UMKC Mathematics Graduate Seminar Series

Graduate Seminar Series

Semester Spring 2009
Location: Royall Hall, Room 213 (Unless otherwise noted)
Day & Time: Wednesdays or Fridays, 2:00-2:50 pm (Unless otherwise noted)
Campus Map for Talks (PDF Format)

Organizer: Dr. Hristo Voulov, 235-5851
Email: voulovh@umkc.edu

Dates, Titles, Speakers (with Abstracts as available)

Wednesday Feb. 18
Transformations of Z₂^∞ in the Study of the Axiom of Choice
Eric Hall, Department of Mathematics and Statistics, UMKC

This talk is about the recent solution to the problem of finding a model of set theory in which the axiom of choice for pairs fails but in which every set of pairs has an infinite partial choice function. Independence results in set theory such as this often reduces to combinatorial problems involving permutation. In this case, it turns out that what is needed is to study permutations which are linear transformations of the vector space Z₂^∞—the countable infinite dimensional vector space over the 2-element field Z₂.

Wednesday Feb. 25
Some Properties of Solutions of Nonlinear Second Order Differential Equations
Lianwen Wang, Department of Mathematics and Computer Science, University of Central Missouri

Wednesday March 11
Miron Bekker, Department of Mathematics and Statistics, UMKC

Friday March 20
Gauss-Seidel Estimation of Generalized Linear Mixed Models with Application to Poisson Modeling of Spatially Varying Disease Rates
Subharup Guha, University of Missouri�Columbia

Generalized linear mixed models (GLMMs) are often fit by computational procedures such as penalized quasi-likelihood. Special cases of GLMMs are generalized linear models, which are often fit using algorithms like iterative weighted least squares (IWLS). High computational costs and memory space constraints make it difficult to apply these iterative procedures to data sets having a very large number of records.

We propose a computationally efficient strategy based on the Gauss-Seidel algorithm that iteratively fits sub-models of the GLMM to collapsed versions of the data. The strategy is applied to investigate the relationship between ischemic heart disease, socioeconomic status and age/gender category in New South Wales, Australia, based on outcome data consisting of approximately 33 million records. For Poisson and binomial regression models, the Gauss-Seidel approach is found to substantially outperform existing methods in terms of maximum analyzable sample size. Remarkably, for both models, the average time per iteration and the total time until convergence of the Gauss-Seidel procedure are less than 0.3% of the corresponding times for the IWLS algorithm. This is joint work with Drs. Louise Ryan and Michele Morara.

Friday April 3
Compound Poisson disorder problems with nonlinear detection delay penalty cost functions
Savas Danayic, Operation Research and Financial Engineering, Princeton University

The quickest detection of the unknown and unobservable disorder time, when the arrival rate and mark distribution of a compound Poisson process suddenly changes, has been formulated in a Bayesian setting, where the detection delay penalty is a general smooth function of the detection delay time. Under suitable conditions, the problem is shown to be equivalent to the optimal stopping of a finite-dimensional piecewise-deterministic strongly Markov sufficient statistic. The solution of the optimal stopping problem is described in detail for the compound Poisson disorder problem with polynomial detection delay penalty function of arbitrary but fixed degree. The results are illustrated for the case of the quadratic detection delay penalty function.

Friday April 10
Automorphic-invariant non-densely defined hermitian contractive operators
Miron Bekker, Dept of Mathematics and Statistics, UMKC

We consider operators with norm not greater than 1, defined on a proper subspace of a Hilbert space that have Hermitian property (non-densely defined Hermitian contractions). In addition we assume that such operators are unitarily equivalent to their linear-fractional transformations (automorphic-invariant operators). We show that any such operator always admits a self-adjoint extension with the same norm that is also automorphic-invariant. A functional characterization of such operators is given in terms of a resolvent of the self-adjoint automorphic-invariant extension. A special attention is given to the case when the codimension of the domain of the non-densely defined Hermitian contraction is one. Examples of automorphic-invariant operators are considered.

Friday April 17, 5:00-5:50 (unusual time)
Kneser's Theorem in Quantum Calculus
Martin Bohner, Dept of Mathematics and Statistics, Missouri University of Science and Technology

While difference equations deal with discrete calculus and differential equations with continuous calculus, so-called q-difference equations are considered when studying q-calculus. In this talk, we present certain oscillation criteria for second-order q-difference equations, among them a q-calculus version of the famous Kneser theorem.

Friday April 24
Robustness of Volatility Estimation
Yingying Li, Operation Research and Financial Engineering, Princeton University

This talk contains three major parts. All are about the market microstructure error and volatility estimation using high frequency data.

In the first part, we consider the case when the market microstructure error is solely due to rounding. Rounding errors affect the estimation of volatility and understanding them is important especially when we use high frequency data. We study the asymptotic behavior of the Realized Volatility (RV) which is commonly used as an estimator of the integrated volatility. We prove the convergence of the RV and scaled RV under different conditions on the rounding level and the number of observations. A bias-corrected volatility estimator is proposed and the associated central limit theorem is shown. Simulation results show that improvement in statistical properties can be substantial.

In the second part, we consider microstructure as an arbitrary contamination of the underlying latent securities price, through a Markov kernel. Special cases include additive error, rounding, and combinations thereof. Our main result is that, subject to smoothness conditions, the Two Scales Realized Volatility is robust to the form of contamination. To push the limits of our result, we show what happens for some models involving rounding and see in this situation how the robustness deteriorates with decreasing smoothness. Our conclusion is that under reasonable smoothness, one does not need to consider too closely how the microstructure is formed, while if severe non-smoothness is suspected, one needs to pay attention to the precise structure and also the use to which the estimator of volatility will be put.

In the third part, we present a generalized pre-averaging approach for estimating the integrated volatility. This approach also provides consistent estimators of other powers of volatility. It gives feasible ways to consistently estimate the asymptotic variance of the estimator of the integrated volatility. This approach, which possesses an intuitive transparency, can generate rate optimal estimators (with convergence rate n^-1/4).