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kde.local.test

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(x): f_1(x) = f_2(x) where f_1, f_2 are the respective density functions. The measure of discrepancy is U(x) = [f_1(x) - f_2(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.

Examples

library(MASS)
x1 <- crabs[crabs$sp=="B", 4]
x2 <- crabs[crabs$sp=="O", 4]
loct <- kde.local.test(x1=x1, x2=x2)
plot(loct)

## see examples in ? plot.kde.loctest

ks

Kernel Smoothing

v1.12.0

GPL-2 | GPL-3

Authors

Tarn Duong [aut, cre], Matt Wand [ctb], Jose Chacon [ctb], Artur Gramacki [ctb]

Initial release

2021-02-06