Statistics Seminar

Speaker Dr. Ming Yuan, University of Wisconsin – Madison
Title           “Rate-Optimal Detection of Very Short Signal Segments”
Date            Thursday, October 23, 2014
Time            4:00 PM – 5:00 PM
Location        269 Everitt

Abstract:
Motivated by a range of applications in engineering and genomics, we consider in this paper detection of very short signal segments in three settings: signals with known shape, arbitrary signals, and smooth signals. Optimal rates of detection are established for the three cases and rate-optimal detectors are constructed. The detectors are easily implementable and are based on scanning with linear and quadratic statistics. Our analysis reveals both similarities and differences in the strategy and fundamental difficulty of detection among these three settings.

Statistics Seminar – Dr. Peter Hoff, University of Washington: “Multilinear Models for Relational Tensor Data”

Date            Thursday, October 16, 2014
Time            4:00 PM – 5:00 PM
Location        269 Everitt

Abstract: A fundamental aspect of relational data, such as from a social network, is the possibility of dependence among the relations: How A relates to B may possibly depend on how C relates to B or even how C relates to D. We investigate the possibility of estimating such dependencies from weekly international relations data among countries from 2004-2013. Such data can be represented as a multiway array, or tensor. We develop a multilinear regression model for analysis of such tensor-valued data, discuss some of its properties, and compare it to some more familiar linear regression models.

Statistics Seminar – Dr. Tobias Kley, Ruhr-Universitaet Bochum
“Quantile-Based Spectral Analysis of Time Series”

Date            Friday, October 03, 2014
Time            3:30 pm – 5:00 pm
Location        7 David Kinley Hall

Abstract:
In this talk an alternative approach to the spectral analysis of time series is presented. First, a “new” spectrum is defined for stationary processes {Y_t}_{t in Z}. We call the Fourier transform of the differences between copulas of the pairs (Y_t,Y_t−k) and the independence copula the copula spectral density kernel. This object allows to separate marginal and serial aspects of a time series. The cop- ula intrinsically provides more information about the conditional distribution than the covariance. Thus, the copula spectral density kernel is more informative than the spectral density obtained from the autocovariances. In particular, a complete description of the distributions of all pairs (Y_t,Y_{t−k}) is possible. Two rank-based estimators of the copula spectral density kernel are provided. We establish their asymptotic properties. The results are applicable without restrictive distributional assumptions such as the existence of finite moments and only weak mixing assumptions are necessary. To allow non-stationarity in the analysis, localized versions of the copula spectrum and of the estimators are introduced. We define the concept of local strict stationarity and show consis- tency of a localized estimator under this condition. The R package quantspec will be introduced. A demonstration on how it can be used in the analysis of an empirical example will be given. This is joint work with H. Dette, M. Hallin, S. Skowronek, and S. Volgushev. Dette, H., Hallin, M., Kley, T. and Volgushev, S. (2014+). Of copulas, quantiles, ranks and spectra: An L1-approach to spectral analysis. To appear in Bernoulli. Kley, T., Volgushev, S., Dette, H. and Hallin, M. (2014). Quantile spectral pro- cesses: Asymptotic analysis and inference (arxiv.org/abs/1401.8104). Skowronek, S, Volgushev, S., Kley, T., Dette, H. and Hallin, M. (2014). Quantile spectral analysis for locally stationary time series (arxiv.org/abs/1404.4605). Kley,T. (2014). Quantile-based spectral analysis in an object-oriented frame- work and a reference implementation in R: The quantspec Package (arxiv.org/abs/1408.6755).

Statistics Seminar – Dr. Frederi Viens: “Parameter estimation for a partially observed Ornstein-Uhlenbeck process with long-memory noise”

Speaker Dr. Frederi Viens (Purdue University)
Date            Thursday, October 02, 2014
Time            4:00 PM – 5:00 PM
Location        269 Everitt

Abstract: Generically, an Ornstein-Uhlenbeck (OU) process is the Gaussian solution $X$ of a stochastic differential equation of the form $dX = -a X dt + dW$, where $a$ is a positive drift parameter (the negative sign turns the term $-a X dt$ into a mean-reversion term) and $W$ is a Gaussian noise term, typically with stationary increments which need not be independent. We consider a system of two such equations, where one equation is autonomous and is driven by a long-memory noise (specifically a fractional Gaussian noise), while the other equation’s noise term is the solution of the first equation. The question is to estimate the two drift parameters based on a single trajectory of the second process. A popular method for this type of task is to use a joint least-squares estimator drawing on continuous-time observation, which happens to coincide with a maximum likelihood estimator in and only in the case of white noise. We show that this approach does work, by proving strong consistency and asymptotic normality for increasing horizon asymptotics, by appealing to some Malliavin calculus computations. Two problems with this approach are that in practice observations are typically discrete, and that the use of Malliavin calculus is non-trivial. We address the first of these two problems by proposing various ways of discretizing the continuous-time estimators, which can be interpreted as generalized methods of moments. Strong consistency and asymptotic normality, for in-fill and increasing horizon asymptotics, are proved by showing how these discrete estimators are related to the continous-time estimator. We study the question of how the partial observation problem affects the speed of convergence and the restrictions on the spacing of observations. For the second issue, that of avoiding the use of the Malliavin calculus, work is currently in progress. This work is joint with Prof. Khalifa Es-sebaiy from the National School of Applied Sciences in Marrakech, Morocco. A third problem with our work and indeed with a large portion of the research to date on statistic of stochastic processes with long memory, is that the issue of estimating the memory length is conveniently swept under the rug. Time allowing, we will briefly mention two techniques for studying this problem: one, jointly with Prof. Alexandra Chronopoulou, which is an application of calibration using high-frequency data in quantitative finance, and another, with Prof. Bo Li, which applies the power of Bayesian estimation in paleoclimatology where calibration cannot rely on high-frequency data.