Seminars and Colloquia by Series

Thursday, April 1, 2010 - 15:00 , Location: Skiles 269 , Hira Koul , Michigan State University , Organizer: Liang Peng
In this talk we shall discuss the problem of fitting a distribution function to the marginal distribution of a long memory process. It is observed that unlike in the i.i.d. set up, classical tests based on empirical process are relatively easy to implement. More importantly, we discuss fitting the marginal distribution of the error process in location, scale and linear regression models. An interesting observation is that the first order difference between the residual empirical process and the null model can not be used to asymptotically to distinguish between the two marginal distributions that differ only in their means. This finding is in sharp contrast to a recent claim of Chan and Ling to appear in the Ann. Statist. that such a process has a Gaussian weak limit. We shall also proposes some tests based on the second order difference in this case and analyze some of their properties. Another interesting finding is that residual empirical process tests in the scale problem are robust against not knowing the scale parameter. The third finding is that in linear regression models with a non-zero intercept parameter the first order difference between the empirical d.f. of residuals and the null d.f. can not be used to fit an error d.f. This talk is based on ongoing joint work with Donatas Surgailis.
Tuesday, March 30, 2010 - 16:00 , Location: Skiles 269 , Boris Rozovsky , Division of Applied Mathematics, Brown University , Organizer:
We consider a stochastic Navier-Stokes equation driven by a space-time Wiener process. This equation is quantized by transformation of the nonlinear term to the Wick product form. An interesting feature of this type of perturbation is that it preserves the mean dynamics: the expectation of the solution of the quantized Navier-Stokes equation solves the underlying deterministic Navier-Stokes equation. From the stand point of a statistician it means that the perturbed model is an unbiased random perturbation of the deterministic Navier-Stokes equation.The quantized equation is solved in the space of generalized stochastic processes using the Cameron-Martin version of the Wiener chaos expansion. A solution of the quantized version is unique if and only if the uniqueness property holds for the underlying deterministic Navier-Stokes equation. The generalized solution is obtained as an inverse of solutions to corresponding quantized equations. We will also demonstrate that it could be approximated by real (non-generalized processes). A solution of the quantized Navier-Stokes equation turns out to be nonanticipating and Markov. The talk is based on a joint work with R. Mikulevicius.
Thursday, March 11, 2010 - 15:00 , Location: Skiles 269 , Evarist Giné , University of Connecticut , Organizer:
 The almost sure rate of convergence in the sup norm  for  linear wavelet density estimators is obtained, as well as a central limit theorem for the distribution functions based on these estimators. These results are then applied to show that the hard thresholding wavelet estimator of Donoho, Johnstone, Kerkyacharian and Picard (1995) is adaptive in sup norm to the smoothness of a density. An alternative adaptive estimator combining Lepski's method with  Rademacher complexities will also be described. This is joint work with Richard Nickl.  
Thursday, March 4, 2010 - 15:00 , Location: Skiles 269 , Dr Juri Lember , Tartu University, Estonia , juri.lember@ut.ee , Organizer: Heinrich Matzinger
Abstract: We consider the hidden Markov model, where the dynamic of theprocess is modelled by a  latent Markov chain Y and the observations X aresuch that: 1) given the realization of Y, the observations areindependent; 2) the distribution of the i-th observations (X_i) depends onthe i-th element of the Y (Y_i), only.The segmentation problem consists of estimating the underlying realization(path) of Y given the n observation. Usually the realization with maximumlikelihood, the so called Viterbi alignment is used. On the other hand, itis easy to see that the Viterbi alignment does not minimize the expectednumber of misclassification errors.We consider the segmentation problem in the framework of statisticallearning. This unified risk-based approach helps to analyse many existingalignments as well as defining many new ones. We also study theasymptotics of the risks and infinite alignments.
Thursday, February 25, 2010 - 15:00 , Location: Skiles 269 , Stanislav Molchanov , UNC Charlotte , Organizer:
The talk will present several limit theorems for the supercritical colony of the particles with masses. Reaction-diffusion equations responsible for the spatial distribution of the species contain the usual random death, birth and migration processes. The evolution of the mass of the individual particle includes (together with the diffusion) the mitosis: the splitting of the mass between the two offspring.  The last process leads to the new effects. The limit theorems give the detailed picture of the space –mass distribution of the particles in the bulk of the moving front of the population.
