R/bw.F.SBC.rt.R

In WData: Statistical Inference for Weighted Data

#' Rule of thumb bandwidth selector for \insertCite{bose2022;textual}{WData} kernel distribution estimator
#'
#' This function computes the bandwidth selector for \insertCite{bose2022;textual}{WData} kernel distribution estimator using the rule of thumb.
#'
#' @param y A numeric vector containing the biased sample.
#' @param w A function representing the bias function applied to the data points. It must be evaluable and positive in each point of the sample `y`. By default, it is set to the length-biased function.
#' @param kernel A character string specifying the kernel function. Available options: `"gaussian"`, `"epanechnikov"`, `"rectangular"`, `"triangular"`, `"biweight"`, `"cosine"` and `"optcosine"`.
#' @return The optimal bandwidth value for \insertCite{bose2022;textual}{WData} kernel distribution estimator based on the rule of thumb.
#' @details
#' The bandwidth is computed as follows:
#' \deqn{\widehat{h}_{F, \mathrm{RT}} =
#' \left(\frac{4 \sqrt{\pi} \widehat{\mu}_{w} \widehat{\bar{\mu}}_{w}\kappa(K)}{n \eta(K)^{2} }\right)^{1/3} \widehat{\sigma}, }
#' where both \eqn{\kappa(K)} and \eqn{\eta(K)} depend only on the kernel and are defined as
#' \deqn{
#' \kappa(K) = \int_{-\infty}^{+\infty} 2 u W (u) K(u) du
#' \quad \text{and} \quad
#' \eta(K) = \int_{-\infty}^{+\infty} u^2 K(u) du,
#' }
#' where \eqn{ W} is the kernel distribution function associated with the kernel density function \eqn{K}.
#' The estimators \eqn{\widehat{\mu}_w} and \eqn{\widehat{\bar{\mu}}_w} are given by
#' \deqn{
#' \widehat{\mu}_w = n \left(\sum_{i=1}^{n}  \frac{1}{w(Y_i)} \right)^{-1}
#' \quad \text{and} \quad
#' \widehat{\bar{\mu}}_w = \frac{\widehat{\mu}_w}{n} \sum_{i=1}^{n} \frac{1}{w(Y_i)^2}.
#' }
#' \eqn{\widehat{\sigma}} is an estimation of the standard deviation of the distribution given by
#' \deqn{\widehat{\sigma}_w=\sqrt{\left(\frac{1}{n} \sum_{i=1}^n \frac{1}{w(Y_i)}\right)^{-1}
#' \left[\left(\frac{1}{n} \sum_{i=1}^n w(Y_i)\right)-\left(\frac{1}{n} \sum_{i=1}^n \frac{1}{w(Y_i)}\right)^{-1}\right]}.}
#' @references \insertAllCited{}
#' @seealso [`cdf.bd`][WData::cdf.bd()]
#' @export
#' @examples
#' bw.F.SBC.rt(shrub.data$Width)
#' bw.F.SBC.rt(shrub.data$Width, kernel = "epanechnikov")
#'
bw.F.SBC.rt <- function(y,
                        w = function(y) {
                          ifelse(y >= 0, y, NA)
                        },
                        kernel = c(
                          "gaussian", "epanechnikov",
                          "rectangular", "triangular",
                          "biweight", "cosine", "optcosine"
                        )) {
  list2env(.check_biased_sample(y, w), envir = environment())
  kernel <- match.arg(kernel)
  list2env(.get_kernel_values(kernel), envir = environment())

  # Compute the bandwidth
  sigma <- sqrt(uw * (mean(yw) - uw))
  sigma * (4 * sqrt(pi) * uw * uwb * kernel_kappa)^(1 / 3) * (n * kernel_eta^2)^(-1 / 3)
}