Jason Schweinsberg, University of California, Berkeley

Coalescents with simultaneous multiple collisions

We will define a family of coalescent processes that undergo "simultaneous multiple collisions," which means that many clusters of particles can merge into a single cluster at one time, and many such mergers can occur simultaneously. This family of processes generalizes the "coalescents with multiple collisions" introduced by Pitman, in which many clusters of particles can merge into a single cluster at one time, but almost surely no two such mergers occur simultaneously. We will discuss how exchangeability considerations make it possible to characterize all coalescents with simultaneous multiple collisions. We will also indicate how these processes can arise as limits of ancestral processes. Finally, we will present a condition under which one of these coalescents "comes down from infinity", meaning that only finitely many clusters remain at all positive times even if the process begins with infinitely many particles at time zero.
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Bhramar Mukherjee, Purdue University

Optimal Designs for Estimating the Path of a Stochastic Process

A second-order random process $Y(t)$, with $E(Y(t)) \equiv 0$, is sampled at a finite number of design points $t_1,t_2,...,t_n$. On the basis of these observations, one wants to estimate the values of the process at unsampled points using the best linear unbiased estimator (BLUE). The performance of the estimator is measured by a weighted integrated mean square error. The goal is to find $t_1,t_2,...,t_n$, such that this integrated mean square error (IMSE) is minimized for a fixed $n$. This design problem arises in statistical communication theory and signal processing as well as in geology and environmental sciences.

This optimization problem depends on the stochastic process only through its covariance structure. For processes with a product type covariance structure, i.e., for $Cov(Y(s),Y(t))=u(s) v(t)$, $s < t$, a set of necessary and sufficient conditions for a design to be exactly optimal will be presented. Explicit calculations of optimal designs for any given $n$ for Brownian Motion, Brownian Bridge and Ornstein-Uhlenbeck process will illustrate the simplicity and usefulness of these conditions. Starting from the set of exact optimality conditions for a fixed $n$, an asymptotic result yielding the density whose percentile points furnish a set of asymptotically optimal design points (in some suitable sense) will be described. Results on the problem when one tries to estimate the integral of $Y(t)$ instead of the path will be discussed briefly. The integral estimation problem is related to certain regression design problems with correlated errors.

For a more general covariance structure, satisfying natural regularity conditions, some interesting asymptotic results will be presented. It will be shown that for processes with no quadratic mean derivative, a much simpler estimator is asymptotically equivalent to the BLUE. This will lead to an intuitively appealing argument in establishing the asymptotic behaviour of the BLUE and also in deriving an analytical expression for the asymptotically optimal design density.

I will omit technical details and focus on explaining the principal ideas.Seminar page

Antonietta Mira, Universita' dell'Insubria (Varese, Italy)

Delaying Rejection in Metropolis-Hastings Algorithms

In a Metropolis-Hastings algorithm, rejection of proposed moves is an intrinsic part of ensuring that the chain converges to the intended target distribution. However, persistent rejection, perhaps in particular parts of the state space, may indicate that locally the proposal distribution is badly calibrated to the target. As an alternative to careful off-line tuning of state-dependent proposals, the basic algorithm can be modified so that on rejection, a second attempt to move is made. A different proposal can be generated from a new distribution, that is allowed to depend on the previously rejected proposal. We generalise this idea of delaying the rejection and adapting the proposal distribution, due to Tierney and Mira (1999), to generate a more flexible class of methods, that in particular applies to a variable dimension setting. The approach is illustrated by a pedagogical example, and a more realistic application, to a change-point analysis for point processes. (This is joint work with Peter Green, U. of Bristol, UK)
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Wolfgang Jank, University of Florida

Efficiency of Monte Carlo EM and Simulated Maximum Likelihood in Generalized Linear Mixed Models

Generalized linear mixed models have become increasingly popular for modeling over-dispersion, clustering and correlation. However, a practical limitation to the use of these models is that the likelihood function often involves an intractable, high-dimensional integral. Analytical approximations to this integral can result in inconsistent parameter estimates and numerical approximations are not generally recommended for high dimensions. In this work we investigate and compare two Monte Carlo approximation methods: simulated maximum likelihood and the Monte Carlo EM algorithm. Replacing an intractable integral by a Monte Carlo approximation introduces a stochastic error, the Monte Carlo error. In this paper we investigate the Monte Carlo error of simulated maximum likelihood and Monte Carlo EM analytically. We show that the Monte Carlo error of simulated maximum likelihood is unbounded relative to that of Monte Carlo EM, suggesting that Monte Carlo EM is the more efficient method to use in many applications. We also discuss practical limitations to the use of simulated maximum likelihood.
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Willa Chen, T. J. Watson Research Center, IBM and Stern School of Business, NYU

