bw.F.SBC.cv: Cross-validation bandwidth selector for...

In WData: Statistical Inference for Weighted Data

Cross-validation bandwidth selector for \insertCitebose2022;textualWData kernel distribution estimator

Description

This function performs bandwidth selection for \insertCitebose2022;textualWData kernel distribution estimator using cross-validation criteria. It iterates through a range of bandwidth values and computes the cross-validation score for each bandwidth. The bandwidth that minimizes the cross-validation function is selected as the optimal bandwidth.

Usage

bw.F.SBC.cv(
  y,
  w = function(y) {
     ifelse(y >= 0, y, NA)
 },
  kernel = c("gaussian", "epanechnikov", "rectangular", "triangular", "biweight",
    "cosine", "optcosine"),
  lower = IQR(y) * length(y)^(-1/3) * 0.05,
  upper = IQR(y) * length(y)^(-1/3) * 5,
  nh = 200L,
  tol = 0.1 * lower,
  plot = TRUE
)

Arguments

y	A numeric vector containing the biased sample.
w	A function representing the bias function applied to the data points. It must be evaluable and positive in each point of the sample the sample y. By default, it is set to the length-biased function.
kernel	A character string specifying the kernel function. Available options: "gaussian", "epanechnikov", "rectangular", "triangular", "biweight", "cosine" and "optcosine".
lower	Numeric value specifying the lower bound for bandwidth selection. Default is computed based on the interquartile range (IQR) and sample size.
upper	Numeric value specifying the upper bound for bandwidth selection. Default is computed based on the interquartile range (IQR) and sample size.
nh	An integer specifying the number of points to evaluate the cross-validation function. Default is 200.
tol	Tolerance value used to check whether the minimum found lies at the boundaries of the interval; that is, the function will return a warning if the window minimizing the cross-validation function lies within ⁠[lower, lower+tol]⁠ or ⁠[upper-tol, upper]⁠. Default is 10% of the lower bound.
plot	A logical value indicating whether to plot the cross-validation function. Default is TRUE.

Details

The optimal bandwidth is obtained as the one that minimizes the cross-validation function, that is,

\widehat{h}_{F, \mathrm{CV}} = \arg \min_{h_{F}>0} \frac{1}{n} \sum_{j=1}^n \left( \frac{\widehat{\mu}_w}{w(Y_j)} \mathbb{I} (y \geq Y_j) - \widehat{F}_{h_{F}, -j}(y)\right)^2 \!\!, \quad \text{with} \quad \widehat{\mu}_w=n \left(\sum_{i=1}^{n} \frac{1}{w(Y_i)} \right)^{-1}

and \widehat{F}_{h_{F}, -j} is the \insertCitebose2022;textualWData kernel distribution estimator without the observation Y_j.

Value

The optimal bandwidth based on cross-validation criteria.

References

\insertAllCited

See Also

cdf.bd

Examples

bw.F.SBC.cv(shrub.data$Width)
bw.F.SBC.cv(shrub.data$Width, kernel = "epanechnikov")

WData documentation built on Nov. 2, 2026, 1:10 a.m.