Paul Rochet

A general procedure to combine estimators

A general method to combine several estimators of the same quantity is investigated. In the spirit of model and forecast averaging, the final estimator is computed as a weighted average of the initial ones, where the weights are constrained to sum to one. In this framework, the optimal weights, minimizing the quadratic loss, are entirely determined by the mean squared error matrix of the vector of initial estimators. The averaging estimator is built using an estimation of this matrix, which can be computed from the same dataset. A non-asymptotic error bound on the averaging estimator is derived, leading to asymptotic optimality under mild conditions on the estimated mean squared error matrix. This method is illustrated on standard statistical problems in parametric and semi-parametric models where the averaging estimator outperforms the initial estimators in most cases.

Algebraic combinatorics on trace monoids: extending number theory to walks on graphs

Partially commutative monoids provide a powerful tool to study graphs, viewingwalks as words whose letters, the edges of the graph, obey a specific commutation rule. A particularclass of traces emerges from this framework, the hikes, whose alphabet is the set of simple cycleson the graph. We show that hikes characterize undirected graphs uniquely, up to isomorphism, andsatisfy remarkable algebraic properties such as the existence and uniqueness of a prime factorization.Because of this, the set of hikes partially ordered by divisibility hosts a plethora of relations in directcorrespondence with those found in number theory. Some applications of these results are presented,including a permanantal extension to MacMahons master theorem and a derivation of the Ihara zetafunction.

Evaluating balance on social networks from their simple cycles

Signed networks have long been used to represent social relations of amity (+) and enmity (-) between individuals. Group of individuals who are cyclically connected are said to be balanced if the number of negative edges in the cycle is even and unbalanced otherwise. In its earliest and most natural formulation, the balance of a social network was thus defined from its simple cycles, cycles which do not visit any vertex more than once. Because of the inherent difficulty associated with finding such cycles on very large networks, social balance has since then been studied via other means. In this article we present the balance as measured from the simple cycles and primitive orbits of social networks. We specifically provide two measures of balance: the proportion

Relations Between Connected and Self-Avoiding Hikes in Labelled Complete Digraphs

R_\ell

Enumerating simple paths from connected induced subgraphs

of negative simple cycles of length

An Hopf algebra for counting simple cycles

\ell

Reconstructing undirected graphs from eigenspaces

for each

Estimating the transition matrix of a Markov chain observed at random times

\ell\leq 20

Hypothesis testing for Markovian models with random time observations

which generalises the triangle index, and a ratio

Adaptive hard-thresholding for linear inverse problems

K_\ell