Package 'kedd'

Kernel Estimator and Bandwidth Selection for Density and Its Derivatives

Description

Smoothing techniques and computing bandwidth selectors of the r'th derivative of a probability density for one-dimensional data.

Details

Package:	kedd
Type:	Package
Version:	1.0.4
Date:	2024-01-27
License:	GPL (>= 2)

There are four main types of functions in this package:

1. Compute the derivatives and convolutions of a kernel function (1-d).

2. Compute the kernel estimators for density and its derivatives (1-d).

3. Computing the bandwidth selectors (1-d).

4. Displaying kernel estimators.

Main Features

Convolutions and derivatives in kernel function:

In non-parametric statistics, a kernel is a weighting function used in non-parametric estimation techniques. The kernels functions $K(x)$ are used in derivatives of kernel density estimator to estimate $\hat{f}^{(r)}_{h}(x)$, satisfying the following three requirements:

1. $\int_{R} K(x) dx = 1$

2. $\int_{R} xK(x) dx = 0$

3. $\mu_{2}(K) = \int_{R}x^{2} K(x) dx < \infty$

Several types of kernel functions $K(x)$ are commonly used in this package: Gaussian, Epanechnikov, Uniform (rectangular), Triangular, Triweight, Tricube, Biweight (quartic), Cosine.

The function kernel.fun for kernel derivative $K^{(r)}(x)$ and kernel.conv for kernel convolution $K^{(r)}\ast K^{(r)} (x)$, where the write formally:

$K^{(r)}(x) = \frac{d^{r}}{d x^{r}} K(x)$

$K^{(r)} \ast K^{(r)} (x) = \int_{-\infty}^{+\infty} K^{(r)}(y)K^{(r)}(x-y)dy$

for $r = 0, 1, 2, \dots$

Estimators of r'th derivative of a density function:

A natural estimator of the r'th derivative of a density function $f(x)$ is:

$\hat{f}^{(r)}_{h}(x)= \frac{d^{r}}{d x^{r}} \frac{1}{nh} \sum_{i=1}^{n} K\left(\frac{x-X_{i}}{h}\right) = \frac{1}{nh^{r+1}}\sum_{i=1}^{n} K^{(r)}\left(\frac{x-X_{i}}{h}\right)$

Here, $X_{1}, X_{2}, \dots,X_{n}$ is an i.i.d, sample of size $n$ from the distribution with density $f(x)$, $K(x)$ is the kernel function which we take to be a symmetric probability density with at least $r$ non zero derivatives when estimating $f^{(r)}(x)$, and $h$ is the bandwidth, this parameter is very important that controls the degree of smoothing applied to the data.

The case $(r=0)$ is the standard kernel density estimator (e.g. Silverman 1986, Wolfgang 1991, Scott 1992, Wand and Jones 1995, Jeffrey 1996, Bowman and Azzalini 1997, Alexandre 2009), properties of such derivative estimators are well known e.g. Sheather and Jones (1991), Jones and Kappenman (1991), Wolfgang (1991). For the case $(r > 0)$, is derivative of kernel density estimator (e.g. Bhattacharya 1967, Schuster 1969, Alekseev 1972, Wolfgang et all 1990, Jones 1992, Stoker 1993) and for applications which require the estimation of density derivatives can be found in Singh (1977).

For r'th derivatives of kernel density estimator one-dimensional, the main function is dkde. For display, its plot method calls plot.dkde, and if to add a plot using lines.dkde.

  R> data(trimodal)
  R> dkde(x = trimodal, deriv.order = 0, kernel = "gaussian")
   
    Data: trimodal (200 obs.);      Kernel: gaussian
    Derivative order: 0;    Bandwidth 'h' = 0.1007
          eval.points           est.fx         
    Min.   :-2.91274   Min.   :0.0000066  
    1st Qu.:-1.46519   1st Qu.:0.0669750  
    Median :-0.01765   Median :0.1682045  
    Mean   :-0.01765   Mean   :0.1723692  
    3rd Qu.: 1.42989   3rd Qu.:0.2484626  
    Max.   : 2.87743   Max.   :0.4157340 
   
  R> dkde(x = trimodal, deriv.order = 1, kernel = "gaussian")
  
    Data: trimodal (200 obs.);      Kernel: gaussian
    Derivative order: 1;    Bandwidth 'h' = 0.09094
          eval.points           est.fx         
    Min.   :-2.87358   Min.   :-1.740447  
    1st Qu.:-1.44562   1st Qu.:-0.343952  
    Median :-0.01765   Median : 0.009057  
    Mean   :-0.01765   Mean   : 0.000000  
    3rd Qu.: 1.41031   3rd Qu.: 0.415343  
    Max.   : 2.83828   Max.   : 1.256891  
  

Bandwidth selectors:

The most important factor in the r'th derivative kernel density estimate is a choice of the bandwidth $h$ for one-dimensional observations. Because of its role in controlling both the amount and the direction of smoothing, this choice is particularly important. We present the popular bandwidth selection (for more details see references) methods in this package:

• Optimal Bandwidth (AMISE); with deriv.order >= 0, name of this function is h.amise.
  For display, its plot method calls plot.h.amise, and to add a plot used lines.h.amise.

