The (S3) generic function h.tcv computes the trimmed
cross-validation bandwidth selector of r'th derivative of
kernel density estimator one-dimensional.
h.tcv(x, ...)
# S3 method for default
h.tcv(x, deriv.order = 0, lower = 0.1 * hos, upper = 2 * hos,
tol = 0.1 * lower, kernel = c("gaussian", "epanechnikov", "uniform",
"triangular", "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 TCV 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.tcv trimmed cross-validation implements for choosing the bandwidth \(h\)
of a r'th derivative kernel density estimator.
Feluch and Koronacki (1992) proposed a so-called trimmed cross-validation (TCV) in kernel
density estimator, a simple modification of the unbiased (least-squares) cross-validation
criterion. We consider the following "trimmed" version of "unbiased", to be minimized with
respect to \(h\):
$$\int \left(\hat{f}_{h}^{(r)}(x)\right)^{2} - 2 \frac{(-1)^{r}}{n(n-1) h^{2r+1}} \sum_{i=1}^{n}\sum_{j=1; j \neq i} K^{(2r)} \left(\frac{X_{j}-X_{i}}{h}\right)\chi\left(|X_{i}-X_{j}| > c_{n}\right)$$
where \(\chi(.)\) denotes the indicator function and \(c_{n}\) is a sequence of positive
constants, \(c_{n}/ h^{2r+1} \rightarrow 0\) as \(n \rightarrow \infty\), and
$$\int \left(\hat{f}_{h}^{(r)}(x)\right)^{2} = \frac{R\left(K^{(r)}\right)}{nh^{2r+1}} + \frac{(-1)^{r}}{n (n-1) h^{2r+1}} \sum_{i=1}^{n}\sum_{j=1;j \neq i}^{n} K^{(r)} \ast K^{(r)} \left(\frac{X_{j}-X_{i}}{h}\right)$$
the trimmed cross-validation function is defined by:
$$TCV(h;r) = \frac{R\left(K^{(r)}\right)}{nh^{2r+1}} + \frac{(-1)^{r}}{n(n-1)h^{2r+1}}\sum_{i=1}^{n} \sum_{j=1;j \neq i}^{n} \varphi^{(r)} \left(\frac{X_{j}-X_{i}}{h}\right)$$
whit $$\varphi^{(r)}(c) = \left(K^{(r)} \ast K^{(r)} - 2 K^{(2r)} \chi\left(|c| > c_{n}/h^{2r+1}\right) \right)(c)$$
here we take \(c_{n} = 1/n\), for assure the convergence. 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).
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).
Feluch, W. and Koronacki, J. (1992). A note on modified cross-validation in density estimation. Computational Statistics and Data Analysis, 13, 143--151.
plot.h.tcv.
## Derivative order = 0
h.tcv(kurtotic,deriv.order = 0)
## Derivative order = 1
h.tcv(kurtotic,deriv.order = 1)
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