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

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

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. Default is FALSE.

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

A kernel two-sample local significance is an object of class
`kde.loctest`

which is a list with fields:

kernel density estimates, objects of class `kde`

chi squared test statistic

matrix of local p-values at each grid point

difference of KDEs

mean of the test statistic

variance of the test statistic

binary matrix to indicate locally signficant fhat1 > fhat2

binary matrix to indicate locally signficant fhat1 < fhat2

sample sizes

bandwidth matrices/scalar bandwidths

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 plugin
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.

Duong, T. (2013) Local signficant 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.

```
# NOT RUN {
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
# }
```

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