For a single sample of symmetric matrices where sum of squared eigenvalues = 1, test eigenvalues of the population mean.
For multiple samples of symmetric matrices where sum of squared eigenvalues = 1, test for equality of the eigenvalues of the population means.
The test statistic is calculated by stat_ss1().
test_ss1(x, evals = NULL, B = 1000, maxit = 25)stat_ss1(x, evals = NULL)
A TFORGE object (see boot_calib() or chisq_calib()) with the eigenvalues of the null hypothesis in the null_evals attribute for t0.
A single sample of symmetric matrices or multiple samples of symmetric matrices. See as_flat().
When x is a single sample, the null hypothesis is that the (extrinsic) mean of the population has eigenvalues equal to evals. For multiple samples evals must be omitted.
Number of bootstrap samples. If B = 'chisq' then a chi-squared calibration is used instead.
The maximum number of Newton steps allowed in empirical likelihood optimisation owen:2013TFORGE.
For a single sample the null hypothesis is that the population (extrinsic) mean has eigenvalues of evals; the alternative hypothesis is that the eigenvalues are not equal to evals.
For multiple samples, evals must be omitted and the null hypothesis is that the population (extrinsic) means have the same eigenvalues.
This function uses a form of weighted bootstrapping called b-boostrapping hall1999inTFORGE. An empirical distribution is defined by sampling weights for each observation in the original sample.
The sampling weights must be such that the (extrinsic) mean of the empirical distribution is $$c\hat{Q} \Lambda \hat{Q}^\top,$$
where \(\hat{Q}\) are the eigenvectors of the sample mean, \(\Lambda\) is a diagonal matrix of eigenvalues specified by either the null hypothesis (for single sample tests) or estimated as the common eigenvalues of multiple populations (for k-sample tests).
In some situations \(c\) is a free scalar to enable projection of the Euclidean mean to the extrinsic mean, otherwise \(c=1\).
If no such sampling weights exist (i.e. the convex hull of the data does not contain \(c\hat{Q} \Lambda \hat{Q}^\top\)), then the test rejects with pval=0 and a warning.
The sampling weights are also optimised to maximise empirical likelihood owen:2013TFORGE.
Test hypotheses described below. The sum of squared eigenvalues constraint forces the set of eigenvalues to lie on a sphere (or circle). The test statistic accounts for this constraint by projecting eigenvalues onto a plane perpendicular to the direction of the sample average's eigenvalues.
Weighted bootstrap calibration is used (see 'Weighted Bootstrapping' below).
Eigenvalues must be distinct.