npbr (version 1.8)

rho_pwm: Probability-weighted moment frontier estimator

Description

This function is an implementation of the Probability-weighted moment frontier estimator developed by Daouia, Florens and Simar (2012).

Usage

rho_pwm(xtab, ytab, x, a=2, lrho=1, urho=Inf)

Value

Returns a numeric vector with the same length as x.

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.

a

a smoothing parameter (integer) larger than or equal to 2.

lrho

a scalar, minimum rho threshold value.

urho

a scalar, maximum rho threshold value.

Author

Abdelaati Daouia and Thibault Laurent.

Details

The function computes the probability-weighted moment (PWM) estimator \(\bar\rho_x\) utilized in the frontier estimate \(\tilde\varphi_{pwm}(x)\)[see dfs_pwm]. This estimator depends on the smoothing parameters \(a\) and \(m\). A simple selection rule of thumb that Daouia et al. (2012) have employed is \(a=2\) [default option in the 4th argument of the function] and \(m=coefm \times N^{1/3}_x\), where \(N_x=\sum_{i=1}^n1_{\{x_i\le x\}}\) and the integer coefm is to be tuned by the user. To choose this parameter in an optimal way for each \(x\), we adapt the automated threshold selection method of Daouia et al. (2010) as follows: We first evaluate the estimator \(\bar\rho_x\) over a grid of values of coefm given by \(c = 1, \cdots, 150\). Then, we select the \(c\) where the variation of the results is the smallest. This is achieved by computing the standard deviation of the estimates \(\bar\rho_x\) over a ``window'' of \(\max([\sqrt{150}],3)\) successive values of \(c\). The value of \(c\) where this standard deviation is minimal defines the value of coefm. The user can also appreciably improve the estimation of the extreme-value index \(\rho_x\) and the frontier function \(\varphi_x\) itself by tuning the choice of the lower limit (default option lrho=1) and upper limit (default option urho=Inf).

References

Daouia, A., Florens, J.-P. and Simar, L. (2010). Frontier estimation and extreme value theory. Bernoulli, 16, 1039-1063.

Daouia, A., Florens, J.-P. and Simar, L. (2012). Regularization of Nonparametric Frontier Estimators. Journal of Econometrics, 168, 285-299.

See Also

dfs_pwm, mopt_pwm.

Examples

data("post")
x.post<- seq(post$xinput[100],max(post$xinput), 
 length.out=100) 
if (FALSE) {
# When rho[x] is unknown and dependent of x, 
# its estimate hat(rho[x]) is obtained via:
rho_pwm <- rho_pwm(post$xinput, post$yprod, x.post,  a=20)
}