Ilya Molchanov

Sharp identification regions in models with convex moment predictions

We provide a tractable characterization of the sharp identification region of the parameters θ in a broad class of incomplete econometric models. Models in this class have set valued predictions that yield a convex set of conditional or unconditional moments for the observable model variables. In short, we call these models with convex moment predictions. Examples include static, simultaneous move finite games of complete and incomplete information in the presence of multiple equilibria; best linear predictors with interval outcome and covariate data; and random utility models of multinomial choice in the presence of interval regressors data. Given a candidate value for θ, we establish that the convex set of moments yielded by the model predictions can be represented as the Aumann expectation of a properly defined random set. The sharp identification region of θ, denoted Θ 1 , can then be obtained as the set of minimizers of the distance from a properly specified vector of moments of random variables to this Aumann expectation. Algorithms in convex programming can be exploited to efficiently verify whether a candidate θ is in Θ 1 . We use examples analyzed in the literature to illustrate the gains in identification and computational tractability afforded by our method. This paper is a revised version of CWP27/09.

Limit theorems for unions of random closed sets

Distributions of random closed sets.- Survey on stability of random sets and limit theorems for Minkowski addition.- Infinite divisibility and stability of random sets with respect to unions.- Limit theorems for normalized unions of random closed sets.- Almost sure convergence of unions of random closed sets.- Multivalued regularly varying functions and their applications to limit theorems for unions of random sets.- Probability metrics in the space of random sets distributions.- Applications of limit theorems.

Averaging of Random Sets Based on Their Distance Functions

A new notion of expectation for random sets (or average of binary images) is introduced using the representation of sets by distance functions. The distance function may be the familiar Euclidean distance transform, or some generalisation. The expectation of a random set X is defined as the set whose distance function is closest to the expected distance function of X. This distance average can be applied to obtain the average of non-convex and non-connected random sets. We establish some basic properties, compute examples, and prove limit theorems for the empirical distance average of independent identically distributed random sets.

Multivariate risks and depth-trimmed regions

Abstract We describe a general framework for measuring risks, where the risk measure takes values in an abstract cone. It is shown that this approach naturally includes the classical risk measures and set-valued risk measures and yields a natural definition of vector-valued risk measures. Several main constructions of risk measures are described in this axiomatic framework. It is shown that the concept of depth-trimmed (or central) regions from multivariate statistics is closely related to the definition of risk measures. In particular, the halfspace trimming corresponds to the Value-at-Risk, while the zonoid trimming yields the expected shortfall. In the abstract framework, it is shown how to establish a both-ways correspondence between risk measures and depth-trimmed regions. It is also demonstrated how the lattice structure of the space of risk values influences this relationship.

Statistics of the Boolean model: from the estimation of means to the estimation of distributions

Non-parametric estimators of the distribution of the grain of the Boolean model are considered. The technique is based on the study of point processes of tangent points in different directions related to the Boolean model. Their second- and higher-order characteristics are used to estimate the mean body and the distribution of the typical grain. Central limit theorems for the improved estimator of the intensity and surface measures of the Boolean model are also proved

A Limit Theorem for Solutions of Inequalities

Let H(p) be the set {x E X: h(x) - p} where h is a real-valued lower semicontinuous function on a locally compact separable metric space X. This paper presents a general limit theorem for the sequence of random sets Hn(p) = {x E X: h.(x) 1, where hn, n - 1, are functions that estimate h.

Variational Analysis of Functionals of Poisson Processes

LetF(II) be a functional of a (generally nonhomogeneous) Poisson process II with intensity measureµ. Considering the expectationEµF(II) as a functional ofµ from the cone M of positive finite measures, we derive closed form expressions for its FrA©chet derivatives of all orders that generalize the perturbation analysis formulae for Poisson processes. Variational methods developed for the space M allow us to obtain first and second order sufficient conditions for various types of constrained optimization problems forEµF. We study in detail optimization over the class of measures with a fixed total massa and develop a technique that often allows us to obtain the asymptotic behavior of the optimal intensity measure in the high intensity setting whena grows to infinity. As a particular application we consider the problem of optimal placement of stations in the Poisson model of a two-layer telecommunication network.

Asymptotic properties of estimators for parameters of the Boolean model

This paper considers estimators of parameters of the Boolean model which are obtained by means of the method of intensities. For an estimator of the intensity of the point process of germ points the asymptotic normality is proved and the corresponding variance is given. The theory is based on a study of second-order characteristics of the point process of lower-positive tangent points of the Boolean model. An estimator of the distribution of a typical grain is also discussed. ASYMPTOTIC NORMALITY; INTENSITY; POINT PROCESS; RANDOM SET; PARTICLE SYSI EM AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 62M30 SECONDARY 60D05, 60G55

Set-Valued Means of Random Particles

Planar images of powder particles or sand grains can be interpreted as “figures”, i.e., equivalence classes of directlycongruent compact sets. The paper introduces a concept of set-valuedmeans and real-valued variances for samples of such figures. Inobtaining these results, the images are registered to have similarlocations and orientations. The method is applied to find a mean figure of a sample of polygonal particles.

Multivariate Risk Measures: A Constructive Approach Based on Selections

Since risky positions in multivariate portfolios can be offset by various choices of capital requirements that depend on the exchange rules and related transaction costs, it is natural to assume that the risk measures of random vectors are set-valued. Furthermore, it is reasonable to include the exchange rules in the argument of the risk measure and so consider risk measures of set-valued portfolios. This situation includes the classical Kabanovs transaction costs model, where the set-valued portfolio is given by the sum of a random vector and an exchange cone, but also a number of further cases of additional liquidity constraints. We suggest a definition of the risk measure based on calling a set-valued portfolio acceptable if it possesses a selection with all individually acceptable marginals. The obtained selection risk measure is coherent (or convex), law invariant and has values being upper convex closed sets. We describe the dual representation of the selection risk measure and suggest efficient ways of approximating it from below and from above. In case of Kabanovs exchange cone model, it is shown how the selection risk measure relates to the set-valued risk measures considered by Kulikov (2008), Hamel and Heyde (2010), and Hamel, Heyde and Rudloff (2013).