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.Seminar page
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
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) Seminar
page
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.Seminar page
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.Seminar page
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. Seminar page
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
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.Seminar page
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.Seminar page
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.Seminar page
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.Seminar page
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.Seminar page
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
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
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.Seminar page
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.Seminar
page
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)Seminar page
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
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.Seminar page