ks (version 1.8.11)

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, 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

fhat1,fhat2

objects of class kde

binned

flag for binned estimation. Default is FALSE.

bgridsize

vector of binning grid sizes

verbose

flag to print out progress information. Default is FALSE.

supp

effective support for normal kernel [-supp, supp]

mean.adj

flag to compute second order correction for mean value of critical sampling distribution. Default is FALSE. Currently implemented for d

signif.level

significance level. Default in 0.05.

min.ESS

minimum effective sample size. See below for details.

Value

• 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.fiff
  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/bandwidths

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 (2012) show that the test statistic obtained, by substituting the KDEs for the true densities, has a null distribution which is asymptotically chi-squared with d d.f, where d is the data dimension.

The required input is either x1,x2 and H1,H2, or fhat1,fhat2. That is, the data values and bandwidths or a KDE object of class kde. In the former case, a kde object is created with plug-in bandwidth Hpi().

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.

References

Duong, T. (2012) Local signficant differences from non-parametric two-sample tests. Submitted. 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

Examples

## univariate example
mus1 <- 0
sigmas1 <- 1
props1 <-1
    
mus2 <- c(0, -1, 1)
sigmas2 <- c(1, 1/4, 1/4)
props2 <- c(1/2, 1/4, 1/4)

x <- rnorm.mixt(n=1000, mus=mus1, sigmas=sigmas1, props=props1)
y <- rnorm.mixt(n=1000, mus=mus2, sigmas=sigmas2, props=props2)
loct <- kde.local.test(x1=x, x2=y, binned=TRUE)
plot(loct, lcol=2)     

## bivariate example
mus1 <- rbind(c(1,-1), c(-1,1))
Sigmas1 <- rbind(invvech(c(4/9, 4/15, 4/9)), invvech(c(4/9, 4/15, 4/9)))
props1 <- c(1,1)/2
    
mus2 <- rbind(c(1,-1), c(-1,1))
Sigmas2 <- rbind(invvech(c(4/9, 14/45, 4/9)), 4/9*diag(2))
props2 <- c(1,1)/2

x <- rmvnorm.mixt(n=10000, mus=mus1, Sigmas=Sigmas1, props=props1)
y <- rmvnorm.mixt(n=10000, mus=mus2, Sigmas=Sigmas2, props=props2)
loct <- kde.local.test(x1=x, x2=y, binned=TRUE)
plot(loct)