High-Dimensional Mean Testing When p Exceeds n

The classical way to test whether a vector of $p$ means is jointly zero — say, whether $p$ different factor strategies each have zero average return, tested together rather than one at a time — is Hotelling's $T^2$ test, which requires inverting the sample covariance matrix of the $p$ variables. That inversion needs the sample size $n$ to exceed $p$; once you have more candidate variables than observations (thousands of factors, only a few hundred months of data), the sample covariance matrix is singular and can't be inverted at all, so the classical test simply doesn't run.

High-dimensional mean tests replace the matrix inversion with statistics that avoid it entirely — a common approach (Bai-Saranadasa and its descendants) uses a test statistic built from the sum of squared standardized means, roughly $\sum_{j=1}^p \bar{x}_j^2$ adjusted for the estimated variances, rather than the full quadratic form that Hotelling's test requires. This sidesteps the singular-matrix problem because it never needs a full covariance inverse, only diagonal or low-rank pieces of it, and its distribution under the null can be derived even as $p$ grows alongside (or faster than) $n$.

The tradeoff is power: these tests are most sensitive to many small effects spread broadly across the $p$ variables, and comparatively weak at detecting a signal concentrated in just a few of them — the opposite profile of a sparsity-based test like one built on Lasso selection.