kde.local.test: Kernel density based local two-sample comparison test

In ks: Kernel Smoothing

Kernel density based local two-sample comparison test

Description

Kernel density based local two-sample comparison test for 1- to 6-dimensional data.

Usage

kde.local.test(x1, x2, H1, H2, h1, h2, fhat1, fhat2, gridsize, binned, 
   bgridsize, verbose=FALSE, supp=3.7, mean.adj=FALSE, signif.level=0.05,
   min.ESS, xmin, xmax)

Arguments

x1, x2	vector/matrix of data values
H1, H2, h1, h2	bandwidth matrices/scalar bandwidths. If these are missing, Hpi or hpi is called by default.
fhat1, fhat2	objects of class kde
binned	flag for binned estimation
gridsize	vector of grid sizes
bgridsize	vector of binning grid sizes
verbose	flag to print out progress information. Default is FALSE.
supp	effective support for normal kernel
mean.adj	flag to compute second order correction for mean value of critical sampling distribution. Default is FALSE. Currently implemented for d<=2 only.
signif.level	significance level. Default is 0.05.
min.ESS	minimum effective sample size. See below for details.
xmin, xmax	vector of minimum/maximum values for grid

Details

The null hypothesis is H_0(\bold{x}): f_1(\bold{x}) = f_2(\bold{x}) where f_1, f_2 are the respective density functions. The measure of discrepancy is U(\bold{x}) = [f_1(\bold{x}) - f_2(\bold{x})]^2. Duong (2013) shows that the test statistic obtained, by substituting the KDEs for the true densities, has a null distribution which is asymptotically chi-squared with 1 d.f.

The required input is either x1,x2 and H1,H2, or fhat1,fhat2, i.e. the data values and bandwidths or objects of class kde. In the former case, the kde objects are created. If the H1,H2 are missing then the default are the plug-in selectors Hpi. Likewise for missing h1,h2.

The mean.adj flag determines whether the second order correction to the mean value of the test statistic should be computed. min.ESS is borrowed from Godtliebsen et al. (2002) to reduce spurious significant results in the tails, though by it is usually not required for small to moderate sample sizes.

Value

A kernel two-sample local significance is an object of class kde.loctest which is a list with fields:

fhat1, fhat2	kernel density estimates, objects of class kde
chisq	chi squared test statistic
pvalue	matrix of local p-values at each grid point
fhat.diff	difference of KDEs
mean.fhat.diff	mean of the test statistic
var.fhat.diff	variance of the test statistic
fhat.diff.pos	binary matrix to indicate locally significant fhat1 > fhat2
fhat.diff.neg	binary matrix to indicate locally significant fhat1 < fhat2
n1, n2	sample sizes
H1, H2, h1, h2	bandwidth matrices/scalar bandwidths

References

Duong, T. (2013) Local significant differences from non-parametric two-sample tests. Journal of Nonparametric Statistics, 25, 635-645.

Godtliebsen, F., Marron, J.S. & Chaudhuri, P. (2002) Significance in scale space for bivariate density estimation. Journal of Computational and Graphical Statistics, 11, 1-22.

See Also

kde.test, plot.kde.loctest

Examples

data(crabs, package="MASS")
x1 <- crabs[crabs$sp=="B", 4]
x2 <- crabs[crabs$sp=="O", 4]
loct <- kde.local.test(x1=x1, x2=x2)
plot(loct, ylim=c(-0.08,0.12))
cols <- hcl.colors(palette="Dark2",2)
plot(loct$fhat1, add=TRUE, col=cols[1])
plot(loct$fhat2, add=TRUE, col=cols[2])

## see examples in ? plot.kde.loctest

ks documentation built on Sept. 30, 2024, 9:15 a.m.