npbr (version 1.8)

mopt_pwm: Threshold selection for the PWM frontier estimator

Description

This function implements the optimal smoothing parameter coefm involved in the probability-weighted moment frontier estimator of Daouia, Florens and Simar (2012).

Usage

mopt_pwm(xtab, ytab, x, a=2, rho, wind.coef=0.1)

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 (2 by default).

rho

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

wind.coef

a scalar coefficient to be selected in the interval (0,1].

Author

Abdelaati Daouia and Thibault Laurent.

Details

This is an implementation of an automated selection of the parameter coefm involved in the probability-weighted moment (PWM) estimator \(\tilde\varphi_{pwm}(x)\) [see dfs_pwm]. It is an adaptation of the experimental method kopt_momt_pick by Daouia et al. (2010). The idea is to select first (for each \(x\)) a grid of values for the parameter coefm given by \(c = 1, \cdots, \min(10,[\sqrt{N_x}])\), where \(N_x=\sum_{i=1}^n1_{\{x_i\le x\}}\), and then select the \(c\) where the variation of the results is the smallest. To achieve this, we compute the standard deviations of \(\tilde\varphi_{pwm}(x)\) over a ``window'' of size \(wind.coef \times \min(10,[\sqrt{N_x}])\), where the coefficient wind.coef should be selected in the interval \((0,1]\) in such a way to avoid numerical instabilities. The default option wind.coef=0.1 corresponds to having a window large enough to cover around \(10\%\) of the possible values of \(c\) in the selected range of values for coefm. The value of \(c\) where the standard deviation is minimal defines the desired coefm.

References

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

See Also

dfs_pwm, kopt_momt_pick.

Examples

data("post")
x.post<- seq(post$xinput[100],max(post$xinput), 
 length.out=100) 
if (FALSE) {
# When rho[x] is known and equal to 2:
best_cm.1<- mopt_pwm(post$xinput, post$yprod, 
 x.post, a=2, rho=2)
}