• Maximum-likelihood cross-validation (MLCV); with deriv.order = 0, name of this function is h.mlcv.
  For display, its plot method calls plot.h.mlcv, and to add a plot used lines.h.mlcv.

• Unbiased cross validation (UCV); with deriv.order >= 0, name of this function is h.ucv.
  For display, its plot method calls plot.h.ucv, and to add a plot used lines.h.ucv.

• Biased cross validation (BCV); with deriv.order >= 0, name of this function is h.bcv.
  For display, its plot method calls plot.h.bcv, and to add a plot used lines.h.bcv.

• Complete cross-validation (CCV); with deriv.order >= 0, name of this function is h.ccv.
  For display, its plot method calls plot.h.ccv, and to add a plot used lines.h.ccv.

• Modified cross-validation (MCV); with deriv.order >= 0, name of this function is h.mcv.
  For display, its plot method calls plot.h.mcv, and to add a plot used lines.h.mcv.

• Trimmed cross-validation (TCV); with deriv.order >= 0, name of this function is h.tcv.
  For display, its plot method calls plot.h.tcv, and to add a plot used lines.h.tcv.

  R> data(trimodal)
  R> h.bcv(x = trimodal, whichbcv = 1, deriv.order = 0, kernel = "gaussian")
  
    Call:           Biased Cross-Validation 1
    Derivative order = 0
    Data: trimodal (200 obs.);      Kernel: gaussian
    Min BCV = 0.004511636;  Bandwidth 'h' = 0.4357812 
	
  R> h.ccv(x = trimodal, deriv.order = 1, kernel = "gaussian")	
  
    Call:           Complete Cross-Validation
    Derivative order = 1 
    Data: trimodal (200 obs.);      Kernel: gaussian
    Min CCV = 0.01985078;   Bandwidth 'h' = 0.5828336
	
  R> h.tcv(x = trimodal, deriv.order = 2, kernel = "gaussian")
  
    Call:           Trimmed Cross-Validation
    Derivative order = 2
    Data: trimodal (200 obs.);      Kernel: gaussian
    Min TCV = -295.563;     Bandwidth 'h' = 0.08908582
	
  R> h.ucv(x = trimodal, deriv.order = 3, kernel = "gaussian")

    Call:           Unbiased Cross-Validation
    Derivative order = 3
    Data: trimodal (200 obs.);      Kernel: gaussian
    Min UCV = -63165.18;    Bandwidth 'h' = 0.1067236  
  

For an overview of this package, see vignette("kedd").

Requirements

R version >= 2.15.0

Licence

This package and its documentation are usable under the terms of the "GNU General Public License", a copy of which is distributed with the package.

References

Alekseev, V. G. (1972). Estimation of a probability density function and its derivatives. Mathematical notes of the Academy of Sciences of the USSR. 12(5), 808–811.

Alexandre, B. T. (2009). Introduction to Nonparametric Estimation. Springer-Verlag, New York.

Bowman, A. W. (1984). An alternative method of cross-validation for the smoothing of kernel density estimates. Biometrika, 71, 353–360.

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

Bowman, A.W. and Azzalini, A. (2003). Computational aspects of nonparametric smoothing with illustrations from the sm library. Computational Statistics and Data Analysis, 42, 545–560.

Bowman, A.W. and Azzalini, A. (2013). sm: Smoothing methods for nonparametric regression and density estimation. R package version 2.2-5.3. Ported to R by B. D. Ripley.

Bhattacharya, P. K. (1967). Estimation of a probability density function and Its derivatives. Sankhya: The Indian Journal of Statistics, Series A, 29, 373–382.

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.

Feluch, W. and Koronacki, J. (1992). A note on modified cross-validation in density estimation. Computational Statistics and Data Analysis, 13, 143–151.

George, R. T. (1990). The maximal smoothing principle in density estimation. Journal of the American Statistical Association, 85, 470–477.

George, R. T. and Scott, D. W. (1985). Oversmoothed nonparametric density estimates. Journal of the American Statistical Association, 80, 209–214.

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.

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.

Jeffrey, S. S. (1996). Smoothing Methods in Statistics. Springer-Verlag, New York.

Jones, M. C. (1992). Differences and derivatives in kernel estimation. Metrika, 39, 335–340.

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.

Jones, M. C. and Kappenman, R. F. (1991). On a class of kernel density estimate bandwidth selectors. Scandinavian Journal of Statistics, 19, 337–349.

Loader, C. (1999). Local Regression and Likelihood. Springer, New York.

Olver, F. W., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (2010). NIST Handbook of Mathematical Functions. Cambridge University Press, New York, USA.

Peter, H. and Marron, J.S. (1987). Estimation of integrated squared density derivatives. Statistics and Probability Letters, 6, 109–115.

