Tracy-Widom statistic for the largest eigenvalue of autoscaled real matrices

Eigenanalysis is common practice in biostatistics, and the largest eigenvalue of a data set contains valuable information about the data. However, to make inferences about the size of the largest eigenvalue, its distribution must be known. Johnstone's theorem states that the largest eigenvalues l₁ of real random covariance matrices are distributed according to the Tracy-Widom distribution of order 1 when properly normalized to L1=l1-ηnpξnp, where η_np and ξ_np are functions of the data matrix dimensions n and p. Very often, data are expressed in terms of correlations (autoscaling) for which case Johnstone's theorem does not work because the normalizing parameters η_np and ξ_np are not theoretically known. In this paper we propose a semi-empirical method based on test-equating theory to numerically approximate the normalization parameters in the case of autoscaled matrices. This opens the way of making inferences regarding the largest eigenvalue of an autoscaled data set. The method is illustrated by means of application to two real-life data sets.