USP {USP}

Permutation test of independence.

Description

Carry out an independence test of the independence of two samples, give two kernel matrices J and K, as described in Section 7.1 of (Berrett et al. 2021). We calculate the test statistic and null statistics using the function KernStat, before comparing them to produce a p-value. For the featured examples considered these matrices can be calculated using FourierKernel or InfKern. Alternatively, if a different basis is to be used, then the kernels can be entered separately.

Usage

USP(J, K, B = 999, ties.method = "standard", nullstats = FALSE)

Arguments

J

n \times n kernel matrix corresponding to first sample.

K

n \times n kernel matrix corresponding to second sample.

B

The number of permutation used to calibrate the test.

ties.method

If "standard" then calculate the p-value as in (5) of (Berrett et al. 2021), which is slightly conservative. If "random" then break ties randomly. This preserves Type I error control.

nullstats

If TRUE, returns a vector of the null statistic values.

Value

Returns the p-value for this independence test and the value of the test statistic, D_n, as defined in (Berrett et al. 2021). If nullstats=TRUE is used, then the function also returns a vector of the null statistics.

References

Berrett TB, Kontoyiannis I, Samworth RJ (2021). “Optimal rates for independence testing via U-statistic permutation tests.” Annals of Statistics, to appear.

Examples

x=runif(100); y=runif(100); M=3
J=FourierKernel(x,M); K=FourierKernel(y,M)
USP(J,K,999)

n=50; r=0.6; Ndisc=1000; t=1/Ndisc
X=matrix(rep(0,Ndisc*n),nrow=n); Y=matrix(rep(0,Ndisc*n),nrow=n)
for(i in 1:n){
 x = rnorm(Ndisc, mean=0, sd= 1)
 se = sqrt(1 - r^2) #standard deviation of error
 e = rnorm(Ndisc, mean=0, sd=se)
 y = r*x + e
 X[i,] = cumsum(x*sqrt(t))
 Y[i,] = cumsum(y*sqrt(t))
}
J=InfKern(X,2,1); K=InfKern(Y,2,1)
USP(J,K,999)