Peter, H. and Marron, J.S. (1991). Local minima in cross-validation functions. Journal of the Royal Statistical Society, Series B, 53, 245–252.

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

Rudemo, M. (1982). Empirical choice of histograms and kernel density estimators. Scandinavian Journal of Statistics, 9, 65–78.

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

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.

Schuster, E. F. (1969) Estimation of a probability density function and its derivatives. The Annals of Mathematical Statistics, 40 (4), 1187–1195.

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

Sheather, S. J. and Jones, M. C. (1991). A reliable data-based bandwidth selection method for kernel density estimation. Journal of the Royal Statistical Society, Series B, 53, 683–690.

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

Singh, R. S. (1977). Applications of estimators of a density and its derivatives to certain statistical problems. Journal of the Royal Statistical Society, Series B, 39(3), 357–363.

Stoker, T. M. (1993). Smoothing bias in density derivative estimation. Journal of the American Statistical Association, 88, 855–863.

Stute, W. (1992). Modified cross validation in density estimation. Journal of Statistical Planning and Inference, 30, 293–305.

Tarn, D. (2007). ks: Kernel density estimation and kernel discriminant analysis for multivariate data in R. Journal of Statistical Software, 21(7), 1–16.

Tristen, H. and Jeffrey, S. R. (2008). Nonparametric Econometrics: The np Package. Journal of Statistical Software,27(5).

Venables, W. N. and Ripley, B. D. (2002). Modern Applied Statistics with S. New York: Springer.

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

Wand, M.P. and Ripley, B. D. (2013). KernSmooth: Functions for Kernel Smoothing for Wand and Jones (1995). R package version 2.23-10.

Wolfgang, H. (1991). Smoothing Techniques, With Implementation in S. Springer-Verlag, New York.

Wolfgang, H., Marlene, M., Stefan, S. and Axel, W. (2004). Nonparametric and Semiparametric Models. Springer-Verlag, Berlin Heidelberg.

Wolfgang, H., Marron, J. S. and Wand, M. P. (1990). Bandwidth choice for density derivatives. Journal of the Royal Statistical Society, Series B, 223–232.

See Also

ks, KernSmooth, sm, np, locfit, feature, GenKern.

---

Datasets

Description

A random sample of size 200 from the claw, bimodal, kurtotic, outlier and trimodal Gaussian density.

Usage

data(claw)
data(bimodal)
data(kurtotic)
data(outlier)
data(trimodal)

Format

Numeric vector with length 200.

Details

Generate 200 random numbers, distributed according to a normal mixture, using rnorMix in package nor1mix.

  ## Claw density
  claw <- rnorMix(n=200, MW.nm10)
  plot(MW.nm10)
  
  ## Bimodal density
  bimodal <- rnorMix(n=200, MW.nm7)
  plot( MW.nm7)
  
  ## Kurtotic density
  kurtotic <- rnorMix(n=200, MW.nm4)
  plot(MW.nm4)
  
  ## Outlier density
  outlier <- rnorMix(n=200, MW.nm5)
  plot( MW.nm5)
  
  ## Trimodal density
  trimodal <- rnorMix(n=200, MW.nm9)
  plot(MW.nm9)
  

Source

Randomly generated a normal mixture with the function rnorMix in package nor1mix.

References

Martin, M. (2013). nor1mix: Normal (1-d) mixture models (S3 classes and methods). R package version 1.1-4.

---

Derivatives of Kernel Density Estimator

Description

The (S3) generic function dkde computes the r'th derivative of kernel density estimator for one-dimensional data. Its default method does so with the given kernel and bandwidth $h$ for one-dimensional observations.

Usage

dkde(x, ...)
## Default S3 method:
dkde(x, y = NULL, deriv.order = 0, h, kernel = c("gaussian", 
         "epanechnikov", "uniform", "triangular", "triweight", 
         "tricube", "biweight", "cosine"), ...)

Arguments

x	the data from which the estimate is to be computed.
y	the points of the grid at which the density derivative is to be estimated; the defaults are $\tau * h$ outside of range($x$), where $\tau = 4$.
deriv.order	derivative order (scalar).
h	the smoothing bandwidth to be used, can also be a character string giving a rule to choose the bandwidth, see h.bcv. The default h.ucv.
kernel	a character string giving the smoothing kernel to be used, with default "gaussian".
...	further arguments for (non-default) methods.

Details

A simple estimator for the density derivative can be obtained by taking the derivative of the kernel density estimate. If the kernel $K(x)$ is differentiable $r$ times then the r'th density derivative estimate can be written as:

$\hat{f}^{(r)}_{h}(x)=\frac{1}{nh^{r+1}}\sum_{i=1}^{n} K^{(r)}\left(\frac{x-X_{i}}{h}\right)$

where,

$K^{(r)}(x) = \frac{d^{r}}{d x^{r}} K(x)$

for $r = 0, 1, 2, \dots$

The following assumptions on the density $f^{(r)}(x)$, the bandwidth $h$, and the kernel $K(x)$:

1. The $(r+2)$ derivative $f^{(r+2)}(x)$ is continuous, square integrable and ultimately monotone.

2. $\lim_{n \to \infty} h = 0$ and $\lim_{n \to \infty}n h^{2r+1} = \infty$ i.e., as the number of samples $n$ is increased $h$ approaches zero at a rate slower than $1/n^{2r+1}$.

