Academic

This talk consists of two parts. The first part, also the main part, of the talk concerns the consistency of estimation criteria based on AIC and BIC model criteria in estimating the number of significant components in principal component analysis (PCA) under a high-dimensional asymptotic framework. The number of significant components corresponds to the number of dominant eigenvalues of the covariance matrix of p variables. Via random matrix techniques, we derive sufficient conditions for the criterion (AIC or BIC) to be strongly consistent when the dominant eigenvalues are bounded, and when the dominant eigenvalues tend to infinity.  Moreover, the asymptotic results are obtained without normality assumption of the population distribution. Simulation results show that the sufficient conditions we derived are essential. The second part concerns strong consistency of variable selection methods of AIC and BIC in multiple linear regression for data under the large sample, large dimension and large model setting.