Thursday, January 28, 2010 - 15:00 , Location: Skiles 269 , Stefan Boettcher , Emory Physics , Organizer:
Finding ground states of spin glasses, a model of disordered materials, has a deep connection to many hard combinatorial optimization problems, such as satisfiability, maxcut, graph-bipartitioning, and coloring. Much insight has been gained for the combinatorial problems from the intuitive approaches developed in physics (such as replica theory and the cavity method), some of which have been proven rigorously recently. I present  a treasure trove of numerical data obtained with heuristic methods that suggest a number conjectures, such as an equivalence between maxcut and bipartitioning for r-regular graphs, a simple relation for their optimal configurations as a function of degree r, and anomalous extreme-value fluctuations in a variety of models, hotly debated in physics currently. For some, such as those related to finite-size effects, not even a physics theory exists, for others theory exists that calls for rigorous methods.
Thursday, November 19, 2009 - 15:00 , Location: Skiles 269 , Lijian Yang , Michigan State University , Organizer:
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.
Thursday, November 12, 2009 - 15:00 , Location: Skiles 269 , Gauri Data , University of Georgia , Organizer: Liang Peng
We consider two problems: (1) estimate a normal mean under a general divergence loss introduced in Amari (1982) and Cressie and Read (1984) and (2) find a predictive density of a new observation drawn independently of the sampled observations from a normal distribution with the same mean but possibly with a different variance under the same loss. The general divergence loss includes as special cases both the Kullback-Leibler and Bhattacharyya-Hellinger losses. The sample mean, which is a Bayes estimator of the population mean under this loss and the improper uniform prior, is shown to be minimax in any arbitrary dimension. A counterpart of this result for predictive density is also proved in any arbitrary dimension. The admissibility of these rules holds in one dimension, and we conjecture that the result is true in two dimensions as well. However, the general Baranchik (1970) class of estimators, which includes the James-Stein estimator and the Strawderman (1971) class of estimators, dominates the sample mean in three or higher dimensions for the estimation problem. An analogous class of predictive densities is defined and any member of this class is shown to dominate the predictive density corresponding to a uniform prior in three or higher dimensions. For the prediction problem, in the special case of Kullback-Leibler loss, our results complement to a certain extent some of the recent important work of Komaki (2001) and George, Liang and Xu (2006). While our proposed approach produces a general class of predictive densities (not necessarily Bayes) dominating the predictive density under a uniform prior, George et al. (2006) produced a class of Bayes predictors achieving a similar dominance. We show also that various modifications of the James-Stein estimator continue to dominate the sample mean, and by the duality of the estimation and predictive density results which we will show, similar results continue to hold for the prediction problem as well. This is a joint research with Professor Malay Ghosh and Dr. Victor Mergel.
Thursday, November 5, 2009 - 15:00 , Location: Skiles 269 , Vlad Vysotsky , University of Delaware , Organizer:
Let $S_n$ be a centered random walk with a finite variance, and define the new sequence $\sum_{i=1}^n S_i$, which we call the {\it integrated random walk}. We are interested in the asymptotics of $$p_N:=\P \Bigl \{ \min \limits_{1 \le k \le N} \sum_{i=1}^k S_i  \ge 0 \Bigr \}$$ as $N \to \infty$. Sinai (1992) proved that $p_N \asymp N^{-1/4}$ if $S_n$ is a simple random walk. We show that $p_N \asymp N^{-1/4}$ for some other types of random walks that include double-sided exponential and double-sided geometric walks (not necessarily symmetric). We also prove that $p_N \le c N^{-1/4}$ for lattice walks and upper exponential walks, i.e., walks such that $\mbox{Law} (S_1 | S_1>0)$ is an exponential distribution.
Tuesday, November 3, 2009 - 16:00 , Location: Skiles 255 (Note unusual time and location) , Ivan NOURDIN , Paris VI , Organizer:
My aim is to explain how to prove multi-dimensional central limit theorems for the spectral moments (of arbitrary degrees) associated with random matrices with real-valued i.i.d. entries, satisfying some appropriate moment conditions. The techniques I will use rely on a universality principle for the Gaussian Wiener chaos as well as some combinatorial estimates. Unlike other related results in the probabilistic literature, I will not require that the law of the entries has a density with respect to the Lebesgue measure. The talk is based on a joint work with Giovanni Peccati, and use an invariance principle obtained in a joint work with G. P. and Gesine Reinert

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