  Most signal processing and statistical applications heavily rely on specific
data distribution models. The Gaussian distributions, although being the most
common choice, are inadequate in most real world scenarios as they fail to
account for data coming from heavy-tailed populations or contaminated by
outliers. Such problems call for the use of Robust Statistics. The robust
models and estimators are usually based on elliptical populations, making the
latter ubiquitous in all methods of robust statistics. To determine whether
such tools are applicable in any specific case, goodness-of-fit (GoF) tests are
used to verify the ellipticity hypothesis. Ellipticity GoF tests are usually
hard to analyze and often their statistical power is not particularly strong.
In this work, assuming the true covariance matrix is unknown we design and
rigorously analyze a robust GoF test consistent against all alternatives to
ellipticity on the unit sphere. The proposed test is based on Tyler's estimator
and is formulated in terms of easily computable statistics of the data. For its
rigorous analysis, we develop a novel framework based on the exchangeable
random variables calculus introduced by de Finetti. Our findings are supported
by numerical simulations comparing them to other popular GoF tests and
demonstrating the significantly higher statistical power of the suggested
technique.
