Description Usage Arguments Details Value Author(s) References See Also Examples
The (S3) generic function h.bcv
computes the biased
cross-validation bandwidth selector of r'th derivative of
kernel density estimator one-dimensional.
1 2 3 4 5 |
x |
vector of data values. |
whichbcv |
method selected, |
deriv.order |
derivative order (scalar). |
lower, upper |
range over which to minimize. The default is
almost always satisfactory. |
tol |
the convergence tolerance for |
kernel |
a character string giving the smoothing kernel to be used, with default
|
... |
further arguments for (non-default) methods. |
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)(h;r) and bar(theta)(h;r) (Peter and Marron 1987, Jones and Kappenman 1991) by:
hat(theta)(h;r) = (-1)^r / n(n-1) h^(2r+1) sum(sum(K(.;r)*K(.;r)(x(j)-x(i)/h)), i=1...n, j=1...n, j != i)
and
bar(theta)(h;r) = (-1)^r / n(n-1) h^(2r+1) sum(sum(K((x(j)-x(i)/h);2r)), i=1...n, j=1...n, j != i)
Scott and George (1987) proposed using hat(theta)(h;r) to estimate f(x;r). Thus, h(r)_(BCV1), say, is the h that minimises:
BCV1(h;r) = R(K(x;r))/ n h^(2r+1) + 0.25 mu(K(x))^2 h^4 hat(theta)(h;r+2)
and we define h(r)_(BCV2) as the minimiser of (Jones and Kappenman 1991):
BCV2(h;r) = R(K(x;r))/ n h^(2r+1) + 0.25 mu(K(x))^2 h^4 bar(theta)(h;r+2)
where K(x;r)*K(x;r) is the convolution of the r'th derivative kernel function K(x;r)
(see kernel.conv
and kernel.fun
); R(K(x;r)) = int K(x;r)^2 dx and mu(K(x)) = int 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).
x |
data points - same as input. |
data.name |
the deparsed name of the |
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. |
Arsalane Chouaib Guidoum acguidoum@usthb.dz
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.
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
.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 |
Call: Biased Cross-Validation 1
Derivative order = 0
Data: x (100 obs.); Kernel: gaussian
Min BCV = 0.00623253; Bandwidth 'h' = 0.5477454
Call: Biased Cross-Validation 2
Derivative order = 0
Data: x (100 obs.); Kernel: gaussian
Min BCV = 0.004457027; Bandwidth 'h' = 0.4619819
Call: Biased Cross-Validation 1
Derivative order = 0
Data: kurtotic (200 obs.); Kernel: gaussian
Min BCV = 0.01403079; Bandwidth 'h' = 0.6190886
Call: Biased Cross-Validation 1
Derivative order = 1
Data: kurtotic (200 obs.); Kernel: gaussian
Min BCV = 0.02761328; Bandwidth 'h' = 0.9717073
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