New Tools for Estimating Long Memory and Fractional Cointegration, with an Application to Interest Rates

Are interest rates stationary or nonstationary processes? Do they have deterministic trend? Do they possess long memory? Is there a cointegrating relationship between two series of interest rates with different maturities? All of these questions are closely tied to the estimation of the memory parameter, $d$. Most theoretical results assume that $d$ lies in the interval $(-0.5,0.5)$, so that the series is stationary and invertible. Furthermore, the potential presence of linear trend is excluded in much of the existing theory. In this talk, we will propose several new frequency domain techniques to provide some answers to the questions posed above, for a data set of daily US interest rates. We will discuss a new family of tapers and their application to memory parameter estimation. These tapers can also be applied to estimation of the cointegration parameter, allowing the memory parameters of the cointegrating series and their residuals to have fractional values instead of the integers, zero and one, assumed in the standard cointegration framework. We find that the degree of cointegration becomes weaker as the difference between the two maturities increases. Furthermore, the degree of cointegration between short and long term interest rates is much weaker than the standard (one-zero) cointegration, contrary to widely held belief. These results are confirmed by a newly-proposed generalized portmanteau goodness-of-fit test for time series models.
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Clyde Schoolfield, Harvard University

Random Walks on Wreath Products of Groups

For a certain random walk on the symmetric group $S_n$ that is generated by random transpositions, Diaconis and Shahshahani (1981) obtained bounds on the rate of convergence to uniformity using group representation theory. Similarly, we bound the rate of convergence to uniformity for a random walk on the hyperoctahedral group $Z_2 \wr S_n$ that is generated by random signed transpositions. Specifically, we determine that, to first order in $n$, $\frac{1}{2} n \log n$ steps are both necessary and sufficient for total variation distance to become small. Moreover, we show that our walk exhibits the so-called ``cutoff phenomenon.'' We extend our results on this random walk to the generalized symmetric groups $Z_m \wr S_n$ and further to the complete monomial groups $G \wr S_n$ for any finite group $G$. As an example, we will describe an application of our results to mathematical biology.
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Hongquan Xu, University of Michigan

Optimal Factor Assignment for Asymmetrical Fractional Factorial Designs: Theory and Applications

Fractional factorial designs have been successfully used in various scientific investigations for many decades. Its practical success is due to its efficient use of experimental runs to study many factors simultaneously. A fundamental and practically important question for factorial designs is the issue of optimal factor assignment to columns of the design matrix.

Aimed at solving this problem, this thesis introduces two new criteria: the generalized minimum aberration and the minimum moment aberration, which are extensions of the minimum aberration and minimum $G_2$-aberration. These new criteria work for symmetrical and asymmetrical designs, regular and nonregular designs, orthogonal and nonorthogonal designs, nonsaturated and supersaturated designs. They are equivalent for symmetrical designs and in a weak sense for asymmetrical designs.

The theory developed for these new criteria covers many existing theoretical results as special cases. In particular, a general complementary design theory is developed for asymmetrical designs and some general optimality results for mixed-level supersaturated designs.

As an application, a two-step approach is proposed for finding optimal designs and some 16-, 27- and 36-run optimal designs are tabulated. As another application, an algorithm is developed for constructing mixed-level orthogonal and nearly orthogonal arrays, which can efficiently construct a variety of small-run arrays with good statistical properties.Seminar page

Radu V. Craiu, The University of Chicago

Multi-process Parallel Antithetic Coupling for Backward and Forward Markov Chain Monte Carlo

The main motivation of this work is given by a novel implementation of the antithetic principle in Markov chain Monte Carlo algorithms. In the past, researchers have typically used antithetic variates in pairs. We argue that the negative coupling of more than two simulation processes significantly increases the variance reductions over the paired case. We use the qualitative and quantitative characterizations of negative dependence, negative association, and respectively, extreme antithesis to mimic some of the optimal properties that made the paired case a favorite for so long. A couple of methods to generate K-tuples of antithetic variates with the desired properties are discussed. Propp and Wilson's coupling from the past is the first medium for implementing our new methods. As a bonus, the theoretical results we obtain for antithetic backward coupling simultaneously answer questions related to the joint stationarity of a coupled chain. As a result, we show that variance savings are guaranteed to occur also in the case of Gibbs samplers for an attractive stationary distribution. We apply our methods to a perfect sampling algorithm devised for a Bayes mixture parameter estimation problem and to a forward slice sampling algorithm.
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Walter W. Piegorsch, University of South Carolina

Extended Logistic Regression via the Heckman-Willis Model, or "When is a Trend a Trend"?

