Linear Statistics with Expectation and Covariance

Strasser-Weber type linear statistics and their expectation and covariance under the independence hypothesis

LinStatExpCov(X, Y, ix = NULL, iy = NULL, weights = integer(0),
              subset = integer(0), block = integer(0), checkNAs = TRUE,
              varonly = FALSE, nresample = 0, standardise = FALSE,
              tol = sqrt(.Machine$double.eps))
lmult(x, object)

Arguments

X

numeric matrix of transformations.

Y

numeric matrix of influence functions.

ix

an optional integer vector expanding X.

iy

an optional integer vector expanding Y.

weights

an optional integer vector of non-negative case weights.

subset

an optional integer vector defining a subset of observations.

block

an optional factor defining independent blocks of observations.

checkNAs

a logical for switching off missing value checks. This included switching off checks for suitable values of subset. Use at your own risk.

varonly

a logical asking for variances only.

nresample

an integer defining the number of permuted statistics to draw.

standardise

a logical asking to standardise the permuted statistics.

tol

tolerance for zero variances.

x

a contrast matrix to be left-multiplied in case X was a factor.

object

an object of class "LinStatExpCov".

Details

The function, after minimal preprocessing, calls the underlying C code and computes the linear statistic, its expectation and covariance and, optionally, nresample samples from its permutation distribution.

When both ix and iy are missing, the number of rows of X and Y is the same, ie the number of observations.

When X is missing and ix a factor, the code proceeds as if X were a dummy matrix of ix without explicitly computing this matrix.

Both ix and iy being present means the code treats them as subsetting vectors for X and Y. Note that ix = 0 or iy = 0 means that the corresponding observation is missing and the first row or X and Y must be zero.

lmult allows left-multiplication of a contrast matrix when X was (equivalent to) a factor.

Value

A list.

References

Strasser, H. and Weber, C. (1999). On the asymptotic theory of permutation statistics. Mathematical Methods of Statistics 8(2), 220–250.

Examples

wilcox.test(Ozone ~ Month, data = airquality, subset = Month %in% c(5, 8),
            exact = FALSE, correct = FALSE)
#> 
#> 	Wilcoxon rank sum test
#> 
#> data:  Ozone by Month
#> W = 127.5, p-value = 0.0001164
#> alternative hypothesis: true location shift is not equal to 0
#> 

aq <- subset(airquality, Month %in% c(5, 8))
X <- as.double(aq$Month == 5)
Y <- as.double(rank(aq$Ozone, na.last = "keep"))
doTest(LinStatExpCov(X, Y))
#> $TestStatistic
#> [1] 3.853635
#> 
#> $p.value
#> [1] 0.0001163773
#>