3. $K(x) \geq 0$ and $\int_{R} K(x) dx = 1$. The kernel function is assumed to be symmetric about the origin i.e., $\int_{R} xK^{(r)}(x) dx = 0$ for even $r$ and has finite second moment i.e., $\mu_{2}(K)=\int_{R}x^{2} K(x) dx < \infty$.

Some theoretical properties of the estimator $\hat{f}^{(r)}_{h}$ have been investigated, among others, by Bhattacharya (1967), Schuster (1969). Let us now turn to the statistical properties of estimator. We are interested in the mean squared error since it combines squared bias and variance.

The bias can be written as:

$E\left[\hat{f}^{(r)}_{h}(x)\right]- f^{(r)}(x) = \frac{1}{2}h^{2}\mu_{2}(K) f^{(r+2)}(x)+o(h^{2})$

The variance of the estimator can be written as:

$VAR\left[\hat{f}^{(r)}_{h}(x)\right]=\frac{f(x) R\left(K^{(r)}\right)}{nh^{2r+1}} + o(1/nh^{2r+1})$

with, $R\left(K^{(r)}\right) = \int_{R} \left(K^{(r)}(x)\right)^{2}dx.$

The MSE (Mean Squared Error) for kernel density derivative estimators can be written as:

$MSE\left(\hat{f}^{(r)}_{h}(x),f^{(r)}(x)\right)=\frac{f(x)R\left(K^{(r)}\right)}{nh^{2r+1}}+\frac{1}{4}h^{4}\mu_{2}^{2}(K) f^{(r+1)}(x)^{2}+o(h^{4}+1/nh^{2r+1})$

It follows that the MSE-optimal bandwidth for estimating $\hat{f}^{(r)}_{h}S(x)$, is of order $n^{-1/(2r+5)}$. Therefore, the estimation of $\hat{f}^{(1)}_{h}(x)$ requires a bandwidth of order $n^{-1/7}$ compared to the optimal $n^{-1/5}$ for estimating $f(x)$ itself. It reveals the increasing difficulty in problems of estimating higher derivatives.

The MISE (Mean Integrated Squared Error) can be written as:

$MISE\left(\hat{f}^{(r)}_{h}(x),f^{(r)}(x)\right)=AMISE\left(\hat{f}^{(r)}_{h}(x),f^{(r)}(x)\right)+o(h^{4}+1/nh^{2r+1})$

where,

$AMISE\left(\hat{f}^{(r)}_{h}(x),f^{(r)}(x)\right)=\frac{1}{nh^{2r+1}}R\left(K^{(r)}\right)+\frac{1}{4}h^{4}\mu_{2}^{2}(K)R\left(f^{(r+2)}\right)$

with: $R\left(f^{(r)}(x)\right) = \int_{R} \left(f^{(r)}(x)\right)^{2}dx.$
The performance of kernel is measured by MISE or AMISE (Asymptotic MISE).

If the bandwidth h is missing from dkde, then the default bandwidth is h.ucv(x,deriv.order,kernel) (Unbiased cross-validation, see h.ucv).
For more details see references.

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	the bandwidth value to use.
eval.points	the coordinates of the points where the density derivative is estimated.
est.fx	the estimated density derivative values.

Note

This function are available in other packages such as KernSmooth, sm, np, GenKern and locfit if deriv.order=0, and in ks package for Gaussian kernel only if 0 <= deriv.order <= 10.

Author(s)

Arsalane Chouaib Guidoum [email protected]

References

Alekseev, V. G. (1972). Estimation of a probability density function and its derivatives. Mathematical notes of the Academy of Sciences of the USSR. 12 (5), 808–811.

Alexandre, B. T. (2009). Introduction to Nonparametric Estimation. Springer-Verlag, New York.

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

Bhattacharya, P. K. (1967). Estimation of a probability density function and Its derivatives. Sankhya: The Indian Journal of Statistics, Series A, 29, 373–382.

Jeffrey, S. S. (1996). Smoothing Methods in Statistics. Springer-Verlag, New York.

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.

Schuster, E. F. (1969) Estimation of a probability density function and its derivatives. The Annals of Mathematical Statistics, 40 (4), 1187–1195.

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

Stoker, T. M. (1993). Smoothing bias in density derivative estimation. Journal of the American Statistical Association, 88, 855–863.

Venables, W. N. and Ripley, B. D. (2002). Modern Applied Statistics with S. New York: Springer.

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.

See Also

plot.dkde, see density in package "stats" if deriv.order = 0, and kdde in package ks.

