Berry-Esseen bounds for nonlinear statistics, and asymptotic relative efficiency between correlation statistics

Four papers, written in collaboration with the author’s graduate school advisor, are presented. In the first paper, uniform and non-uniform Berry-Esseen (BE) bounds on the convergence to normality of a general class of nonlinear statistics are provided; novel applications to specific statistics, including the non-central Student’s, Pearson’s, and the non-central Hotelling’s, are also stated. In the second paper, a BE bound on the rate of convergence of the F-statistic used in testing hypotheses from a general linear model is given. The third paper considers the asymptotic relative efficiency (ARE) between the Pearson, Spearman, and Kendall correlation statistics; conditions sufficient to ensure that the Spearman and Kendall statistics are equally (asymptotically) efficient are provided, and several models are considered which illustrate the use of such conditions. Lastly, the fourth paper proves that, in the bivariate normal model, the ARE between any of these correlation statistics possesses certain monotonicity properties; quadratic lower and upper bounds on the ARE are stated as direct applications of such monotonicity patterns.

Light Transport in PT-invariant Photonic Structures with Hidden Symmetries

We introduce a recursive bosonic quantization technique for generating classical PT photonic structures that possess hidden symmetries and higher order exceptional points. We study light transport in these geometries and we demonstrate that perfect state transfer is possible only for certain initial conditions. Moreover, we show that for the same propagation direction, left and right coherent transports are not symmetric with field amplitudes following two different trajectories. A general scheme for identifying the conservation laws in such PT-symmetric photonic networks is also presented.