Article Hudecová and Šiman (2022) describes usage of randomized hyperplane-based ranks and signs for one sample location test for an elliptical distribution. The necessary functions for the computation of interdirections and randomized lift-interdirections and the one sample test statistic are available in the file oneSampleTest.R.

source("oneSampleTest.R")

# One sample test

We illustrate the usage of the test with a random sample $$\boldsymbol{X}_1,\dots,\boldsymbol{X}_n$$ for $$n=100$$ from a $$p$$-variate normal distribution with zero mean and identity variance matrix for $$p=4$$.

library(mvtnorm)
n=100
p=4
X=rmvnorm(n,mean=rep(0,p),sigma=diag(p))

The data are stored in a $$n\times p$$ matrix.

Function one.sample.int.test computes test statistic $$S$$ from equation (3) in Proposition 2 of Hudecová and Šiman (2022) and the corresponding p-value computed from the asymptotic $$\chi^2$$ distribution is reported.

The user needs to specify the score function $$K$$ and the number of hyperplanes from which incomplete interdirections and randomized lift interdirection are computed. The score functions $$K$$ is specified via the input KFUN as

• wilcoxon for the Wilcoxon score function $$K(x)=x$$,
• gaussian for the van der Waerden score function $$K(x)=\sqrt{F^{-1}(x)}$$, where $$F$$ stands for the CDF function of $$\chi^2_p$$ distribution,
• sign for $$K(x)=1$$.

The following code computes interdirections using a design set $$\mathcal{Q}_C$$ of hyperplanes chosen independently with replacement such that $$|\mathcal{Q}_C|=4n$$ (parameter DesI). Randomized lift-interdirections use a design set $$\mathcal{Q}_L$$ chosen independently with replacement with size $$|\mathcal{Q}_L|=5n$$ (parameter DesL). Since the design matrices are chosen randomly, different results are obtained if the same code is run with different set.seed.

one.sample.int.test(X, DesI=4,DesL=5, KFUN="wilcox")
## $S ## [,1] ## [1,] 9.67349 ## ##$pval
##            [,1]
## [1,] 0.04630192

The test statistic based on the van der Waerden score function is obtained as:

one.sample.int.test(X, DesI=4,DesL=5, KFUN="gaussian")
## $S ## [,1] ## [1,] 7.235502 ## ##$pval
##           [,1]
## [1,] 0.1239542

The test based on “signs” only ($$K(x)=1$$) is obtained as:

one.sample.int.test(X, DesI=4,DesL=5, KFUN="sign")
## $S ## [,1] ## [1,] 6.108064 ## ##$pval
##          [,1]
## [1,] 0.191222

# Usage in other tests

If one wants to replace the elliptical signs and ranks by the hyperplane based couterparts, these can be obtained from function getSignsAndRanks, which returns a vector of normalized ranks, i.e. $$R_i/(n+1)$$, $$i=1,\dots,n$$, and $$n\times n$$ matrix of cosines $$\cos(\pi\cdot C_{\boldsymbol{X}_i,\boldsymbol{X}_j}(\mathcal{X}_n)/|\mathcal{Q}_C|)$$, for pairs of observations $$(i,j)$$.

The output is illustrated on a small data of size $$n=5$$ with dimension $$2$$.

X=rmvnorm(n<-5,rep(0,2))

o=getSignsAndRanks(X,DesI=10,DesL=10,IsRank=1)
o
## $SignMat ## [,1] [,2] [,3] [,4] [,5] ## [1,] 0.9602937 0.3090170 -0.48175367 0.84432793 0.8910065 ## [2,] 0.3090170 0.9177546 0.68454711 0.77051324 -0.1564345 ## [3,] -0.4817537 0.6845471 0.91775463 0.06279052 -0.8270806 ## [4,] 0.8443279 0.7705132 0.06279052 0.96029369 0.5090414 ## [5,] 0.8910065 -0.1564345 -0.82708057 0.50904142 0.9822873 ## ##$RVec
##  0.8333333 0.1666667 0.3333333 0.6666667 0.5000000
##
## $Status ##  0 By default, both interdirections and randomized lift interdirections are computed. If one wants to obtain only interdirections (i.e. if sign score function is used) then the computation can be fastened by specifying IsRank=0. The value of Status indicates whether the computation was successful. Negative values correpond to an error in the input: o=getSignsAndRanks(X,DesI="a",DesL=10,IsRank=0) o ##$Status
##  -1

Note that only some possible input problems are checked.