.momt.1[,2], lty = 3, col = "magenta")
lines(x.post, res.momt.1[,3], lty = 3, col = "magenta")
## Not run:
# 2. rho[x] is unknown and estimated by
# the Pickands estimator tilde(rho[x])
rho_momt <- rho_momt_pick(post$xinput, post$yprod,
x.post)
best_kn.2 <- kopt_momt_pick(post$xinput, post$yprod,
x.post, rho = rho_momt)
res.momt.2 <- dfs_momt(post$xinput, post$yprod, x.post,
rho = rho_momt, k = best_kn.2)
# 3. rho[x] is unknown independent of x and estimated
# by the (trimmed) mean of tilde(rho[x])
rho_trimmean <- mean(rho_momt, trim=0.00)
best_kn.3 <- kopt_momt_pick(post$xinput, post$yprod,
x.post, rho = rho_trimmean)
res.momt.3 <- dfs_momt(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.momt.2[,1], lty = 1, lwd = 2, col = "cyan")
lines(x.post, res.momt.2[,2], lty = 3, lwd = 4, col = "magenta")
lines(x.post, res.momt.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.momt.3[,1], lty = 1, lwd = 2, col = "cyan")
lines(x.post, res.momt.3[,2], lty = 3, lwd = 4, col = "magenta")
lines(x.post, res.momt.3[,3], lty = 3, lwd = 4, col = "magenta")
## End(Not run)
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