arXiv subjects

Mathieu Rosenbaum

Publications and source records attributed to Mathieu Rosenbaum.

77 records · Page 5Linked to original sources

arXiv2012

Testing the finiteness of the support of a distribution: a statistical look at Tsirelson's equation

We consider the following statistical problem: based on an i.i.d.sample of size n of integer valued random variables with common law m, is it possible to test whether or not the support of m is finite as n goes to infinity? This question is in particular connected to a simple case of Tsirelson's equation, for which it is natural to distinguish between two main configurations, the first one leading only to laws with finite support, and the second one including laws with infinite support. We show that it is in fact not possible to discriminate between the two situations, even using a very weak notion of statistical test.

math.PR↗

arXiv2011

Improved Matrix Uncertainty Selector

We consider the regression model with observation error in the design: y=Xθ* + e, Z=X+N. Here the random vector y in R^n and the random n*p matrix Z are observed, the n*p matrix X is unknown, N is an n*p random noise matrix, e in R^n is a random noise vector, and θ* is a vector of unknown parameters to be estimated. We consider the setting where the dimension p can be much larger than the sample size n and θ* is sparse. Because of the presence of the noise matrix N, the commonly used Lasso and Dantzig selector are unstable. An alternative procedure called the Matrix Uncertainty (MU) selector has been proposed in Rosenbaum and Tsybakov (2010) in order to account for the noise. The properties of the MU selector have been studied in Rosenbaum and Tsybakov (2010) for sparse θ* under the assumption that the noise matrix N is deterministic and its values are small. In this paper, we propose a modification of the MU selector when N is a random matrix with zero-mean entries having the variances that can be estimated. This is, for example, the case in the model where the entries of X are missing at random. We show both theoretically and numerically that, under these conditions, the new estimator called the Compensated MU selector achieves better accuracy of estimation than the original MU selector.

math.ST↗

arXiv2010

Sparse recovery under matrix uncertainty

We consider the model {eqnarray*}y=Xθ^*+ξ, Z=X+Ξ,{eqnarray*} where the random vector $y\in\mathbb{R}^n$ and the random $n\times p$ matrix $Z$ are observed, the $n\times p$ matrix $X$ is unknown, $Ξ$ is an $n\times p$ random noise matrix, $ξ\in\mathbb{R}^n$ is a noise independent of $Ξ$, and $θ^*$ is a vector of unknown parameters to be estimated. The matrix uncertainty is in the fact that $X$ is observed with additive error. For dimensions $p$ that can be much larger than the sample size $n$, we consider the estimation of sparse vectors $θ^*$. Under matrix uncertainty, the Lasso and Dantzig selector turn out to be extremely unstable in recovering the sparsity pattern (i.e., of the set of nonzero components of $θ^*$), even if the noise level is very small. We suggest new estimators called matrix uncertainty selectors (or, shortly, the MU-selectors) which are close to $θ^*$ in different norms and in the prediction risk if the restricted eigenvalue assumption on $X$ is satisfied. We also show that under somewhat stronger assumptions, these estimators recover correctly the sparsity pattern.

math.ST↗

arXiv2010

Asymptotic results and statistical procedures for time-changed Lévy processes sampled at hitting times

We provide asymptotic results and develop high frequency statistical procedures for time-changed Lévy processes sampled at random instants. The sampling times are given by first hitting times of symmetric barriers whose distance with respect to the starting point is equal to $\varepsilon$. This setting can be seen as a first step towards a model for tick-by-tick financial data allowing for large jumps. For a wide class of Lévy processes, we introduce a renormalization depending on $\varepsilon$, under which the Lévy process converges in law to an $α$-stable process as $\varepsilon$ goes to $0$. The convergence is extended to moments of hitting times and overshoots. In particular, these results allow us to construct consistent estimators of the time change and of the Blumenthal-Getoor index of the underlying Lévy process. Convergence rates and a central limit theorem are established under additional assumptions.

math.PR↗

arXiv2009

Integrated volatility and round-off error

We consider a microstructure model for a financial asset, allowing for price discreteness and for a diffusive behavior at large sampling scale. This model, introduced by Delattre and Jacod, consists in the observation at the high frequency $n$, with round-off error $α_n$, of a diffusion on a finite interval. We give from this sample estimators for different forms of the integrated volatility of the asset. Our method is based on variational properties of the process associated with wavelet techniques. We prove that the accuracy of our estimation procedures is $α_n\vee n^{-1/2}$. Using compensated estimators, limit theorems are obtained.

math.ST↗