this paper, we consider the asymptotic behaviour of regression estimators that minimize the residual sum of squares plus a penalty proportional to

Abstract. Remarkable progress has been made in the development of algorithmic procedures and the availability of software for stochastic programming problems. However, some fundamental questions have remained unexplored. This paper identifies the more challenging open questions in the field of stochastic programming. Some are purely technical in nature, but many also go to the foundations of designing models for decision making under uncertainty.

: Epi-convergence in distribution is a useful tool in establishing limiting distributions of "argmin" estimators; however, it is not always easy to find the epi-limit of a given sequence of objective functions. In this paper, we define the notion of stochastic equi-lower-semicontinuity of a sequence of random objective functions. It is shown that epi-convergence in distribution and finite dimensional convergence in distribution (to a given limit) of a sequence of random objective functions are equivalent under this condition. Key words and phrases: argmin estimators, convergence in distribution, epi-convergence, equi-semicontinuity AMS 1991 subject classifications: Primary 62F12, 60F05; Secondary 62E20, 60F17. Running head: Stochastic equi-semicontinuity 1 Introduction Many statistical estimators are defined as the minimizer (or maximizer) of some objective function; common examples include maximum likelihood estimation and M-estimation. Since any maximization problem can be re-exp...

We propose estimating density functions by means of a constrained optimization problem whose criterion function is the maximum likelihood function, and whose constraints model any (prior) information that might be available. The asymptotic justification for such an approach relies on the theory of epiconvergence. A simple numerical example is used to signal the potential of such an approach.

In this paper, we address low-rank matrix completion problems with fixed basis coefficients, which include the low-rank correlation matrix completion in various fields such as the financial market and the low-rank density matrix completion from the quantum state tomography. For this class of problems, the efficiency of the common nuclear norm penalized estimator for recovery may be challenged. Here, with a reasonable initial estimator, we propose a rank-corrected procedure to generate an estimator of high accuracy and low rank. For this new estimator, we establish a non-asymptotic recovery error bound and analyze the impact of adding the rank-correction term on improving the recoverability. We also provide necessary and sufficient conditions for rank consistency in the sense of Bach [3], in which the concept of constraint nondegeneracy in matrix optimization plays an important role. As a byproduct, our results provide a theoretical foundation for the majorized penalty method of Gao and Sun [25] and Gao [24] for structured low-rank matrix optimization problems.

Abstract not found

We consider set-valued mappings defined on a linear normed space with convex closed images in R^n. Our aim is to construct selections which are (Hadamard) directionally differentiable using some approximation of the multifunction. The constructions suggested assume existence of a cone approximation given by a certain "derivative" of the mapping. The first one makes use of the properties of Steiner points. The notion of Steiner center is generalized for a class of unbounded sets, which include the polyhedral sets. The second construction defines a continuous selection through a given point of the graph of the multifunction and being Hadamard directionally differentiable at that point with derivatives belonging to the corresponding "derivative" of the multifunction. Both constructions lead to a directionally differentiable Castaing representation of measurable multifunctions with the required differentiability properties. The results are applied to obtain statements about the asymptotic behaviour of measurable selections of random sets via the delta-approach. Particularly, random sets of this kind build the solutions of two-stage stochastic programs.

Abstract not found

The Chebyshev or L ∞ estimator minimizes the maximum absolute residual and is useful in situations where the error distribution has bounded support. In this paper, we derive the asymptotic distribution of this estimator in cases where the error distribution has bounded and unbounded support. We also consider the asymptotics of set-membership estimators such as the Chebyshev centre and maximum inscribed ellipsoid estimators.

We consider some asymptotic distribution theory for M-estimators of the parameters of a linear model whose errors are non-negative; these estimators are the solutions of constrained optimization problems and their asymptotic theory is non-standard. Under weak conditions on the distribution of the errors and on the design, we show that a large class of estimators have the same asymptotic distributions in the case of i.i.d. errors; however, this invariance does not hold under non-i.i.d. errors.