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Series: Stochastics Seminar

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.

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

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