Stanislav Molchanov, UNC Charlotte
Title: A Review of the Anderson Parabolic Model: Known Results and Open Problems
Zhiyi Zhang UNC Charlotte
Title: Statistical Implications of Turing’s Formula
Abstract: This talk is organized into three parts.
1. Turing’s formula is introduced. Given an iid sample from an countable alphabet under a probability distribution, Turing’s formula (introduced by Good (1953), hence also known as the Good-Turing formula) is a mind-bending non-parametric estimator of total probability associated with letters of the alphabet that are NOT represented in the sample. Many of its statistical properties were not clearly known for a stretch of nearly sixty years until recently. Some of the newly established results, including various asymptotic normal laws, are described.
2. Turing’s perspective is described. Turing’s formula brought about a new perspective (or
a new characterization) of probability distributions on general countable alphabets. The
new perspective in turn provides a new way to do statistics on alphabets, where the usual statistical concepts associated with random variables (on the real line) no longer exist, for example, moments, tails, coefficients of correlation, characteristic functions don’t exist on alphabets (a major challenge of modern data sciences). The new perspective, in the form of entropic basis, is introduced.
3. Several applications are presented, including estimation of information entropy and diversity indices.
Isaac Sonin, UNC Charlotte
Title: A Continuous-Time Model of Financial Clearing. (Banks as Tanks).
Abstract: We present a simple and transparent model of clearing in financial networks in continuous time, in which firms are represented by reservoirs filled with “liquid money,” flowing in and out of each firm. The model gives a simple recursive solution to a classical static model of financial clearing by Eisenberg and Noe (2001). The dynamic structure of our model opens the way to handle more complicated real financial networks dynamic in nature. Our approach also provides a useful tool to solve nonlinear equations involving linear system and max min operations similar to Bellman equation for the optimal stopping of Markov chains and other optimization problems.