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.

informs doi 10.1287/moor.1060.0222

We study the convergence in distribution of M--estimators over a convex kernel. Under convexity, the limit distribution of M--estimators can be obtained under minimal assumptions. We consider the case when the limit is arbitrary, not necessarily normal. If some Taylor expansions hold, the limit distribution is stable. As an application, we examine the limit distribution of M--estimators for the multivariate linear regression model. We obtain the distributional convergence of M--estimators for the multivariate linear regression model for a wide range of sequences of regressors and different types of conditions on the sequence of errors. 1. Introduction. There exists an extensive literature in estimators which are defined as the minimizer of certain stochastic process. For example, a maximum likelihood estimator ` n is a value satisfying n X j=1 g(X j ; ` n ) = inf `2\Theta n X j=1 g(X j ; `); where e \Gammag(x;`) , ` 2 \Theta, is a family of densities. Huber (1964) cons...

We consider set-valued mappings defined on a linear normed space with convex closed images in IR 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 asympto...

Smith (1994) proposes estimation in linear regression models with non-negative errors by maximizing the sum of fitted values subject to the constraint that the fitted values can be no larger than the corresponding response value. In this paper, we consider the limiting distribution of these estimators under very general conditions. Some extensions to local polynomial estimation are also considered. Keywords: epi-convergence, extreme value distribution, linear programming estimator. 1 Introduction Consider the linear regression model Y i = x T i fi +W i (i = 1; \Delta \Delta \Delta ; n) (1) where x i is a vector of covariates (of length p) whose first component is always 1, fi is a vector of parameters and W 1 ; \Delta \Delta \Delta ; W n are i.i.d. random variables. (The assumption that the model (1) has an intercept is not always necessary in the sequel but will be assumed throughout as its inclusion reflects common practice.) Smith (1994) considers the case where the W i 's are no...

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.

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