Examples

## EXAMPLE 1:  Simple example of a Gaussian density derivative

x <- rnorm(100)
dkde(x,deriv.order=0)  ## KDE of f
dkde(x,deriv.order=1)  ## KDDE of d/dx f
dkde(x,deriv.order=2)  ## KDDE of d^2/x^2 f
dkde(x,deriv.order=3)  ## KDDE of d^3/x^3 f
oldpar <- par(no.readonly = TRUE)
dev.new()
par(mfrow=c(2,2))
plot(dkde(x,deriv.order=0))
plot(dkde(x,deriv.order=1))
plot(dkde(x,deriv.order=2))
plot(dkde(x,deriv.order=3))
par(oldpar)

## EXAMPLE 2: Bimodal Gaussian density derivative
## show the kernels in the dkde parametrization

fx  <- function(x) 0.5 * dnorm(x,-1.5,0.5) + 0.5 * dnorm(x,1.5,0.5)
fx1 <- function(x) 0.5 *(-4*x-6)* dnorm(x,-1.5,0.5) + 0.5 *(-4*x+6) * 
                   dnorm(x,1.5,0.5)
				   
## 'h = 0.3' ; 'Derivative order = 0'

kernels <- eval(formals(dkde.default)$kernel)
dev.new()
plot(dkde(bimodal,h=0.3),sub=paste("Derivative order = 0",";",
     "Bandwidth =0.3 "),ylim=c(0,0.5), main = "Bimodal Gaussian Density")
for(i in 2:length(kernels))
   lines(dkde(bimodal, h = 0.3, kernel =  kernels[i]), col = i)
curve(fx,add=TRUE,lty=8)
legend("topright", legend = c(TRUE,kernels), col = c("black",seq(kernels)),
          lty = c(8,rep(1,length(kernels))),cex=0.7, inset = .015)
	   
## 'h = 0.6' ; 'Derivative order = 1'

kernels <- eval(formals(dkde.default)$kernel)[-3]
dev.new()
plot(dkde(bimodal,deriv.order=1,h=0.6),main = "Bimodal Gaussian Density Derivative",sub=paste
         ("Derivative order = 1",";","Bandwidth =0.6"),ylim=c(-0.6,0.6))
for(i in 2:length(kernels))
   lines(dkde(bimodal,deriv.order=1, h = 0.6, kernel =  kernels[i]), col = i)
curve(fx1,add=TRUE,lty=8)
legend("topright", legend = c(TRUE,kernels), col = c("black",seq(kernels)),
          lty = c(8,rep(1,length(kernels))),cex=0.7, inset = .015)

---

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 [email protected]

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)

---

Biased Cross-Validation for Bandwidth Selection

Description

The (S3) generic function h.bcv computes the biased cross-validation bandwidth selector of r'th derivative of kernel density estimator one-dimensional.

Usage

h.bcv(x, ...)
## Default S3 method:
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"), ...)

Arguments

x	vector of data values.
whichbcv	method selected, 1 = BCV1 or 2 = BCV2, see details.
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.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).

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.
whichbcv	method selected.
h	value of bandwidth parameter.
min.bcv	the minimal BCV value.

Author(s)

Arsalane Chouaib Guidoum [email protected]

References

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.

See Also

plot.h.bcv, see bw.bcv in package "stats" and bcv in package MASS for Gaussian kernel only if deriv.order = 0, Hbcv for bivariate data in package ks for Gaussian kernel only if deriv.order = 0, kdeb in package locfit if deriv.order = 0.

Examples

## 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)

---

Complete Cross-Validation for Bandwidth Selection

Description

The (S3) generic function h.ccv computes the complete cross-validation bandwidth selector of r'th derivative of kernel density estimator one-dimensional.

Usage

h.ccv(x, ...)
## Default S3 method:
h.ccv(x, deriv.order = 0, lower = 0.1 * hos, upper = hos, 
         tol = 0.1 * lower, kernel = c("gaussian", "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.ccv complete cross-validation implements for choosing the bandwidth $h$ of a r'th derivative kernel density estimator.

Jones and Kappenman (1991) proposed a so-called complete cross-validation (CCV) in kernel density estimator. This method can be extended to the estimation of derivative of the density, basing our estimate of integrated squared density derivative (Peter and Marron 1987) on the $\bar{\theta}_{r}(h)$'s, we get the following, start from $R\left(\hat{f}_{h}^{(r)}\right) - \bar{\theta}_{r}(h)$ as an estimate of MISE. Thus, $\hat{h}^{(r)}_{CCV}$, say, is the $h$ that minimises:

$CCV(h;r)=R\left(\hat{f}_{h}^{(r)}\right)-\bar{\theta}_{r}(h)+\frac{1}{2}\mu_{2}(K) h^{2} \bar{\theta}_{r+1}(h)+\frac{1}{24}\left(6\mu_{2}^{2}(K) -\delta(K)\right)h^{4}\bar{\theta}_{r+2}(h)$

with

$R\left(\hat{f}_{h}^{(r)}\right) = \int \left(\hat{f}_{h}^{(r)}(x)\right)^{2} dx = \frac{R\left(K^{(r)}\right)}{nh^{2r+1}} + \frac{(-1)^{r}}{n (n-1) h^{2r+1}} \sum_{i=1}^{n}\sum_{j=1;j \neq i}^{n} K^{(r)} \ast K^{(r)} \left(\frac{X_{j}-X_{i}}{h}\right)$

and

$\bar{\theta}_{r}(h)= \frac{(-1)^r}{n(n-1) h^{2r+1}} \sum_{i=1}^{n} \sum_{j=1;j \neq i}^{n} K^{(2r)} \left(\frac{X_{j}-X_{i}}{h}\right)$

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 $\mu_{2}(K) = \int_{R}x^{2} K(x) dx$, $\delta(K) = \int_{R}x^{4} K(x) dx$.

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.
min.ccv	the minimal CCV value.

Author(s)

Arsalane Chouaib Guidoum [email protected]

References

Jones, M. C. and Kappenman, R. F. (1991). On a class of kernel density estimate bandwidth selectors. Scandinavian Journal of Statistics, 19, 337–349.

Peter, H. and Marron, J.S. (1987). Estimation of integrated squared density derivatives. Statistics and Probability Letters, 6, 109–115.

See Also

plot.h.ccv.

Examples

## Derivative order = 0

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

## Derivative order = 1

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

---

Modified Cross-Validation for Bandwidth Selection

Description

The (S3) generic function h.mcv computes the modified cross-validation bandwidth selector of r'th derivative of kernel density estimator one-dimensional.

Usage

h.mcv(x, ...)
## Default S3 method:
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"), ...)

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.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).

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.
min.mcv	the minimal MCV value.

Author(s)

Arsalane Chouaib Guidoum [email protected]

References

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.

See Also

plot.h.mcv.

Examples

## Derivative order = 0

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

## Derivative order = 1

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

---

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

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.

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 $MLCV(h)$:

$h_{mlcv} = \arg \max_{h} MLCV(h) = \arg \max_{h} \left(n^{-1} \sum_{i=1}^{n} \log\left[\sum_{j \neq i} K \left(\frac{X_{j}-X_{i}}{h}\right)\right]-\log[(n-1)h]\right)$

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.

Author(s)

Arsalane Chouaib Guidoum [email protected]

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")

---

Trimmed Cross-Validation for Bandwidth Selection

Description

The (S3) generic function h.tcv computes the trimmed cross-validation bandwidth selector of r'th derivative of kernel density estimator one-dimensional.

Usage

h.tcv(x, ...)
## Default S3 method:
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"), ...)

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.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).

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.
min.tcv	the minimal TCV value.

Author(s)

Arsalane Chouaib Guidoum [email protected]

References

Feluch, W. and Koronacki, J. (1992). A note on modified cross-validation in density estimation. Computational Statistics and Data Analysis, 13, 143–151.

See Also

plot.h.tcv.

Examples

## Derivative order = 0

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

## Derivative order = 1

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

---

Unbiased (Least-Squares) Cross-Validation for Bandwidth Selection

Description

The (S3) generic function h.ucv computes the unbiased (least-squares) cross-validation bandwidth selector of r'th derivative of kernel density estimator one-dimensional.

Usage

h.ucv(x, ...)
## Default S3 method:
h.ucv(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"), ...)

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.ucv unbiased (least-squares) cross-validation implements for choosing the bandwidth $h$ of a r'th derivative kernel density estimator.

Rudemo (1982) and Bowman (1984) proposed a so-called unbiased (least-squares) cross-validation (UCV) in kernel density estimator. An adaptation of unbiased cross-validation is proposed by Wolfgang et al. (1990) for bandwidth choice in the r'th derivative of kernel density estimator. The essential idea of this methods, for the estimation of $f^{(r)}(x)$ ($r$ is derivative order), is to use the bandwidth $h$ which minimizes the function:

$UCV(h;r) = \int \left(\hat{f}_{h}^{(r)}(x)\right)^{2} - 2n^{-1}(-1)^{r}\sum_{i=1}^{n} \hat{f}_{h,i}^{(2r)}(X_{i})$

The bandwidth minimizing this function is:

$\hat{h}^{(r)}_{ucv} = \arg \min_{h^{(r)}} UCV(h;r)$

for $r = 0, 1, 2, \dots$
where

$\int \left(\hat{f}_{h}^{(r)}(x)\right)^{2} = \frac{R\left(K^{(r)}\right)}{nh^{2r+1}} + \frac{(-1)^{r}}{n (n-1) h^{2r+1}} \sum_{i=1}^{n}\sum_{j=1;j \neq i}^{n} K^{(r)} \ast K^{(r)} \left(\frac{X_{j}-X_{i}}{h}\right)$

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).
The estimate $\hat{f}_{h,i}^{(2r)}(x)$ on the subset $\{X_{j}\}_{j \neq i}$ denoting the leave-one-out estimator, can be written:

