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Speaker: Liang Peng, School of Mathematics Georgia Tech

Title: Inference for double AR(l) models and near integrated AR models

Abstract: First, the weighted least absolute deviations estimate is studied for an AR(1) process with ARCH(1) errors. Unlike the quasi maximum likelihood estimate, the limiting distribution of the weighted least absolute deviations estimate is shown to be normal even when the underlying noise does not have a finite fourth moment. Second, the limiting distribution of quantile estimate of the near integrated first order autoregressive parameter with infinite variance errors is studied, and an empirical likelihood method is applied to construct a confidence interval for the near unit root.

Speaker: Alexander Yu. Mitrophanov, School of Biology, Georgia Tech

Title: Inequality-based perturbation theory for Markov chains: recent advances

Abstract: Perturbation bounds are useful when it is necessary to bound the uncertainty in the output of a mathematical model given the uncertainty in the parameter values. Such situations arise when numerical values for the parameters are taken from experimental data. In this talk I overview my results concerning perturbation bounds for Markov chains, focusing on the connection between the sensitivity to perturbations and the rate of convergence to stationarity. I have developed a new convergence-based approach which produces much tighter uniform sensitivity bounds than the techniques previously known. Originally, my results in this direction were obtained for finite, homogeneous, continuous-time Markov chains having a unique stationary distribution, but they can be generalized to nonhomogeneous chains, as well as discrete-time chains with general state space. I demonstrate the important role of the spectral gap of the generator in the sensitivity analysis for continuous-time Markov chains. The spectral gap is the primary factor which governs the sensitivity of the chain Rs distribution in the stationary state; for reversible chains, this is also true for the transient state. I also show that in some cases (e.g., for chains with a strongly accessible state) one can obtain sensitivity bounds in terms of the entries of the generator.

Speaker: Serge Guillas, School of Mathematics, Georgia Tech

Title: Functional samples and bootstrap for predicting sulfur dioxide levels

Abstract: In this study, enhancements of several functional techniques are given in order to forecast sulfur dioxide levels near a power plant. The data are considered as a time series of curves. Assuming a lag one dependence, the predictions are computed using the functional kernel (with local bandwith) and the linear autoregressive Hilbertian model. We carry out the estimation with a so-called historical matrix, which is a subsample that emphasizes uncommon shapes. A bootstrap method is introduced to evaluate the range of the forecasts, which uses Fraiman and Muniz's order for functional data. Finally, we compare our functional techniques with neural networks and semi-parametric methods, and find that the former models are often more effective. (Joint work with B. Fernandez de Castro and W. Gonzalez Manteiga)

Speaker: Alex J. Koning, Econometric Institute, Erasmus University Rotterdam

Title: Pre-monitoring by Comparing Empirical Cumulative Distribution Functions

Abstract: This talk discusses stochastic processes which enable pre-monitoring the possible changes of probability distributions over time. These processes may in particular be used to test the null hypothesis of no change. The pre-monitoring processes are bivariate functions, of time and position at the measurement scale, and are approximated with zero mean Gaussian processes under the constancy hypothesis. One may then form Kolmogorov-Smirnov or other type of tests as functionals of the processes.

We study the behaviour of (bootstrapped versions of) such "derived" test statistics under the constancy hypothesis (where we employ KMT-type inequalities to derive Cramér-type deviation results), as well as under alternatives in the vicinity of the constancy hypothesis. This enables us to compute efficiencies of the corresponding tests. The discussion of some examples yield guidelines for the choice of the test statistic, and hence for the underlying pre-monitoring process.

(Joint work with Nils Lid Hjort, University of Oslo)

Speaker: Dominique Spehner, Theoretical Physics Universit\"at Duisburg-Essen

Title: Mott law as lower bound for a random walk in a random environment

Abstract: We consider a random walk on the support of a stationary simple point process on R^d, d\geq 2 which satisfies a mixing condition w.r.t.the translations or has a strictly positive density uniformly on large enough cubes. Furthermore the point process is furnished with independent random bounded energy marks. The transition rates of the random walk decay exponentially in the jump distances and depend on the energies through a factor of the Boltzmann-type. This is an effective model for the phonon-induced hopping of electrons in disordered solids within the regime of strong Anderson localization. We show that the rescaled random walk converges to a Brownian motion whose diffusion coefficient is bounded below by Mott's law for the variable range hopping conductivity at zero frequency. The proof of the lower bound involves estimates for the supercritical regime of an associated site percolation problem.

Fall 2003
Spring 2003
Fall 2002

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