The (S3) generic function h.bcv computes the biased
cross-validation bandwidth selector of r'th derivative of
kernel density estimator one-dimensional.
h.bcv(x, ...)
# S3 method for default
h.bcv(x, whichbcv = 1, deriv.order = 0, lower = 0.1 * hos, upper = 2 * hos,
tol = 0.1 * lower, kernel = c("gaussian","epanechnikov",
"triweight","tricube","biweight","cosine"), ...)
data points - same as input.
the deparsed name of the x argument.
the sample size after elimination of missing values.
name of kernel to use
the derivative order to use.
method selected.
value of bandwidth parameter.
the minimal BCV value.
vector of data values.
method selected, 1 = BCV1 or 2 = BCV2, see details.
derivative order (scalar).
range over which to minimize. The default is
almost always satisfactory. hos (Over-smoothing) is calculated internally
from an kernel, see details.
the convergence tolerance for optimize.
a character string giving the smoothing kernel to be used, with default
"gaussian".
further arguments for (non-default) methods.
Arsalane Chouaib Guidoum acguidoum@usthb.dz
h.bcv biased cross-validation implements for choosing the bandwidth \(h\) of a
r'th derivative kernel density estimator. if whichbcv = 1 then BCV1 is selected
(Scott and George 1987), and if whichbcv = 2 used BCV2 (Jones and Kappenman 1991).
Scott and George (1987) suggest a method which has as its immediate target the AMISE
(e.g. Silverman 1986, section 3.3). We denote \(\hat{\theta}_{r}(h)\) and
\(\bar{\theta}_{r}(h)\) (Peter and Marron 1987, Jones and Kappenman 1991) by:
$$\hat{\theta}_{r}(h)= \frac{(-1)^{r}}{n(n-1)h^{2r+1}} \sum_{i=1}^{n} \sum_{j=1;j \neq i}^{n} K^{(r)} \ast K^{(r)} \left(\frac{X_{j}-X_{i}}{h}\right)$$
and
$$\bar{\theta}_{r}(h)= \frac{(-1)^r}{n(n-1) h^{2r+1}} \sum_{i=1}^{n} \sum_{j=1;j \neq i}^{n} K^{(2r)} \left(\frac{X_{j}-X_{i}}{h}\right)$$
Scott and George (1987) proposed using \(\hat{\theta}_{r}(h)\) to estimate \(f^{(r)}(x)\).
Thus, \(\hat{h}^{(r)}_{BCV1}\), say, is the \(h\) that minimises:
$$BCV1(h;r)= \frac{R\left(K^{(r)}\right)}{nh^{2r+1}} + \frac{1}{4} \mu_{2}^{2}(K) h^{4} \hat{\theta}_{r+2}(h)$$
and we define \(\hat{h}^{(r)}_{BCV2}\) as the minimiser of (Jones and Kappenman 1991):
$$BCV2(h;r)= \frac{R\left(K^{(r)}\right)}{nh^{2r+1}} + \frac{1}{4} \mu_{2}^{2}(K) h^{4} \bar{\theta}_{r+2}(h)$$
where \(K^{(r)} \ast K^{(r)} (x)\) is the convolution of the r'th derivative kernel function \(K^{(r)}(x)\)
(see kernel.conv and kernel.fun); \(R\left(K^{(r)}\right) = \int_{R} K^{(r)}(x)^{2} dx\) and \(\mu_{2}(K) = \int_{R}x^{2} K(x) dx\).
The range over which to minimize is hos Oversmoothing bandwidth, the default is almost always
satisfactory. See George and Scott (1985), George (1990), Scott (1992, pp 165), Wand and Jones (1995, pp 61).
Jones, M. C. and Kappenman, R. F. (1991). On a class of kernel density estimate bandwidth selectors. Scandinavian Journal of Statistics, 19, 337--349.
Jones, M. C., Marron, J. S. and Sheather,S. J. (1996). A brief survey of bandwidth selection for density estimation. Journal of the American Statistical Association, 91, 401--407.
Peter, H. and Marron, J.S. (1987). Estimation of integrated squared density derivatives. Statistics and Probability Letters, 6, 109--115.
Scott, D.W. and George, R. T. (1987). Biased and unbiased cross-validation in density estimation. Journal of the American Statistical Association, 82, 1131--1146.
Sheather,S. J. (2004). Density estimation. Statistical Science, 19, 588--597.
Tarn, D. (2007). ks: Kernel density estimation and kernel discriminant analysis for multivariate data in R. Journal of Statistical Software, 21(7), 1--16.
Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing. Chapman and Hall, London.
Wolfgang, H. (1991). Smoothing Techniques, With Implementation in S. Springer-Verlag, New York.
## EXAMPLE 1:
x <- rnorm(100)
h.bcv(x,whichbcv = 1, deriv.order = 0)
h.bcv(x,whichbcv = 2, deriv.order = 0)
## EXAMPLE 2:
## Derivative order = 0
h.bcv(kurtotic,deriv.order = 0)
## Derivative order = 1
h.bcv(kurtotic,deriv.order = 1)
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