USPFourier {USP}

Independence test for continuous data

Description

Performs a permutation test of independence between two univariate continuous random variables, using the Fourier basis to construct the test statistic, as described in (Berrett et al. 2021).

Usage

USPFourier(x, y, M, B = 999, ties.method = "standard", nullstats = FALSE)

Arguments

x

A vector containing the first sample, with each entry in [0,1].

y

A vector containing the second sample, with each entry in [0,1].

M

The maximum frequency to use in the Fourier basis.

B

The number of permutation to use when calibrating 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(10); y=x^2
USPFourier(x,y,1,999)

n=100; w=2; x=integer(n); y=integer(n); m=300
unifdata=matrix(runif(2*m,min=0,max=1),ncol=2); x1=unifdata[,1]; y1=unifdata[,2]
unif=runif(m); prob=0.5*(1+sin(2*pi*w*x1)*sin(2*pi*w*y1)); accept=(unif<prob);
Data1=unifdata[accept,]; x=Data1[1:n,1]; y=Data1[1:n,2]
plot(x,y)
USPFourier(x,y,2,999)

x=runif(100); y=runif(100)
test=USPFourier(x,y,3,999,nullstats=TRUE)
plot(density(test$NullStats,from=min(test$NullStats),to=max(max(test$NullStats),test$TestStat)),
         xlim=c(min(test$NullStats),max(max(test$NullStats),test$TestStat)),main="Test Statistics")
abline(v=test$TestStat,col=2); TestStats=c(test$TestStat,test$NullStats)
abline(v=quantile(TestStats,probs=0.95),lty=2)