The (S3) generic function h.mcv computes the modified
cross-validation bandwidth selector of r'th derivative of
kernel density estimator one-dimensional.
h.mcv(x, ...)
# S3 method for default
h.mcv(x, 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.
value of bandwidth parameter.
the minimal MCV value.
vector of data values.
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.mcv modified cross-validation implements for choosing the bandwidth \(h\)
of a r'th derivative kernel density estimator.
Stute (1992) proposed a so-called modified cross-validation (MCV) in kernel
density estimator. This method can be extended to the estimation of derivative
of a density, the essential idea based on approximated the problematic term
by the aid of the Hajek projection (see Stute 1992). The minimization criterion is defined by:
$$MCV(h;r) = \frac{R\left(K^{(r)}\right)}{nh^{2r+1}} + \frac{(-1)^{r}}{n(n-1)h^{2r+1}}\sum_{i=1}^{n} \sum_{j=1;j \neq i}^{n} \varphi^{(r)} \left(\frac{X_{j}-X_{i}}{h}\right)$$
whit $$\varphi^{(r)}(c) = \left(K^{(r)} \ast K^{(r)} - K^{(2r)} - \frac{\mu_{2}(K)}{2}K^{(2r+2)} \right)(c)$$
and \(K^{(r)} \ast K^{(r)} (x)\) 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).
Heidenreich, N. B., Schindler, A. and Sperlich, S. (2013). Bandwidth selection for kernel density estimation: a review of fully automatic selectors. Advances in Statistical Analysis.
Stute, W. (1992). Modified cross validation in density estimation. Journal of Statistical Planning and Inference, 30, 293--305.
plot.h.mcv.
## Derivative order = 0
h.mcv(kurtotic,deriv.order = 0)
## Derivative order = 1
h.mcv(kurtotic,deriv.order = 1)
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