Évènements

This presentation is a summary of my PhD work. I focus on the topic of hypothesis testing, extensively studied in statistics and theoretical computer science.

I start with presenting the classical identity testing problem, in which an independent sample set X ~ q is given and one would like to determine whether q=p for some fixed known p. This problem is very related to that of estimating a distribution from a given sample set. The study of testing is relevant, because for the same fixed sample size, it is possible to test against a distribution up to a smaller separation distance than what is possible in estimation. This will give me the opportunity to describe the minimax framework which proves the theoretical optimality of statistical methods in the worst case.

I will refine the study of minimax identity testing by adding a local differential privacy condition and the interest will be in the quantitative effect of ensuring privacy. The presentation will largely be on the topic of privacy, because it bears similarities with ensuring fairness conditions.

We will also shortly consider the neighboring problem of closeness testing, where the goal remains to determine whether p=q, but only an independent sample set Y ~ p is given instead of p directly. In this context, we will go beyond a simple worst-case analysis and develop instance optimal results instead. This will highlight the interplay between one-sample testing and two-sample testing, the latter being a harder problem.

Soient $\theta$ et $\rho$ des nombres réels avec $0 \le \theta, \rho < 1$ et $\theta$ irrationnel. Pour $n \ge 1$, posons $$ s_n := s_n (\theta, \rho) = \big\lfloor n \theta + \rho \big\rfloor – \big\lfloor (n-1) \theta + \rho \big\rfloor $$ Alors, le mot infini $$ {\bf s}_{\theta, \rho} := s_1 s_2 s_3 \ldots $$ est le mot sturmien (inférieur) de pente $\theta$ et d’intercept $\rho$, écrit sur l’alphabet $\{0, 1\}$. Nous explicitons le développement en fraction continue du nombre réel $$ \xi_{b, \theta, \rho} = (b-1) \, \sum_{n \ge 1} \, {s_n (\theta, \rho) \over b^n}. $$ Cela nous permet d’obtenir une formule donnant son exposant d’irrationalité en fonction de $\theta$ et du développement d’Ostrowski de $\rho$ en base $\theta$. Nous étendons ainsi un résultat classique de Böhmer (1927) qui ne couvre que le cas où $\rho = \theta$ et contient par exemple la surprenante égalité $$ \sum_{j \ge 1} {1 \over 2^{\lfloor j \gamma \rfloor} } = [0; 1, 2, 2, 2^2, 2^3, 2^5, 2^8, 2^{13}, 2^{21}, \ldots ], \quad \gamma = {1 + \sqrt{5} \over 2}. $$ Il s’agit d’un travail en commun avec Michel Laurent.