MaxBETs {BET}

Binary Expansion Testing up to a Certain Depth

Description

MaxBETs is used for nonparametric dependence detection. Extended from BET, for a chosen maximal depth d.max, MaxBETs does a sequential test up to d.max and avoids overlapping symmetry statistics in different depths, for all 2 \le d \le d.max. The adjustment is done by multiplying the number of interactions which are in the \sigma-field generated by marginal binary expansions at depth d but not in that at depth d-1.

Usage

MaxBETs(
  X,
  d.max = 4,
  unif.margin = FALSE,
  asymptotic = TRUE,
  plot = FALSE,
  index = list(c(1:ncol(X)))
)

Arguments

X

a matrix to be tested. When X has only one column, BETs will test whether X is [0, 1]-uniformly distributed (an error will be given if data is out of range [0, 1]). When X has two or more columns, BETs tests the independence among those column vectors.

d.max

the maximal depth of the binary expansion for BETs.

unif.margin

logicals. If TRUE the marginal distribution is known to be Uniform[0,1]. Default is FALSE, and empirical cdf transformation will be applied to each marginal distribution.

asymptotic

logicals. If TRUE the p-value is computed by asymptotic distribution. Default to be TRUE. Ignored if X has three or more columns.

plot

logicals. If TRUE, make the plot of cross interaction of the strongest asymmetry. Default to be FALSE. This option only works for X with two columns.

index

a list of indices. If provided, test the independence among two or more groups of variables, for example, (X_1, X_2) and X_3.

Value

bet.s.pvalue.bonf

the overall p-value on the test.

bet.s.index

the interaction that the p-value is minimal.

bet.s.zstatistic

normal approximation of the test statistic.

Examples

##test mutual independence
v <- runif(128, -pi, pi)
X1 <- cos(v) + 2.5 * rnorm(128, 0, 1/20)
X2 <- sin(v) + 2.5 * rnorm(128, 0, 1/20)
MaxBETs(cbind(X1, X2), 3, asymptotic = FALSE, index = list(1,2))

##test independence between (x1, x2) and y
x1 = runif(128)
x2 = runif(128)
y = sin(4*pi*(x1 + x2)) + 0.4*rnorm(128)
MaxBETs(cbind(x1, x2, y), 3, index = list(c(1,2), c(3)))

##test uniformity
x1 = rbeta(128, 2, 4)
x2 = rbeta(128, 2, 4)
x3 = rbeta(128, 2, 4)
MaxBETs(cbind(x1, x2, x3), 3)