bw.F.SBC.pi: Plug-in bandwidth selector for...

In WData: Statistical Inference for Weighted Data

Plug-in bandwidth selector for \insertCitebose2022;textualWData kernel distribution estimator

Description

This function computes the bandwidth selector for \insertCitebose2022;textualWData kernel distribution estimator using the plug-in method.

Usage

bw.F.SBC.pi(
  y,
  w = function(y) {
     ifelse(y >= 0, y, NA)
 },
  kernel = c("gaussian", "epanechnikov", "rectangular", "triangular", "biweight",
    "cosine", "optcosine")
)

Arguments

y	A numeric vector containing the biased sample.
w	A function representing the bias function to be used. 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".

Details

The bandwidth is given by:

\widehat{h}_{F, \mathrm{PI}} = \left(\frac{\widehat{\mu}_w \widehat{\bar{\mu}}_w \kappa(K) }{n \eta(K)^2 R \left(\widehat{f}_{\mathrm{J},\widehat{h}_{F,0, \mathrm{opt}}}^{(1)}\right)}\right)^{1/3},

where \kappa(K) and \eta(K) depend only on the kernel and are defined as

\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 W is the kernel distribution function associated with the kernel density function K. The estimators \widehat{\mu}_w and \widehat{\bar{\mu}}_w are given by

\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}.

\widehat{h}_{F,0, \mathrm{opt}} is an estimator of

h_{F,0, \mathrm{opt}} = \left(\frac{3 \mu_w \bar{\mu}_w R\left(L^{(1)}\right)}{2 n \eta(L) R\left( f^{(2)} \right)^2}\right)^{1/5},

where R\left(f^{(2)}\right) is estimated assuming that f follows a gaussian distribution and \mu_w and \bar{\mu}_w are estimated by \widehat{\mu}_w and \widehat{\bar{\mu}}_w as defined above.

Value

The bandwidth value using the plug-in method.

References

\insertAllCited

See Also

cdf.bd

Examples

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

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