MaxBET {BET}

Binary Expansion Testing at a Certain Depth

Description

MaxBET stands for Binary Expansion Testing. It is used for nonparametric detection of nonuniformity or dependence. It can be used to test whether a column vector is [0, 1]-uniformly distributed. It can also be used to detect dependence between columns of a matrix X, if X has more than one column.

Usage

MaxBET(
  X,
  dep,
  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, BET 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, BET tests the independence among those column vectors.

dep

depth of the binary expansion for the BET.

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, index = list(c(1,2), c(3))) refers to test the independence between (X_1, X_2) and X_3. Default to be list(c(1:p)), where p = ncol(X), then test data uniformity.

Details

MaxBET tests the independence or uniformity by considering the maximal magnitude of the symmetry statistics in the sigma-field generated from marginal binary expansions at the depth d.

Value

Interaction

a dataframe with p columns, where p is the number of columns of X. It displays the interactions where the extreme symmetry statistics happens. For each column in X, we use a binary index to indicate binary variables involved in the extreme symmetry statistic.

Extreme.Asymmetry

the extreme asymmetry statistics.

p.value.bonf

p-value of the test with Bonferroni adjustment.

z.statistic

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)
MaxBET(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)
MaxBET(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)
MaxBET(cbind(x1, x2, x3), 3)