kedd (version 1.0.4)

h.mlcv: Maximum-Likelihood Cross-validation for Bandwidth Selection

Description

The (S3) generic function h.mlcv computes the maximum likelihood cross-validation (Kullback-Leibler information) bandwidth selector of a one-dimensional kernel density estimate.

Usage

h.mlcv(x, ...)
# S3 method for default
h.mlcv(x, lower = 0.1, upper = 5, tol = 0.1 * lower, 
         kernel = c("gaussian", "epanechnikov", "uniform", "triangular", 
         "triweight", "tricube", "biweight", "cosine"), ...)

Value

x

data points - same as input.

data.name

the deparsed name of the x argument.

n

the sample size after elimination of missing values.

kernel

name of kernel to use

h

value of bandwidth parameter.

mlcv

the maximal likelihood CV value.

Arguments

x

vector of data values.

lower, upper

range over which to maximize. The default is almost always satisfactory.

tol

the convergence tolerance for optimize.

kernel

a character string giving the smoothing kernel to be used, with default "gaussian".

...

further arguments for (non-default) methods.

Author

Arsalane Chouaib Guidoum

Details

h.mlcv maximum-likelihood cross-validation implements for choosing the optimal bandwidth \(h\) of kernel density estimator.

This method was proposed by Habbema, Hermans, and Van den Broeck (1971) and by Duin (1976). The maximum-likelihood cross-validation (MLCV) function is defined by: $$MLCV(h) = n^{-1} \sum_{i=1}^{n} \log\left[\hat{f}_{h,i}(x)\right]$$ the estimate \(\hat{f}_{h,i}(x)\) on the subset \(\{X_{j}\}_{j \neq i}\) denoting the leave-one-out estimator, can be written: $$\hat{f}_{h,i}(X_{i}) = \frac{1}{(n-1) h} \sum_{j \neq i} K \left(\frac{X_{j}-X_{i}}{h}\right)$$ Define that \(h_{mlcv}\) as good which approaches the finite maximum of \(MLCV(h)\): $$h_{mlcv} = \arg \max_{h} MLCV(h) = \arg \max_{h} \left(n^{-1} \sum_{i=1}^{n} \log\left[\sum_{j \neq i} K \left(\frac{X_{j}-X_{i}}{h}\right)\right]-\log[(n-1)h]\right)$$

References

Habbema, J. D. F., Hermans, J., and Van den Broek, K. (1974) A stepwise discrimination analysis program using density estimation. Compstat 1974: Proceedings in Computational Statistics. Physica Verlag, Vienna.

Duin, R. P. W. (1976). On the choice of smoothing parameters of Parzen estimators of probability density functions. IEEE Transactions on Computers, C-25, 1175--1179.

See Also

plot.h.mlcv, see lcv in package locfit.

Examples

h.mlcv(bimodal)
h.mlcv(bimodal, kernel ="epanechnikov")