$\hat{f}_{h,i}^{(2r)}(X_{i}) = \frac{1}{(n-1) h^{2r+1}} \sum_{j \neq i} K^{(2r)} \left(\frac{X_{j}-X_{i}}{h}\right)$

The function $UCV(h;r)$ is unbiased cross-validation in the sense that $E[UCV]=MISE[\hat{f}_{h}^{(r)}(x)]-R(f^{(r)}(x))$ (see, Scott and George 1987). Can be simplified to give the computationally:

$UCV(h;r) = \frac{R\left(K^{(r)}\right)}{nh^{2r+1}} + \frac{(-1)^{r}}{n (n-1) h^{2r+1}} \sum_{i=1}^{n}\sum_{j=1 ;j \neq i}^{n} \left(K^{(r)} \ast K^{(r)} -2K^{(2r)}\right) \left(\frac{X_{j}-X_{i}}{h}\right)$

where $R\left(K^{(r)}\right) = \int_{R} K^{(r)}(x)^{2} dx$.

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.
min.ucv	the minimal UCV value.

Author(s)

Arsalane Chouaib Guidoum [email protected]

References

Bowman, A. (1984). An alternative method of cross-validation for the smoothing of kernel density estimates. Biometrika, 71, 353–360.

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.

Rudemo, M. (1982). Empirical choice of histograms and kernel density estimators. Scandinavian Journal of Statistics, 9, 65–78.

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.

Wolfgang, H., Marron, J. S. and Wand, M. P. (1990). Bandwidth choice for density derivatives. Journal of the Royal Statistical Society, Series B, 223–232.

See Also

plot.h.ucv, see bw.ucv in package "stats" and ucv in package MASS for Gaussian kernel only if deriv.order = 0, hlscv in package ks for Gaussian kernel only if 0 <= deriv.order <= 5, kdeb in package locfit if deriv.order = 0.

Examples

## Derivative order = 0

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

## Derivative order = 1

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

---

Convolutions of r'th Derivative for Kernel Function

Description

The (S3) generic function kernel.conv computes the convolution of r'th derivative for kernel function.

Usage

kernel.conv(x, ...)
## Default S3 method:
kernel.conv(x = NULL, deriv.order = 0,kernel = c("gaussian","epanechnikov", 
             "uniform", "triangular", "triweight", "tricube", 
             "biweight", "cosine", "silverman"), ...)

Arguments

x	points at which the convolution of kernel derivative is to be evaluated.
deriv.order	derivative order (scalar).
kernel	a character string giving the smoothing kernel to be used, with default "gaussian".
...	further arguments for (non-default) methods.

Details

The convolution of r'th derivative for kernel function is written $K^{(r)}\ast K^{(r)}$. It is defined as the integral of the product of the derivative for kernel. As such, it is a particular kind of integral transform:

$K^{(r)} \ast K^{(r)}(x) = \int_{-\infty}^{+\infty} K^{(r)}(y)K^{(r)}(x-y)dy$

where:

$K^{(r)}(x) = \frac{d^{r}}{d x^{r}} K(x)$

for $r = 0, 1, 2, \dots$

Value

kernel	name of kernel to use.
deriv.order	the derivative order to use.
x	the n coordinates of the points where the convolution of kernel derivative is evaluated.
kx	the convolution of kernel derivative values.

Author(s)

Arsalane Chouaib Guidoum [email protected]

References

Olver, F. W., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (2010). NIST Handbook of Mathematical Functions. Cambridge University Press, New York, USA.

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

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.

Wolfgang, H. (1991). Smoothing Techniques, With Implementation in S. Springer-Verlag, New York.

See Also

plot.kernel.conv, kernapply in package "stats" for computes the convolution between an input sequence, and convolve use the Fast Fourier Transform (fft) to compute the several kinds of convolutions of two sequences.

Examples

kernels <- eval(formals(kernel.conv.default)$kernel)
kernels

## gaussian
kernel.conv(x = 0,kernel=kernels[1],deriv.order=0)
kernel.conv(x = 0,kernel=kernels[1],deriv.order=1)

## silverman
kernel.conv(x = 0,kernel=kernels[9],deriv.order=0)
kernel.conv(x = 0,kernel=kernels[9],deriv.order=1)

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Derivatives of Kernel Function

Description

The (S3) generic function kernel.fun computes the r'th derivative for kernel density.

Usage

kernel.fun(x, ...)
## Default S3 method:
kernel.fun(x = NULL, deriv.order = 0, kernel = c("gaussian","epanechnikov", 
            "uniform", "triangular", "triweight", "tricube", 
            "biweight", "cosine", "silverman"), ...)

Arguments

x	points at which the derivative of kernel function is to be evaluated.
deriv.order	derivative order (scalar).
kernel	a character string giving the smoothing kernel to be used, with default "gaussian".
...	further arguments for (non-default) methods.

