# Mathematical Statistics Seminar

## Statistical inference in possibly misspecified nonregular models

Speaker(s):
Natalia Bochkina (University of Edinburgh, UK)
Date:
Wednesday, April 15, 2015 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

Finite dimensional statistical models are usually called nonregular if the regularity assumptions (e.g. of the Cramer-Rao inequality) do not hold. For such models, it is possible to construct an estimator with the rate of convergence that is faster than the parametric root-n rate. I will give an overview of such models with the corresponding rates of convergence in the frequentist setting under the assumption that they are well-specified.

## Nonparametric estimation in the presence of complex nuisance components

Speaker(s):
Martin Wahl (Universität Mannheim)
Date:
Wednesday, February 4, 2015 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

We consider the nonparametric random regression model $Y=f_1(X_1)+f_2(X_2)+\epsilon$ and address the problem of estimating the function $f_1$. The term $f_2(X_2)$ is regarded as a nuisance term which can be considerably more complex than $f_1(X_1)$. Under minimal assumptions, we prove several nonasymptotic risk bounds for our estimators of $f_1$. Our approach is geometric and based on considerations in Hilbert spaces. It shows that the performance of our estimators is closely related to geometric quantities, such as minimal angles and Hilbert-Schmidt norms.

## Extreme value analysis of frame coefficients and applications

Speaker(s):
Markus Haltmeier (Universität Innsbruck)
Date:
Wednesday, January 28, 2015 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

Consider the problem of estimating a high-dimensional vector from linear observations that are corrupted by additive Gaussian white noise. Many solution approaches for such problems construct an estimate as the most regular element satisfying a bound on the coefficients of the residuals with respect to some frame. In order that the true parameter is feasible, the coefficients of the noise must satisfy the bound. For that purpose we compute the asymptotic distribution of these coefficients.

## Invariant Prediction and Causal Inference

Speaker(s):
Jonas Peters (ETH Zürich)
Date:
Wednesday, January 21, 2015 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

Why are we interested in the causal structure of a data-generating process? In a classical regression problem, for example, we include a variable into the model if it improves the prediction; it seems that no causal knowledge is required. In many situations, however, we are interested in the system's behavior under a change of environment. Here, causal models become important because they are usually considered invariant under those changes.

## Statistics of discretely observed semi-martingales under noise

Speaker(s):
Markus Bibinger (HU Berlin)
Date:
Wednesday, January 14, 2015 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

We consider statistical inference with main focus on volatility estimation from discrete noisy observations of a semi-martingale. In the prominent model with market microstructure noise, we discuss asymptotically efficient estimation of the integrated volatility matrix in a multivariate setup under high-frequency asymptotics. The estimation methodology along with stable central limit theorems for a general framework, also in presence of jumps, provide a valid approach for analyzing high-frequency financial data.

## Goodness-of-Fit Test for Model Specification

Speaker(s):
Qian (Michelle) Zhou (Simon Fraser University)
Date:
Wednesday, December 17, 2014 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

In this talk, I will introduce information ratio (IR) statistic to test for model misspecification in various models. The IR test was first proposed in my Ph.D. thesis to test for model misspecification of variance/covariance structure in quasi-likelihood inference for cross-sectional data or longitudinal data. The statistic is constructed via a contrast between two forms of information matrix: the negative sensitivity matrix and variability matrix.

## Gaussian Approximations, Bootstrap, and Z-estimators when p >> n

Speaker(s):
Viktor Chernozhukov (MIT)
Date:
Wednesday, December 10, 2014 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

We show that central limit theorems hold for high-dimensional normalized means hitting high-dimensional rectangles. These results apply even when p>> n. These theorems provide Gaussian distributional approximations that are not pivotal, but they can be consistently estimated via Gaussian multiplier methods and the empirical bootstrap. These results are useful for building confidence bands and for multiple testing via the step-down methods. Moreover, these results hold for approximately linear estimators.

## Nonparametric instrumental variable estimation under monotonicity

Speaker(s):
Denis Chetverikov (UCLA, USA)
Date:
Wednesday, December 3, 2014 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

The ill-posedness of the inverse problem of recovering a regression function in a nonparametric instrumental variable model leads to estimators that may suffer from a very slow, logarithmic rate of convergence. In this paper, we show that restricting the problem to models with monotone regression functions and monotone instruments significantly weakens the ill-posedness of the problem.

## On distributional properties of Lasso-type estimators

Speaker(s):
Ulrike Schneider (Wien)
Date:
Wednesday, November 26, 2014 - 10:00am
Location:
WIAS, Hausvogteiplatz 11a, 10117 Berlin, Raum 4.13

Penalized least-squares estimators, such as the famous Lasso estimator, have been studied intensively in the statistics literature in the past decade. While many aspects of these estimators are well-understood, still, relatively little is known about their distributional properties, such as finite- and large-sample distributions, uniform convergence rates and, in particular, confidence sets. We present exemplary results for the adaptive Lasso estimator and discuss why the approach often taken in the literature only gives partial answers.

## Asymptotic equivalence for discretely or continously observed Lévy processes and Gaussian white noise

Speaker(s):
Ester Mariucci (Grenoble)
Date:
Wednesday, November 12, 2014 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

When looking for asymptotic results for some statistical model it is often useful to dispose of a global asymptotic equivalence, in the Le Cam sense, in order to be allowed to work in a simpler model. In this talk, after giving an introduction to the main characters involved in the Le Cam theory, I will focus on equivalence results for Lévy processes. I will discuss global asymptotic equivalences between the experiments generated by the discrete (high frequency) or continuous observation of a path of a Lévy process and a Gaussian white noise experiment.