
Statistics and Probability Seminar Series  Fall 2008
Thursday 4:005:00pm, Room MCS 148 (Tea served from 3:304:00pm, Room MCS 153)
September 11
School of Operations Research and Information Engineering
Cornell University
Capital regulation is an important instrument to maintain a safe banking system and relies on proper risk measurement procedures. In the current talk, we review recent advances in the theory of risk measures. This includes their efficient computational implementation in practice, possible adjustments for liquidity risk, and a theoretical analysis of the economic consequences of regulation.
October 16
Department of Statistics
Yale University
In recent years there has been a resurgence of interest in nonlinear
dimension reduction
methods, which concern with constructing a nonlinear low dimensional
embedding of a hypothetical manifold near which the data fall. In this
talk we will introduce a family of new nonlinear dimension reduction
methods called "Local Multidimensional Scaling" or LMDS. Like other
methods in the area, LMDS only uses local information from userchosen
neighborhoods, but it differs from them in that it uses ideas from the
area of "graph layout". We approach the force paradigm, which is
commonly used in "graph layout", by proposing a parametrized family of
stress or energy functions inspired by BoxCox power transformations.
This family provides users with considerable flexibility for achieving
desirable embeddings, and it comprises most energy functions proposed
in the past.
Facing an embarrassment of riches of energy functions, we propose a metacriterion that measures how well the sets of Knearest neighbors agree between the original highdimensional space and the lowdimensional embedding space. This metacriterion has intuitive appeal, and it performs well in creating faithful embeddings.
October 23
Department of Statistics
Harvard University
Robust Parameter Design is a strategy of making products and
processes least sensitive to the effect of noise (uncontrollable)
variables by exploiting interactions between control factors and noise
factors. The talk will describe the development of comprehensive
frameworks for designing and analyzing such experiments for two different
systems. The first part will focus on automatically controlled dynamic
processes with correlated disturbances and the development of an
experimental approach to optimize such systems. The second part will
describe an experimental approach for estimation and reduction of
measurement variation (and its components) using a random coefficients
model.
October 30
Department of Statistics
Yale University
We will show a class of model selection procedures are asymptotically
sharp minimax to recover sparse signals over a wide range of parameter spaces.
Connections to Bayesian model selection, MDL principle and wavelet estimation
will be discussed.
November 4 ( Tuesday 4:005:00 pm )
Laboratoire de Mathématiques
Université de Franche Comté
Lagrangian relaxation is a general methodology for
approximating hard combinatorial problems and
has been applied with success to various instances such as the MaxCut
problem. The compressed sensing problem
of recovering the sparsest solution to a system of linear equations is
known to be NPhard and is often relaxed by
means of L_{1} minimization, i.e. a simple linear program. The goal of
this talk is to present
an approximate Lagrangian relaxation of the compressed sensing problem
leading to a better relaxation scheme than
L_{1} minimization. The resulting iterative algorithm is called Alternating L_{1}.
One of the standard features of the Lagrangian approach for hard
combinatorial problems is to provide provably efficient approximation
schemes. In the case of L_{1} minimization, E. Candès, T. Tao and their
collaborators were able to to show that with high probability, L_{1}
minimization does in fact better in that it recovers the solution of
the original sparsest recovery problem exactly. We will show
that the Alternating L_{1} algorithm allows to recover the sparsest
solution with fewer observations than L_{1} norm minimization. As many
recent proposals for the compressed sensing problem, an additional
parameter has to be tuned in order to obtain significant improvements
over the plain L_{1} strategy. A nice feature of our approach is that
this additional parameter is nothing but a Lagrange multiplier and the
best value is simply the one that optimizes the dual function. We will
also show how to circumvent the difficult task of computing the dual
optimizer exactly by proposing a meaningful value of this parameter
allowing for significant improvements in the first two steps of the
method, hence avoiding fastidious empirical tuning as is usually done
in the case of other methods such as the Reweighted L_{1} algorithm.
November 6
Department of Mathematics
Queen's University, Kingston, ON Canada
A new spectrum estimator is introduced. The new estimator exploits quadratic inverse theory, see [1][2], to attain improved meansquare error performance over the standard, eigenvalue weighted, nonadaptive spectrum estimator. In [3], the standard, consistent, multitaper estimator is obtained by averaging a highresolution, inconsistent, spectrum estimator over the estimator bandwidth. The improved performance of the proposed estimator results from replacing this average with a weighted average, computed in the space spanned by the quadratic inverse basis. The weighting, determined analytically, is chosen such that the resulting estimator minimizes the sum of the variance and the square of the inband bias; neglecting bias due to spectral leakage and potential bias due to the possible incompleteness of the quadratic inverse basis. The relative reduction of the meansquare error of the proposed spectrum estimator is validated, by simulation, for an ARMA(4,2) process. The reduction is found to be approximately equal to the square of the spectrum at frequencies relatively unaffected by spectral leakage. At leakage prone frequencies, performance increase is reduced, qualitatively consistent with the effect of the eigenvalue
weighting in the standard multitaper estimator.
