ICtest (version 0.3-5)

SIRboot: Testing the Subspace Dimension for Sliced Inverse Regression Using Bootstrapping.

Description

Using the two scatter matrices approach (SICS) for sliced inversion regression (SIR) the function tests if the last p-k components have zero eigenvalues, where p is the number of explaining variables. Hence the assumption is that the first k components are relevant for modelling the response y and the remaining components are not. The function performs bootstrapping to obtain a p-value.

Usage

SIRboot(X, y, k, h = 10, n.boot = 200, ...)

Arguments

X

a numeric data matrix of explaining variables.

y

a numeric vector specifying the response.

k

the number of relevant components under the null hypothesis.

h

the number of slices used in SIR. Passed on to function covSIR.

n.boot

number of bootstrapping samples.

…

other arguments passed on to covSIR.

Value

A list of class ictest inheriting from class htest containing:

statistic

the value of the test statistic.

p.value

the p-value of the test.

parameter

the number of boostrapping samples used to compute the p-value.

method

character string which test was performed.

data.name

character string giving the name of the data.

alternative

character string specifying the alternative hypothesis.

k

the number of non-zero eigenvalues used in the testing problem.

W

the transformation matrix to the underlying components.

S

data matrix with the centered underlying components.

D

the underlying eigenvalues.

MU

the location of the data which was substracted before calculating the components.

Details

Under the null hypthesis the last p-k eigenvalue as given in D are zero. The test statistic is then the sum of these eigenvalues.

Denote W as the transformation matrix to the supervised invariant coordinates (SIC) \(s_i\), \(i=1,\ldots,n\), i.e. $$s_i = W (x_i-MU),$$ where MU is the location.

Let \(S_1\) be the submatrix of the SICs which are relevant and \(S_2\) the submatrix of the SICs which are irrelevant for the response y under the null.

The boostrapping has then the following steps:

1. Take a boostrap sample \((y^*, S_1^*)\) of size \(n\) from \((y, S_1)\).

2. Take a boostrap sample \(S_2^*\) of size \(n\) from \(S_2\).

3. Combine \(S^*=(S_1^*, S_2^*)\) and create \(X^*= S^* W\).

4. Compute the test statistic based on \(X^*\).

5. Repeat the previous steps n.boot times.

References

Nordhausen, K., Oja, H. and Tyler, D.E. (2022), Asymptotic and Bootstrap Tests for Subspace Dimension, Journal of Multivariate Analysis, 188, 104830. <doi:10.1016/j.jmva.2021.104830>.

See Also

covSIR, SIRasymp

Examples

# NOT RUN {
X <- matrix(rnorm(1000), ncol = 5)
eps <- rnorm(200, sd = 0.1)
y <- 2 + 0.5 * X[, 1] + 2 * X[, 3] + eps
  
SIRboot(X, y, k = 0) 
SIRboot(X, y, k = 1)    
# }