Georgia Institute of Technology College of Sciences

We explore the connection between Scenery Reconstruction and Optimal Alignments. We present some new algorithms which work in practise and not just in theory, to solve the Scenery Reconstruction problem

This work began in collaboration with C.Heitsch. I will briefly discuss the biological motivation. Then I will introduce Gibbs random trees and study their asymptotics as the tree size grows to infinity. One of the results is a "thermodynamic limit" allowing to introduce a limiting infinite random tree which exhibits a few curious properties. Under appropriate scaling one can obtain a diffusion limit for the process of generation sizes of the infinite tree. It also turns out that one can approach the study the details of the geometry of the tree by tracing progenies of subpopulations. Under the same scaling the limiting continuum random tree can be described as a solution of an SPDE w.r.t. a Brownian sheet.

In this approach to the Gaussian Correlation Conjecture we must check the log-concavity of the moment generating function of certain measures pulled down by a particular Gaussian density.

So far, likelihood-based interval estimate for quantiles has not been studied in literature for interval censored Case 2 data and partly interval-censored data, and in this context the use of smoothing has not been considered for any type of censored data. This article constructs smoothed weighted empirical likelihood ratio confidence intervals (WELRCI) for quantiles in a unified framework for various types of censored data, including right censored data, doubly censored data, interval censored data and partly interval-censored data. The 4th-order expansion of the weighted empirical log-likelihood ratio is derived, and the 'theoretical' coverage accuracy equation for the proposed WELRCI is established, which generally guarantees at least the 'first-order' accuracy. In particular for right censored data, we show that the coverage accuracy is at least O(n^{-1/2}), and our simulation studies show that in comparison with empirical likelihood-based methods, the smoothing used in WELRCI generally gives a shorter confidence interval with comparable coverage accuracy. For interval censored data, it is interesting to find that with an adjusted rate n^{-1/3}, the weighted empirical log-likelihood ratio has an asymptotic distribution completely different from that by the empirical likelihood approach, and the resulting WELRCI perform favorably in available comparison simulation studies.