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.

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 user-chosen 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 Box-Cox 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 meta-criterion that measures how well the sets of K-nearest neighbors agree between the original high-dimensional space and the low-dimensional embedding space. This meta-criterion has intuitive appeal, and it performs well in creating faithful embeddings.

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.

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.

Lagrangian relaxation is a general methodology for approximating hard combinatorial problems and has been applied with success to various instances such as the Max-Cut problem. The compressed sensing problem of recovering the sparsest solution to a system of linear equations is known to be NP-hard and is often relaxed by means of L₁ 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 L₁ minimization. The resulting iterative algorithm is called Alternating L₁. One of the standard features of the Lagrangian approach for hard combinatorial problems is to provide provably efficient approximation schemes. In the case of L₁ minimization, E. Candès, T. Tao and their collaborators were able to to show that with high probability, L₁ minimization does in fact better in that it recovers the solution of the original sparsest recovery problem exactly. We will show that the Alternating L₁ algorithm allows to recover the sparsest solution with fewer observations than L₁ 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 L₁ 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 L₁ algorithm.

A new spectrum estimator is introduced. The new estimator exploits quadratic inverse theory, see [1][2], to attain improved mean-square error performance over the standard, eigenvalue weighted, non-adaptive spectrum estimator. In [3], the standard, consistent, multitaper estimator is obtained by averaging a high-resolution, 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 in-band 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 mean-square 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 time-series 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. 317-394, Cambridge Univ. Press, 2001

[3] "Spectrum Estimation and Harmonic Analysis", D. J. Thomson, Proceedings of the IEEE, Vol. 70, pp. 1055-1096, 1982

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.

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.

This talk describes a computationally feasible approach for maximum likelihood estimation of parameters based on long realizations from vector autoregressive fractionally integrated moving-average (VARFIMA) processes. VARFIMA models can capture both short-term correlation structure and long-range 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 block-Toeplitz 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.

Truncated Toeplitz operators arise commonly in the statistical analysis of continuous-time 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 continuous-time stationary process with possibly unbounded or vanishing spectral density, that is, the model displays long-memory or is an anti-persistent process.
In particular, we obtain an explicit second-order asymptotic expansion for the trace of product of two truncated Toeplitz operators generated by the spectral densities of continuous-time stationary fractional Riesz-Bessel motions. We show that the order of magnitude of the second term in this expansion depends on the long-memory parameters of the underlying processes, and the singularity in the first-order approximation is removed by the second-order term, which provides a substantially improved approximation to the original functional.