Methods are discussed for modeling trends in proportions that exhibit extra-binomial variability. Motivation is taken from experiments in developmental toxicology, where correlations among embryos within a litter lead to the extra-binomial variability. Appeal is made to the well-known beta-binomial distribution to represent the excess variability. An exponential function is used to model the dose-response relationship, which leads to a form of logistic regression for the mean proportion response. The exponential relation also induces a functional model for the intra-litter correlation. The model is common in the econometric literature, and is known there as the Heckman-Willis beta-logistic model. In the biological and environmental sciences, however, the model is virtually unused. An example is presented to illustrate how easily the model is applied to developmental toxicity data. An associated question of how to define an increasing trend in the mean response under non-constant correlation will be discussed.
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Persi Diaconis, Stanford University

Statistical Analysis of the Zeros of Riemann's Zeta Function

I will model and compare 50,000 zeros of Riemann's Zeta Function with the predications of Random Matrix Theory. This gives rise to interesting questions in Bayesian testing and testing in the presence of dependence.
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Luis Pericchi, Universidad Simon Bolivar (Venezuela)

Objective Bayes Factors and Posterior Model Probabilities: A Host of Potential Applications

In recent years, there has been developments within the Bayesian approach to statistics, that have made possible to tackle ever more complex problems with milder prior assumptions. The conjunction of these two directions, is certainly enabling Bayesian statistics to share a wider market of the practice of statistics. One fundamental area of statistics as a whole, falls under the term of model comparisons, hypothesis testing and inferences under model uncertainty, an area on which, arguably, Bayesian methods are useful to improve the practice of statistics. In this area, Objective and Intrinsic Bayes Factors, Intrinsic priors and related methods, are studied in Berger and Pericchi (2000), showing how to produce Bayes Factors and model posterior probabilities with minimal prior inputs. These theories have attracted attention, and they are now used and studied by both practitioners and theoreticians of Statistics. In this talk we briefly present some theory and introduce applications to Automatic Robust Statistical Inference and to Model Comparisons of Dynamic Linear Models. There are many other potential applications that I hope to discuss during my visit to the University of Florida at Gainesville.
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Rob Kass, Carnegie Mellon University

Bayesian Curve Fitting and Neuronal Firing Patterns

A central problem in cognitive neuroscience is to understand the way neuronal firing patterns represent information---this is usually discussed under the rubric of ``neural coding.'' From a statistical point of view, the issues involved have to do with the analysis of single and multiple point process data. Together with colleagues in our Department and at the joint Carnegie Mellon and University of Pittsburgh Center for the Neural Basis of Cognition, I have been working on spline-based methods of fitting inhomogeneous Poisson processes, and a non-Poisson generalization of these, to single and multiple neuron data. In my talk, I will briefly describe the neurophysiological setting, and some results using standard smoothing methods, and then use this as a background to discuss a general approach to curve fitting with free-knot splines and reversible-jump Markov chain Monte Carlo. Simulation results indicate this to be a powerful methodology. I will introduce it in the setting of Normal observations, then show how it may be applied to Poisson and other non-Normal data, so that we will ultimately arrive at a fairly general approach to fitting neuronal ``spike train'' point process intensity functions.
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Jeanne Kowalski, Harvard School of Public Health

A Non-Parametric Approach to Testing and Characterizing Gene Region Heterogeneity Associated with Phenotype

An important focus of genetic research is to study mutation patterns through their impact on phenotype. Of particular interest is the identification of mutations at locations within a gene region that together depict phenotype. This task includes statistical issues of high dimensionality coupled with small sample sizes. For retroviral genomes, such as HIV, these issues are further compounded by the existence of distinct but genetically related viral variants.

In this talk I describe two non-parametric approaches, one for comparing and another for characterizing, distributions of a gene region (sequence pair) heterogeneity measure between groups with similar phenotype. For comparing distributions, hypotheses are constructed for testing differential between-group heterogeneity and within-group homogeneity. Group comparisons are made based on either developed asymptotics (extending U-statistic theory to a correlated multivariate two-sample setting) or permutation tests. For characterizing gene region heterogeneity, a method is constructed for identifying potentially important locations and their mutation patterns. The relative importance of locations is evaluated through estimation of their contribution to observed gene region differences; mutation patterns are discerned through location descriptive statistics. As motivation for the methods, I examine the problem of altered HIV drug susceptibility and illustrate their use in testing and characterizing protease region differences associated with a phenotypic treatment response.Seminar page

Trevor Sweeting , University of Surrey (UK)

Some applications of higher-order asymptotics in Bayesian inference

I will describe some applications of higher-order asymptotic techniques in two related areas: Bayesian computation and sampling properties of Bayesian procedures. Specific topics will include the use of directed likelihood methods in simulation-based computation and the development of objective Bayesian methods via considerations of coverage probability bias. The talk will begin with a review of higher-order asymptotics in Bayesian statistics, particularly methods based on signed-root log-likelihood ratios, followed by an account of some of my recent and current work on applications of these methods.Seminar page

