# Testing the Sphericity of a covariance matrix when the dimension is much larger than the sample size

arXiv: https://arxiv.org/abs/1508.02498 · HTML: https://mlsanity.com/paper/1508.02498/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Aug 2015.

## Built on by (most cited first, 2 of 2)

- [Deformed semicircle law and concentration of nonlinear random matrices for ultra-wide neural networks](https://mlsanity.com/paper/2109.09304/) (2021)
- [When Noise Estimation Hides Basis Misspecification in Repeated Bayesian Inverse Problems](https://mlsanity.com/paper/2609.37762/) (2026)

## Abstract (arXiv)

This paper focuses on the prominent sphericity test when the dimension p is much lager than sample size n. The classical likelihood ratio test(LRT) is no longer applicable when pgg n. Therefore a Quasi-LRT is proposed and asymptotic distribution of the test statistic under the null when p/nrightarrow∞, nrightarrow∞ is well established in this paper. Meanwhile, John's test has been found to possess the powerful it dimension-proof property, which keeps exactly the same limiting distribution under the null with any (n,p)-asymptotic, i.e. p/nrightarrow[0,∞], nrightarrow∞. All asymptotic results are derived for general population with finite fourth order moment. Numerical experiments are implemented for comparison.