Georgia Institute of Technology College of Sciences

Recently functional data analysis has received considerable attention in statistics research and a number of successful applications have been reported, but there has been no results on the inference of the global shape of the mean regression curve. In this paper, asymptotically simultaneous confidence band is obtained for the mean trajectory curve based on sparse longitudinal data, using piecewise constant spline estimation. Simulation experiments corroborate the asymptotic theory.

Several interesting models of random partial orders can be described via a process that builds the partial order one step at a time, at each point adding a new maximal element. This process therefore generates a linear extension of the partial order in tandem with the partial order itself. A natural condition to demand of such processes is that, if we condition on the occurrence of some finite partial order after a given number of steps, then each linear extension of that partial order is equally likely. This condition is called "order-invariance". The class of order-invariant processes includes processes generating a random infinite partial order, as well as those that amount to taking a random linear extension of a fixed infinite poset. Our goal is to study order-invariant processes in general. In this talk, I shall focus on some of the combinatorial problems that arise. (joint work with Malwina Luczak)

It is well known that a needle thrown at random has zero probability of intersecting any given irregular planar set of finite 1-dimensional Hausdorff measure. Sharp quantitative estimates for fine open coverings of such sets are still not known, even for such sets as the Sierpinski gasket and the 4-corner Cantor set (with self-similarities 1/4 and 1/3). In 2008, Nazarov, Peres, and Volberg provided the sharpest known upper bound for the 4-corner Cantor set. Volberg and I have recently used the same ideas to get a similar estimate for the Sierpinski gasket. Namely, the probability that Buffon's needle will land in a 3^{-n}-neighborhood of the Sierpinski gasket is no more than C_p/n^p, where p is any small enough positive number.