Lei Li, Florida State University

DNA sequencing and parametric deconvolution

One of the key practice of Human genome project is Sanger DNA sequencing. Most DNA sequencing errors come from diffusion effect in electrophoresis, and deconvolution is the tool to get over this problem. Deconvolution is one of the so called ill-posed problems if no further knowledge on the unknowns can be assumed. In this talk, we discuss general ideas of well-defined statistical models as the counterparts of the concepts of well-posedness. Consequently, we address that knowledge of the unknowns such as non-negativity and sparsity can help a great deal to get over the ill-posedness in deconvolution. This is illustrated by the parametric deconvolution method based on the spike-convolution model. Not only this knowledge together with the choice of the measure of goodness of fit helps people think of data---models, but also determines the way people compute with data---algorithms. This is illustrated by taking a fresh look at two deconvolvers: the widely used Jansson's method, and another one which is to minimize the Kullback-Leibler distance between the observations and the fitted values. We compare the performance of these deconvolvers using data simulated from a spike-convolution model and real DNA sequencing data.
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Erich Lehmann, University of California, Berkeley

The assumption of normality. Should we trust it? Do we need it?

The talk considers the robustness of the one- and two-sample t-tests against non-normality, and -- in the two-sample case -- heterogeneity of variance. The t-test is compared with its permutation version and with the Wilcoxon test. Some open problems will be mentioned.
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Stan Uryasev, University of Florida (Industrial and Systems Engineering)

Mathematical Programming Approaches to Classification: Application to Credit Cards Scoring

We consider a new approach for classification of objects using mathematical programming algorithms. Although the approach is quite general, our study is focused on financial engineering applications. In particular, we consider the so called "credit cards scoring" problem. With this approach we are able to split credit card applications into three classes: "give credit card", "don't give credit card", and "don't know". The last class includes applications for which a credit professional should make a decision based on his/her experience. The approach is based on optimization of the utility function, which is quadratic with respect to the information parameters and is linear with respect to the control parameters, which need to be identified. A new feature is that we are able to incorporate expert judgments in the model. We include qualitative characteristics of the utility function, such as monotonicity, using additional constraints. For instance, we are able to consider the following constraint: give more preference to customers who have higher income. Numerical experiments have shown that including such constraints improve performance of the algorithm. We have completed a case study for a real life dataset obtained from a bank. (This is joint work with Vlad Bugera)
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Probal Chaudhuri, Indian Statistical Institute

Coelacanth vs. Lung-fish : A Fishy Story

The transition of life from water to land leading towards the development of land vertebrates during the Devonian period (approximately 350--400 million years ago) is one of the most significant events in the evolutionary history of vertebrates. The origin of terrestrial vertebrates from the aquatic ones involved complex morphological changes as well as physiological innovations, and the scarcity of fossil records has created a great deal of controversies among paleontologists, comparative morphologists and evolutionary biologists for several decades. It is generally agreed upon that the coelacanth together with different types of lungfish and the extinct rhipidistians formed the class of {\it lobe-finned fish} (Sarcopterygii) from which the tetrapods originated, and different varieties of {\it ray-finned fish} (Actinopterygii) are only distantly related to tetrapods. An extensive amount of research work has been carried out and published in recent issues of leading science journals by geneticists based on mitochondrial DNA data from the lungfish, the coelacanth and various other vertebrates such as mammals, birds, fish and amphibia to determine the relative phylogenetic positions of the coelacanth and the lungfish in the evolutionary tree of vertebrates. However, their findings are not unambiguous, and different statistical methods applied to different parts of the mitochondrial DNA pointed towards different possibilities.

In this talk, I will present some interesting results obtained from statistical analysis of complete mitochondrial genomes of coelacanth, lungfish and several mammals, birds, amphibia, reptiles and ray-finned fish based on {\bf distributions of DNA words}. I will demonstrate the use of a software called {\bf SWORDS}, which has been developed at Indian Statistical Institute for statistical analysis of large DNA sequences based on distributions of DNA words. This software has been specifically designed for handling very large DNA sequences (for instance, the size of a full mitochondrial genome of any vertebrate lies between 15,000 and 18,000 base pairs) to compare them for phylogenetic analysis.Seminar page

Amy Cantrell , University of Florida

On the Strong Law of Large Numbers for Sums of Random Elements in Banach Spaces

We present strong laws of large numbers for Banach space valued random variables under various conditions. The main result extends a well-known strong law for independent random variable result due to Heyde (1968) and yields, as a corollary, Feller's (1946) strong law. A version of the main result is obtained for which the assumption of independence is not needed. The results presented are new even when the Banach space is chosen to be the real line.
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