Equilibrium existence theorems in KKM spaces

An abstract convex space satisfying the KKM principle is called a KKM space. This class of spaces contains G$G$-convex spaces properly. In this work, we show that a large number of results in KKM theory on G$G$-convex spaces also hold on KKM spaces. Examples of such results are theorems of Sperner and Alexandroff–Pasynkoff, Fan–Browder type fixed point theorems, Horvath type fixed point theorems, Ky Fan type minimax inequalities, variational inequalities, von Neumann type minimax theorems, Nash type equilibrium theorems, and Himmelberg type fixed point theorems.     

An example showing that A-lower semi-continuity is essential for minimax continuity theorems

Recently the authors have established continuity properties of minimax values and solution sets for a function of two variables depending on a parameter. Some of these properties hold under the assumption that the multifunction, defining the domains of the second variable, is $A$-lower semi-continuous. This property is stronger than lower semi-continuity, but in several important cases these two properties coincide. This note provides an example demonstrating that in general the $A$-lower semi-continuity assumption cannot be relaxed to lower semi-continuity.     

Multiplicity results for quasi-linear problems

Applying the minimax arguments and Morse theory, we establish some results on the existence of multiple nontrivial solutions for a class of p$p$-Laplacian elliptic equations.     

Statistical estimation in a randomly structured branching population

We consider a binary branching process structured by a stochastic trait that evolves according to a diffusion process that triggers the branching events, in the spirit of Kimmel’s model of cell division with parasite infection. Based on the observation of the trait at birth of the first $n$ generations of the process, we construct nonparametric estimator of the transition of the associated bifurcating chain and study the parametric estimation of the branching rate. In the limit $n\to \infty$, we obtain asymptotic efficiency in the parametric case and minimax optimality in the nonparametric case.     

Estimation de la densité de probabilité améliorée par pre-testing

We consider the nonparametric problem of multidimensional probability density estimate. Using concept of minimax risk with random normalizing factor introduced by Lepski [Math. Methods Statist. 8 (1999) 441–486], by considering an independence hypothesis, we build an estimator which can be adaptive and whose accuracy, depending on the observation, is better than the minimax estimate, $n^{?\frac{\beta }{2\beta +d}}$, with prescribed confidence level. To cite this article: A.F. Yode, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Minimax estimators that shift towards a hypersphere for location vectors of spherically symmetric distributions

Let X be a p-dimensional random vector with density f(6X?θ6) where θ is an unknown location vector. For p ≥ 3, conditions on f are given for which there exist minimax estimators θ?(X) satisfying 6Xt6 · 6θ?(X) ? X6 ≤ C, where C is a known constant depending on f. (The positive part estimator is among them.) The loss function is a nondecreasing concave function of 6θ?? θ6². If θ is assumed likely to lie in a ball in $\text{R}$^p, then minimax estimators are given which shrink from the observation X outside the ball in the direction of P(X) the closest point on the surface of the ball. The amount of shrinkage depends on the distance of X from the ball.     

Minimax estimation of location parameters for certain spherically symmetric distributions

Families of minimax estimators are found for the location parameters of a p-variate distribution of the form

, where G(·) is a known c.d.f. on (0, ∞), p ≥ 3 and the loss is sum of squared errors. The estimators are of the form $\text{(1 ? ar(X′X)}\text{E}{}_0\text{(}\text{1}\text{X′X}\text{)X′X)X}$ where 0 ≤ a ≤ 2, r(X′X) is nondecreasing, and $\text{r(X′X)}\text{X′X}$ is nonincreasing. Generalized Bayes minimax estimators are found for certain G(·)'s.     

Existence of three nontrivial solutions for nonlinear Neumann hemivariational inequalities

In this paper we consider a nonlinear Neumann problem driven by the p$p$-Laplacian with a nonsmooth potential (hemivariational inequality). Using minimax methods based on the nonsmooth critical point theory together with suitable truncation techniques, we show that the problem has at least three nontrivial smooth solutions. Two of these solutions have constant sign (one is positive, the other negative).     

On maximal element theorems,variants of Ekeland’s variational principle and their applications

In this paper, we establish several different versions of generalized Ekeland’s variational principle and maximal element theorem for τ$\tau$-functions in ?$?$ complete metric spaces. The equivalence relations between maximal element theorems, generalized Ekeland’s variational principle, generalized Caristi’s (common) fixed point theorems and nonconvex maximal element theorems for maps are also proved. Moreover, we obtain some applications to a nonconvex minimax theorem, nonconvex vectorial equilibrium theorems and convergence theorems in complete metric spaces.     

Minimax inequalities and fixed points of expansive set-valued mappings with noncompact and nonconvex domains and ranges in topological spaces

Some minimax inequalities involving two bifunctions with noncompact and nonconvex domains are first proved in finite continuous topological spaces (in short, FC$FC$-spaces) without convexity structure. As applications some new Fan–Browder type fixed point theorems for expansive set-valued maps with noncompact and nonconvex domains and ranges are obtained in general topological spaces. These results generalize some known results in the recent literature.     

Learning from MOM’s principles: Le Cam’s approach

New robust estimators are introduced, derived from median-of-means principle and Le Cam’s aggregation of tests. Minimax sparse rates of convergence are obtained with exponential probability, under weak moment’s assumptions and possible contamination of the dataset. These derive from general risk bounds of the following informal structure $max\left(\text{minimax rate in the i.i.d. setup},\frac{\text{number of outliers}}{\text{number of observations}}\right).$In this result, the number of outliers may be as large as (number of data)$\times$(minimax rate) without affecting the rates. As an example, minimax rates $slog(ed∕s)∕N$ of recovery of $s$-sparse vectors in ${\mathbb{R}}^d$ holding with exponentially large probability, are deduced for median-of-means versions of the LASSO when the noise has $q_0$ moments for some $q_0>2$, the entries of the design matrix have $C_{0}log\left(ed\right)$ moments and the dataset is corrupted by up to $C_{1}slog(ed∕s)$ outliers.     

The game of “N questions” on a tree

We consider the minimax number of questions required to determine which leaf in a finite binary tree T your opponent has chosen, where each question may ask if the leaf is in a specified subtree of T. The requisite number of questions is shown to be approximately the logarithm (base &0slash;) of the number of leaves in T as T becomes large, where Ø = 1.61803… is the “golden ratio”. Specifically, q questions are sufficient to reduce the number of possibilities by a factor of $\text{2}\text{F}{}_{\mathrm{q+3}}$ (where F, is the ith Fibonacci number), and this is the best possible.     

Inégalités d'oracle pour l'estimation d'une densité de probabilité

We study the problem of the nonparametric estimation of a probability density in $L_2(\mathbb{R})$. Expressing the mean integrated squared error in the Fourier domain, we show that it is close to the mean squared error in the Gaussian sequence model. Then, applying a modified version of Stein's blockwise method, we obtain a linear monotone oracle inequality and a kernel oracle inequality. As a consequence, the proposed estimator is sharp minimax adaptive (i.e. up to a constant) on a scale of Sobolev classes of densities. To cite this article: Ph. Rigollet, C. R. Acad. Sci. Paris, Ser. I 340 (2005).