Seminars and Colloquia by Series

Estimation, Prediction and the Stein Phenomenon under Divergence Loss

Series
Stochastics Seminar
Time
Thursday, November 12, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Gauri Data – University of Georgia
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.

Integrated random walks: the probability to stay positive

Series
Stochastics Seminar
Time
Thursday, November 5, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Vlad Vysotsky – University of Delaware
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.

Universal Gaussian fluctuations of non-Hermitian matrix ensembles

Series
Stochastics Seminar
Time
Tuesday, November 3, 2009 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 255 (Note unusual time and location)
Speaker
Ivan NOURDIN – Paris VI
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

Interacting particles, series Jackson networks, and non-crossing probabilities

Series
Stochastics Seminar
Time
Thursday, October 22, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Ton Dieker – (ISyE, Georgia Tech)
In this talk, we study an interacting particle system arising in the context of series Jackson queueing networks. Using effectively nothing more than the Cauchy-Binet identity, which is a standard tool in random-matrix theory, we show that its transition probabilities can be written as a signed sum of non-crossing probabilities. Thus, questions on time-dependent queueing behavior are translated to questions on non-crossing probabilities. To illustrate the use of this connection, we prove that the relaxation time (i.e., the reciprocal of the ’spectral gap’) of a positive recurrent system equals the relaxation time of a single M/M/1 queue with the same arrival and service rates as the network’s bottleneck station. This resolves a 1985 conjecture from Blanc on series Jackson networks. Joint work with Jon Warren, University of Warwick.

Nonuniqueness for some stochastic partial differential equations

Series
Stochastics Seminar
Time
Friday, October 9, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 154 (Unusual time and room)
Speaker
Carl Mueller – University of Rochester
One of the most important stochastic partial differential equations, known as the superprocess, arises as a limit in population dynamics. There are several notions of uniqueness, but for many years only weak uniqueness was known. For a certain range of parameters, Mytnik and Perkins recently proved strong uniqueness. I will describe joint work with Barlow, Mytnik and Perkins which proves nonuniqueness for the parameters not included in Mytnik and Perkins' result. This completely settles the question for strong uniqueness, but I will end by giving some problems which are still open.

Arbitrage ­free option pricing models 

Series
Stochastics Seminar
Time
Thursday, October 1, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Denis Bell – University of North Florida
The Black‐Scholes model for stock price as geometric Brownian motion, and the corresponding European option pricing formula, are standard tools in mathematical finance. In the late seventies, Cox and Ross developed a model for stock price based on a stochastic differential equation with fractional diffusion coefficient. Unlike the Black‐Scholes model, the model of Cox and Ross is not solvable in closed form, hence there is no analogue of the Black‐Scholes formula in this context. In this talk, we discuss a new method, based on Stratonovich integration, which yields explicitly solvable arbitrage‐free models analogous to that of Cox and Ross. This method gives rise to a generalized version of the Black‐Scholes partial differential equation. We study solutions of this equation and a related ordinary differential equation.

The dynamics of moving interfaces in a random environment

Series
Stochastics Seminar
Time
Thursday, September 24, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Jim Nolen – Duke University
I will describe recent work on the behavior of solutions to reaction diffusion equations (PDEs) when the coefficients in the equation are random. The solutions behave like traveling waves moving in a randomly varying environment. I will explain how one can obtain limit theorems (Law of Large Numbers and CLT) for the motion of the interface. The talk will be accessible to people without much knowledge of PDE.

Simultaneous Asymptotics for the Shape of Young Tableaux: Tracy-Widom and beyond.

Series
Stochastics Seminar
Time
Thursday, September 10, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Christian Houdré – Georgia Tech

Please Note: Given a random word of size n whose letters are drawn independently from an ordered alphabet of size m, the fluctuations of the shape of the corresponding random RSK Young tableaux are investigated, when both n and m converge together to infinity. If m does not grow too fast and if the draws are uniform, the limiting shape is the same as the limiting spectrum of the GUE. In the non-uniform case, a control of both highest probabilities will ensure the convergence of the first row of the tableau, i.e., of the length of the longest increasing subsequence of the random word, towards the Tracy-Widom distribution.

Sparsity pattern aggregation in generalized linear models.

Series
Stochastics Seminar
Time
Thursday, September 3, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Philippe Rigollet – Princeton University
The goal of this talk is to present a new method for sparse estimation which does not use standard techniques such as $\ell_1$ penalization. First, we introduce a new setup for aggregation which bears strong links with generalized linear models and thus encompasses various response models such as Gaussian regression and binary classification. Second, by combining maximum likelihood estimators using exponential weights we derive a new procedure for sparse estimations which satisfies exact oracle inequalities with the desired remainder term. Even though the procedure is simple, its implementation is not straightforward but it can be approximated using the Metropolis algorithm which results in a stochastic greedy algorithm and performs surprisingly well in a simulated problem of sparse recovery.

Penalized orthogonal-components regression for large p small n data

Series
Stochastics Seminar
Time
Thursday, August 27, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Dabao Zhang – Purdue University
We propose a penalized orthogonal-components regression (POCRE) for large p small n data. Orthogonal components are sequentially constructed to maximize, upon standardization, their correlation to the response residuals. A new penalization framework, implemented via empirical Bayes thresholding, is presented to effectively identify sparse predictors of each component. POCRE is computationally efficient owing to its sequential construction of leading sparse principal components. In addition, such construction offers other properties such as grouping highly correlated predictors and allowing for collinear or nearly collinear predictors. With multivariate responses, POCRE can construct common components and thus build up latent-variable models for large p small n data. This is an joint work with Yanzhu Lin and Min Zhang

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