Quadratic scan statistic (${\max }_{S}u(S{)}^{\top }\hat{\Sigma }u(S)$)

A quadratic scan statistic maximizes a directional empirical second moment over candidate unit vectors. For a Gaussian random vector sample with a rank-one covariance spike in one of those directions, the matching statistic has its scale multiplied by the spike's eigenvalue. A chi-squared concentration inequality controls each direction.