# Mathematical Statistics Seminar

## Local asymptotic equivalence for quantum models

Speaker(s):
Cristina Butucea (Université Marne-la-Vallée)
Date:
Wednesday, May 3, 2017 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

Quantum statistics is concerned with inference for physical systems described by quantum mechanics. After an introduction to the main notions of quantum statistics: quantum states, measurements, channels, we describe nonparametric quantum models. We prove the local asymptotic equivalence (LAE) of i.i.d. quantum pure states and a quantum Gaussian state, in the sense of Le Cam theory. As an application, we show the optimal rates for the estimation of pure states, for the estimation of some quadratic functionals and for the testing of pure states.

## Optimal rates of estimation for the multi-reference alignment problem

Speaker(s):
Jonathan Weed (MIT)
Date:
Wednesday, April 26, 2017 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

How should one estimate a signal, given only access to noisy versions of the signal corrupted by unknown circular shifts? This simple problem has surprisingly broad applications, in fields from structural biology to aircraft radar imaging. We describe how this model can be viewed as a multivariate Gaussian mixture model whose centers belong to an orbit of a group of orthogonal transformations. This enables us to derive matching lower and upper bounds for the optimal rate of statistical estimation for the underlying signal.

## The role of machine learning in the nonparametric prediction of time

Speaker(s):
László Györfi (Budapest University)
Date:
Wednesday, February 15, 2017 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

The main purpose of this paper is to consider the prediction of stationary time series for various losses: squared loss (regression problem), $0, 1$ loss (pattern recognition) and log utility (growth optimal portfolio selection). We are interested in universal prediction rules, which are consistent for all possible stationary and ergodic processes. Such rules can be constructed using aggregation techniques of machine learning by combining elementary rules (experts) in data dependent way.

## Multiscale scanning in inverse problems - With applications to nanobiophotonics

Speaker(s):
Katharina Proksch (Universität Göttingen)
Date:
Wednesday, February 8, 2017 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

We propose a multiscale scanning method to determine active components of a quantity $f$ w.r.t. a dictionary $\mathcal U$ from observations $Y$ in an inverse regression model $Y = Tf + \xi$ with operator $T$ and general random error $\xi$. To this end, we provide uniform confidence statements for the coefficients $(\varphi, f),\varphi\in\mathcal U$, under the assumption that $(T^{\star})^{-1}(\mathcal U)$ is of wavelet-type. Based on this we obtain a decision rule that allows to identify the active components of $\mathcal U$, i.e.

## Laguerre basis for inverse problems related to nonnegative random variables

Speaker(s):
Fabienne Comte (Université Paris Descartes)
Date:
Wednesday, February 1, 2017 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

I will present, through two main examples, the specific properties of the Laguerre basis and show that it is a very convenient tool to solve estimation problems on R+. The first example is the regression-convolution model: an estimator of the unknown underlying function is built in two steps (deconvolution step, regression step) which are explained and discussed. Then, a risk study is conducted, that shows as usual that a bias-variance tradeoff must be performed. A model selection device is shown to solve this question.

## Family-Wise separation rates for multiple testing

Speaker(s):
Magalie Fromont-Renoir (Université Rennes)
Date:
Wednesday, January 25, 2017 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

This joint work with Matthieu Lerasle (Univ. Paris-Saclay, France) and Patricia Reynaud-Bouret (Univ. Cote d'Azur, France) is devoted to the question of the theoretical evaluation of multiple testing procedures.Where as many first kind error-related evaluation criteria have been defined, as generalizations or relaxations of the historical Family-Wise Error Rate (FWER), very few second kind error-related criteria have been proposed in the multiple testing literature.

## Bootstrap Confidence Sets for Spectral Projectors of Sample Covariance

Speaker(s):
Alexey Naumov (Skoltech, Moscow)
Date:
Wednesday, January 18, 2017 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

Let X_1, ... ,X_n be i.i.d. sample in R^p with zero mean and the covariance matrix S. The problem of recovering the projector onto the eigenspace of S from these observations naturally arises in many applications. Recent technique from [Koltchinskii and Lounici, 2015b] helps to study the asymptotic distribution of the distance in the Frobenius norm between the true projector P_r on the subspace of the r th eigenvalue and its empirical counterpart \hat{P}_r in terms of the effective trace of S.

## Estimating latent asset-pricing factors

Speaker(s):
Markus Pelger (Stanford University)
Date:
Wednesday, January 11, 2017 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

We develop an estimator for latent factors in a large-dimensional panel of financial data that can explain expected excess returns. Statistical factor analysis based on Principal Component Analysis (PCA) has problems identifying factors with a small variance that are important for asset pricing. Our estimator searches for factors with a high Sharpe-ratio that can explain both the expected return and covariance structure.

Speaker(s):
Date:
Wednesday, December 7, 2016 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

In this talk we discuss a new method of unsupervised learning for high dimensional data based on the idea of adaptive weights from Polzehl and Spokoiny (2000). The procedure recovers the unknown clustering structure without any prior information about the number of clusters, their size, distance between clusters, etc. The approach extends the popular k-mean and density based clustering procedures by using dynamically updated local weights. Theoretical results describe two major features of the method: propagation within a homogeneous region and separation between two different regions.

## Estimation of linear and nonlinear functionals in nonparametric boundary models

Speaker(s):
Gwennaëlle Mabon und Markus Reiß (HU Berlin)
Date:
Wednesday, November 30, 2016 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

For nonparametric regression with one-sided errors and a boundary curve model for Poisson point R processes we consider first the problem of efficient estimation for linear functionals of the form \int f(x)w(x)dx with unknown f and known w. We propose a simple blockwise estimator and then build up a nonparametric maximum-likelihood approach. Both methods allow for estimation with optimal rate n^{-(\beta+1/2)/(\beta+1) under \beta-Hölder smoothness or monotonicity constraints (analogue of \beta = 1).