MaxBET {BET} R Documentation

## 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(c(1), c(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)
```

[Package BET version 0.4.1 Index]