# Kernel density based local two-sample comparison test

### Description

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

### Usage

1 2 3 | ```
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)
``` |

### Arguments

`x1,x2` |
vector/matrix of data values |

`H1,H2,h1,h2` |
bandwidth matrices/scalar bandwidths. If these are missing, |

`fhat1,fhat2` |
objects of class |

`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. |

### 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 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.

### 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 |

`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 signficant fhat1 > fhat2 |

`fhat.diff.neg` |
binary matrix to indicate locally signficant fhat1 < fhat2 |

`n1,n2` |
sample sizes |

`H1,H2,h1,h2` |
bandwidth matrices/scalar bandwidths |

### References

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.

### See Also

`kde.test`

, `plot.kde.loctest`

### Examples

1 2 3 4 5 6 7 | ```
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
``` |