h.amise {kedd}

AMISE for Optimal Bandwidth Selectors

Description

The (S3) generic function h.amise evaluates the asymptotic mean integrated squared error AMISE for optimal smoothing parameters h of r'th derivative of kernel density estimator one-dimensional.

Usage

h.amise(x, ...)
## Default S3 method:
h.amise(x, deriv.order = 0, lower = 0.1 * hos, upper = 2 * hos, 
         tol = 0.1 * lower, kernel = c("gaussian", "epanechnikov", "triweight", 
         "tricube", "biweight", "cosine"), ...)

Arguments

x

vector of data values.

deriv.order

derivative order (scalar).

lower, upper

range over which to minimize. The default is almost always satisfactory. hos (Over-smoothing) is calculated internally from an kernel, see details.

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.

Details

h.amise asymptotic mean integrated squared error implements for choosing the optimal bandwidth h of a r'th derivative kernel density estimator.

We Consider the following AMISE version of the r'th derivative of f the r'th derivative of the kernel estimate (see Scott 1992, pp 131):

AMISE(h;r)= \frac{R\left(K^{(r)}\right)}{nh^{2r+1}} + \frac{1}{4} h^{4} \mu_{2}^{2}(K) R\left(f^{(r+2)}\right)

The optimal bandwidth minimizing this function is:

h_{(r)}^{\ast} = \left[\frac{(2r+1)R\left(K^{(r)}\right)}{\mu_{2}^{2}(K) R\left(f^{(r+2)}\right)}\right]^{1/(2r+5)} n^{-1/(2r+5)}

whereof

\inf_{h > 0} AMISE(h;r) = \frac{2r+5}{4} R\left(K^{(r)}\right)^{\frac{4}{(2r+5)}} \left[ \frac{\mu_{2}^{2}(K)R\left(f^{(r+2)}\right)}{2r+1} \right]^{\frac{2r+1}{2r+5}} n^{-\frac{4}{2r+5}}

which is the smallest possible AMISE for estimation of f^{(r)}(x) using the kernel K(x), where 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).

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

deriv.order

the derivative order to use.

h

value of bandwidth parameter.

amise

the AMISE value.

Author(s)

Arsalane Chouaib Guidoum [e-mail]

References

Bowman, A. W. and Azzalini, A. (1997). Applied Smoothing Techniques for Data Analysis: the Kernel Approach with S-Plus Illustrations. Oxford University Press, Oxford.

Radhey, S. S. (1987). MISE of kernel estimates of a density and its derivatives. Statistics and Probability Letters, 5, 153–159.

Scott, D. W. (1992). Multivariate Density Estimation. Theory, Practice and Visualization. New York: Wiley.

Sheather, S. J. (2004). Density estimation. Statistical Science, 19, 588–597.

Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman & Hall/CRC. London.

Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing. Chapman and Hall, London.

See Also

plot.h.amise, see nmise in package sm this function evaluates the mean integrated squared error of a density estimate (deriv.order = 0) which is constructed from data which follow a normal distribution.

Examples

## Derivative order = 0

h.amise(kurtotic,deriv.order = 0)

## Derivative order = 1

h.amise(kurtotic,deriv.order = 1)