# Probability-weighted moment frontier estimator This function is an implementation of the probability-weighted moment frontier estimator developed by Daouia, Florens and Simar (2012). ```r dfs_pwm(xtab, ytab, x, coefm, a=2, 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. - `coefm`: a tuning parameter (integer) larger than or equal to 1. - `a`: a smoothing parameter (integer) larger than or equal to 2. - `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 The regularized frontier estimator introduced by Daouia et al. (2012) is based on the unregularized probability-weighted moment estimator $$ \hat{\varphi}_m(x) = \varphi_{fdh}(x) - \int_{0}^{\varphi_{fdh}(x)} \hat{F}^m(y|x)dyhat(varphi[m])(x)=varphi[fdh](x)-int[0 to varphi[fdh](x)]hat(F)^m(y|x)dy $$ where the trimming order $m>=1$ is an integer such that $m=m[n]->Infinity$ as $n->Infinity$, and $hat(F)(y|x)=sum[i=1]^n 1[(x[i] <= x, y[i] <=y)]/(sum[i=1]^n 1[(x[i] <= x])$. The implemented estimator of $varphi(x)$ is then defined as $$ \tilde{\varphi}_m(x) = \hat{\varphi}_m(x) + \Gamma\left(1 + 1/\bar\rho_x\right)\left( 1/m\,\hat\ell_x\right)^{1/\bar\rho_x}tilde(varphi[m])(x) = hat(varphi[m])(x) + Gamma(1 + 1/hat(rho)[x])(1/(m.hat(l)[x]))^(1/hat(rho)[x]) $$ where $$ \bar{\rho}_x = \log (a)\left\{ \log\Big( \frac{\hat\varphi_{m}(x)-\hat\varphi_{am}(x)}{\hat\varphi_{am}(x)-\hat\varphi_{a^2m}(x)}\Big) \right\}^{-1} , \quad\hat{\ell}_x = \frac {1}{m}\left[\frac{\Gamma(1+ 1/\bar\rho_x)\big(1-a^{-1/\bar\rho_x}\big)}{\hat\varphi_{m}(x)-\hat\varphi_{am}(x)}\right]^{\bar\rho_x},hat(rho)[x] = log(a){log((hat(varphi)[m](x)-hat(varphi)[am](x))/(hat(varphi)[am](x)-hat(varphi)[a^2m](x))}^{-1},hat(l)[x] = 1/m ((Gamma(1+ 1/hat(rho)[x])(1-a^(-1/hat(rho)[x])))/(hat(varphi)[m](x)-hat(varphi)[am](x)))^(hat(rho)[x]), $$ with $a >= 2$ being a fixed integer. If the true tail-index $rho[x]=beta[x]+2$ is known, we set $hat(rho)[x]=rho[x]$ in the expressions above. The two smoothing parameters $m$ and $a$ have to be fixed by the user in the 4th and 5th arguments of the function. The pointwise $95\%$ confidence interval of $varphi(x)$ derived from the asymptotic normality of $tilde(varphi)[m](x)$ is given by $[tilde(varphi)[m](x) +- 1.96 hat(sigma)(m,x)/sqrt(n)]$ where $$ \hat\sigma^2(m,x)= \frac{2m^2}{ \hat F_X(x)}\int_0^{\varphi_{fdh}(x)}\int_0^{\varphi_{fdh}(x)} \hat F^{m}(y|x)\hat F^{m-1}(u|x)(1-\hat F(u|x))1_{(y\le u)}\, dy\,du,hat(sigma)^2(m,x)= (2m^2)/( hat(F)[X](x))int[0 -> varphi[fdh](x)]int[0 -> varphi[fdh](x)] hat(F)^m(y|x)hat(F)^(m-1)(u|x)(1-hat(F)(u|x)) 1[(y <= u)]dydu, $$ with $hat(F)[X](x) =(1/n)sum[i=1 -> n] 1[(x[i] <= x)]$. Note that the standard deviation $sigma(m,x)/sqrt(n)$ of the bias-corrected estimator $tilde(varphi)[m](x)$ is adjusted by a bootstrap estimator in the numerical illustrations of Daouia et al. (2012), whereas the exact estimate $hat(sigma)(m,x)/sqrt(n)$ is utilized in the implemented function. A practical choice of $m$ that Daouia et al. (2012) have employed is the simple rule of thumb $m=coefm N^(1/3)[x]$, where $N[x]=sum[i=1 -> n] 1[{x[i] <= x}]$, and the integer `coefm` as well as the second smoothing parameter `a` are to be tuned by the user to avoid numerical instabilities in the pointwise estimates of the tail-index $rho[x]$ and the frontier function $varphi(x)$. The user may start with the values `coefm=5` and `a=2` [respectively, `coefm=10` and `a=20`] for computing the estimator $tilde(varphi)[m](x)$ [respectively, $hat(rho)[x]$]. Note that tail-index estimation and frontier estimation are conducted separately. ## Returns Returns a numeric vector with the same length as `x`. ## Note The computational burden here is demanding, so be forewarned. ## References Daouia, A., Florens, J.-P. and Simar, L. (2012). Regularization of Nonparametric Frontier Estimators. **Journal of Econometrics**, 168, 285-299. ## Author(s) Abdelaati Daouia and Thibault Laurent (converted from Abdelaati Daouia's Matlab code). ## See Also `rho_pwm`, `mopt_pwm`. ## Examples ```r data("post") x.post<- seq(post$xinput[100],max(post$xinput), length.out=100) ## Not run: # 1. When rho[x] is known and equal to 2, we set: rho<-2 res.pwm.1<- dfs_pwm(post$xinput, post$yprod, x.post, coefm=5, a=2, rho, ci=TRUE) # 2. 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, coefm=10, a=20) # and the corresponding frontier estimator via: res.pwm.2<- dfs_pwm(post$xinput, post$yprod, x.post, coefm=5, a=2, rho_pwm, ci=TRUE) # 3. When rho[x] is unknown but independent of x, # a robust estimation strategy is by using the (trimmed) mean # over the estimates hat(rho[x]): rho_trimmean<-mean(rho_pwm, trim=0.00) res.pwm.3<- dfs_pwm(post$xinput, post$yprod, x.post, coefm=5, a=2, rho_trimmean, ci=TRUE) ## End(Not run) ```

dfs_pwm.rd function