bwJf: Plug-in Method by Tsuruta and Sagae with additive...

In circularKDE: Recent Methods for Kernel Density Estimation of Circular Data

Plug-in Method by Tsuruta and Sagae with additive Jones-Foster approach

Description

This function computes the optimal smoothing parameter (bandwidth) for circular data using the plug-in method introduced by Tsuruta and Sagae (see \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.spl.2017.08.003")}) with the additive method from Jones and Foster (1993) to form higher-order kernel functions.

Usage

bwJf(x, verbose = FALSE)

Arguments

x	Data from which the smoothing parameter is to be computed. The object is coerced to a numeric vector in radians using circular. Can be a numeric vector or an object of class circular.
verbose	Logical indicating whether to print intermediate computational values for debugging and teaching purposes. Shows kappa_hat, r_hat, and component calculations. Default is FALSE.

Details

The plug-in approach estimates the optimal bandwidth through the following steps:

1. Apply the additive method from Jones and Foster (1993) to construct a p-th order kernel function.

2. Derive expression for asymptotic mean integrated squared error (AMISE) expression.

3. Solving for the bandwidth that minimizes the AMISE. The optimal bandwidth for the additive Jones-Foster method is given by:

   \hat{\kappa}_{JF} = \left[\frac{16\sqrt{\pi}}{3} \hat{R}_{\hat{\tau}}\left(\frac{5f_{VM}^{(2)} + 2f_{VM}^{(4)}}{12}\right)n\right]^{2/9}

   where the functional \hat{R}_{\hat{\tau}} is computed as a weighted linear combination under the von Mises assumption and \hat{\tau} is the MLE estimate of the von Mises concentration parameter used as the initial value.

Value

The computed optimal smoothing parameter kappa, a numeric concentration parameter (analogous to inverse radians) derived from the circular version of the additive method for circular kernel density estimation. Higher values indicate sharper, more concentrated kernels and less smoothing; lower values indicate broader kernels and more smoothing.

References

Tsuruta, Yasuhito & Sagae, Masahiko (2017). Higher order kernel density estimation on the circle. Statistics & Probability Letters, 131:46–50. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.spl.2017.08.003")}

Jones, M. C. & Foster, P. J. (1993). Generalized jackknifing and higher-order kernels. Journal of Nonparametric Statistics, 3:81–94. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/10485259308832573")}

See Also

bwScv, bwLscvg, bwCcv

Examples

# Example with circular data
library(circular)
set.seed(123)
x <- rvonmises(100, mu = circular(0), kappa = 2)
bw <- bwJf(x)
print(bw)

x <- rwrappednormal(100, mu = circular(1), rho = 0.7)
bw <- bwJf(x)
print(bw)

circularKDE documentation built on Jan. 7, 2026, 9:06 a.m.