Throughout the talk, an effort will be made to provide background on spectral timeseries analysis, on the multitaper method of spectrum estimation, and on quadratic inverse theory. [1] "Quadratic Inverse Spectrum Estimates: Applications to Palaeoclimatology", D. J. Thomson, Phil. Trans. R. Soc. Lond. A, 1990 [2] "Multitaper Analysis of Nonstationary and Nonlinear Time Series Data", D. J. Thomson in "Nonlinear and Nonstationary Signal Processing", W. Fitzgerald, R. Smith, A. Walden, P. Young, Eds., pp. 317394, Cambridge Univ. Press, 2001 [3] "Spectrum Estimation and Harmonic Analysis", D. J. Thomson, Proceedings of the IEEE, Vol. 70, pp. 10551096, 1982
November 13
Department of Mathematics and Statistics
University of Massachusetts, Amherst
Parameter identifiability is a principal assumption of statistical models in order to make meaningful inferences.
There are two nonidentifiabilities in finite mixture models: boundary nonidentifiability
and label nonidentifiability. Although parameters are not identifiable in the strict
sense, there is a form of asymptotic identifiability which can
provide reasonable answers as the sample size grows.
The reason why asymptotic identifiability occurs is because the parameters are locally identifiable.
In this talk, I address the role of the two key identifiabilities and nonidentifiabilities on finite mixture inference, and investigate estimation and labelling of parameter estimators when the sample size is not large relative to the separation of the components. I then propose new methods which can solve several drawbacks of existing methods.
November 20
Department of Biostatistics
Boston University
Queuing models of the National Airspace System require demand information
about when aircraft intend to land. Empirical data typically reflect when flights actually landed. Published schedules encapsulate delay expectations and actual landing times encapsulate actual delays, so neither is directly applicable. Instead, one might infer nominal arrival times from actual upstream departure times and unimpeded travel times. This paper presents a statistical aproach to estimate unimpeded flight times. The underlying idea is that the observed flight time is a mixture of two unobserved distributions  the unimpeded flight time and the delay. The statistical approach estimates these unobserved distributions, taking into account the seasonal periodicity of the data. The parameters of the model are estimated through an integrated EM algorithm and Kalman filter.
December 4
Department of Statistics
University of Connecticut
This talk describes a computationally feasible approach for maximum likelihood estimation of parameters based on long realizations from vector autoregressive fractionally integrated movingaverage (VARFIMA) processes. VARFIMA models can capture both shortterm correlation structure and longrange dependence characteristics of the individual series, as well as interdependence and feedback relationships between the series. Our approach is based on a multivariate preconditioned conjugate gradient (MPCG) algorithm, involving solution of a blockToeplitz system, and Monte Carlo integration over the process history. An application to pricing financial derivatives related to weather is discussed. This involves as a first step the modeling of univariate and multivariate daily average temperatures at selected measurement sites in the US using the long memory models, paying attention to accommodating some degree of volatility exhibited by such series.
December 11
Department of Mathematics and Statistics
Boston University
Truncated Toeplitz operators arise commonly in the
statistical analysis of continuoustime stationary
processes: asymptotic distributions and large
deviations of Toeplitz type quadratic functionals,
estimation of the spectral parameters and functionals,
hypotheses testing about the spectrum, etc.
In this talk we discuss the problem of approximation of the traces of products of truncated Toeplitz operators generated by integrable real symmetric functions defined on the real line, and bounding the corresponding errors. These approximations and the corresponding error bounds are of particular importance in the cases where the underlying model is a continuoustime stationary process with possibly unbounded or vanishing spectral density, that is, the model displays longmemory or is an antipersistent process. In particular, we obtain an explicit secondorder asymptotic expansion for the trace of product of two truncated Toeplitz operators generated by the spectral densities of continuoustime stationary fractional RieszBessel motions. We show that the order of magnitude of the second term in this expansion depends on the longmemory parameters of the underlying processes, and the singularity in the firstorder approximation is removed by the secondorder term, which provides a substantially improved approximation to the original functional. Information on seminars from previous semesters may be found here: Fall 2005  Spring 2006  Fall 2006 Spring 2007 Fall2007 Spring 2008.
