USPFourierAdapt {USP}

Adaptive permutation test of independence for continuous data.

Description

We implement the adaptive version of the independence test for univariate continuous data using the Fourier basis, as described in Section 4 of (Berrett et al. 2021). This applies USPFourier with a range of values of M, and a properly corrected significance level.

Usage

USPFourierAdapt(x, y, alpha, B = 999, ties.method = "standard")

Arguments

x

The vector of data points from the first sample, each entry belonging to [0,1].

y

The vector of data points from the second sample, each entry belonging to [0,1].

alpha

The desired significance level of the test.

B

Controls the number of permutations to be used. With a sample size of n each test uses B \log_2 n permutations. If B+1 < 1/\alpha then it is not possible to reject the null hypothesis.

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.

Value

Returns an indicator with value 1 if the null hypothesis of independence is rejected and 0 otherwise. If the null hypothesis is rejected, the function also outputs the value of M at the which the null was rejected and the value of the test statistic.

References

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

Examples

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)
USPFourierAdapt(x,y,0.05,999)