Details

We give a short survey of some kernels functions $K(x;r)$; where $r$ is derivative order,

• Gaussian: $K(x;\infty) =\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{x^{2}}{2}\right)1_{]-\infty,+\infty[}$

• Epanechnikov: $K(x;2)=\frac{3}{4}(1-x^{2})1_{(|x| \leq 1)}$

• uniform (rectangular): $K(x;0)=\frac{1}{2}1_{(|x| \leq 1)}$

• triangular: $K(x;1)=(1-|x|)1_{(|x| \leq 1)}$

• triweight: $K(x;6)=\frac{35}{32}(1-x^{2})^{3} 1_{(|x| \leq 1)}$

• tricube: $K(x;9)=\frac{70}{81}(1-|x|^{3})^{3} 1_{(|x| \leq 1)}$

• biweight: $K(x;4)=\frac{15}{16}(1-x^{2})^{2} 1_{(|x| \leq 1)}$

• cosine: $K(x;\infty)=\frac{\pi}{4}\cos\left(\frac{\pi}{2}x\right) 1_{(|x| \leq 1)}$

• Silverman: $K(x;r \bmod 8)=\frac{1}{2}\exp\left(-\frac{|x|}{\sqrt{2}}\right)\sin\left(\frac{|x|}{\sqrt{2}}+\frac{\pi}{4}\right)1_{]-\infty,+\infty[}$

The r'th derivative for kernel function $K(x)$ is written:

$K^{(r)}(x) = \frac{d^{r}}{d x^{r}} K(x)$

for $r = 0, 1, 2, \dots$
The r'th derivative of the Gaussian kernel $K(x)$ is given by:

$K^{(r)}(x) = (-1)^{r} H_{r}(x) K(x)$

where $H_{r}(x)$ is the r'th Hermite polynomial. This polynomials are set of orthogonal polynomials, for more details see, hermite.h.polynomials in package orthopolynom.

Value

kernel	name of kernel to use.
deriv.order	the derivative order to use.
x	the n coordinates of the points where the derivative of kernel function is evaluated.
kx	the kernel derivative values.

Author(s)

Arsalane Chouaib Guidoum [email protected]

References

Jones, M. C. (1992). Differences and derivatives in kernel estimation. Metrika, 39, 335–340.

Olver, F. W., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (2010). NIST Handbook of Mathematical Functions. Cambridge University Press, New York, USA.

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

See Also

plot.kernel.fun, deriv and D in package "stats" for symbolic and algorithmic derivatives of simple expressions.

Examples

kernels <- eval(formals(kernel.fun.default)$kernel)
kernels

## gaussian
kernel.fun(x = 0,kernel=kernels[1],deriv.order=0)
kernel.fun(x = 0,kernel=kernels[1],deriv.order=1)

## silverman
kernel.fun(x = 0,kernel=kernels[9],deriv.order=0)
kernel.fun(x = 0,kernel=kernels[9],deriv.order=1)

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Plot for Kernel Density Derivative Estimate

Description

The plot.dkde function loops through calls to the dkde function. Plot for kernel density derivative estimate for 1-dimensional data.

Usage

## S3 method for class 'dkde'
plot(x, fx = NULL, ...)
## S3 method for class 'dkde'
lines(x, ...)

Arguments

x	object of class dkde (output from dkde).
fx	add to graphics the true density derivative (class :function), to compare it by the density derivative to estimate.
...	other graphics parameters, see par in package "graphics".

Details

The 1-d plot is a standard plot of a 1-d curve. If !is.null(fx) then a true density derivative is added.

Value

Plot of 1-d kernel density derivative estimates are sent to graphics window.

Author(s)

Arsalane Chouaib Guidoum [email protected]

See Also

dkde, plot.density in package "stats" if deriv.order = 0.

Examples

plot(dkde(kurtotic,deriv.order=0,kernel="gaussian"),sub="")
lines(dkde(kurtotic,deriv.order=0,kernel="biweight"),col="red")

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Plot for Asymptotic Mean Integrated Squared Error

Description

The plot.h.amise function loops through calls to the h.amise function. Plot for asymptotic mean integrated squared error function for 1-dimensional data.

Usage

## S3 method for class 'h.amise'
plot(x, seq.bws=NULL, ...)
## S3 method for class 'h.amise'
lines(x,seq.bws=NULL, ...)

Arguments

x	object of class h.amise (output from h.amise).
seq.bws	the sequence of bandwidths in which to compute the AMISE function. By default, the procedure defines a sequence of 50 points, from 0.15*hos to 2*hos (Over-smoothing).
...	other graphics parameters, see par in package "graphics".

Value

Plot of 1-d AMISE function are sent to graphics window.

kernel	name of kernel to use.
deriv.order	the derivative order to use.
seq.bws	the sequence of bandwidths.
amise	the values of the AMISE function in the bandwidths grid.

Author(s)

Arsalane Chouaib Guidoum [email protected]

See Also

h.amise.

Examples

plot(h.amise(bimodal,deriv.order=0))

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