rho_pwm {npbr}

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)

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.

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).

Value

Returns a numeric vector with the same length as x.

Note

The computational burden here is demanding, so be forewarned.

Author(s)

Abdelaati Daouia and Thibault Laurent.

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) 
## Not run: 
# 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)

## End(Not run)