Asymptotic results in robust quasi-bayesian estimation

Let Θ be a real random variable with a known (a priori) distribution. Let the observations given Θ be iid with a common distributionF₀(·)=F(·−0)F is known to belong to some family of distributions on the real line. We find estimators of Θ which are asymptotically minimax for two types of loss functions.     

Expansions of algebraically closed fields in o-minimal structures

We develop a notion of differentiability over an algebraically closed field K of characteristic zero with respect to a maximal real closed subfield R. We work in the context of an o-minimal expansion ? \cal {R} of the field R and obtain many of the standard results in complex analysis in this setting. In doing so we use the topological approach to complex analysis developed by Whyburn and others. We then prove a model theoretic theorem that states that the field R is definable in every proper expansion of the field K all of whose atomic relations are definable in ? \cal {R} . One corollary of this result is the classical theorem of Chow on projective analytic sets.     

Homogeneous Customers Renege from Invisible Queues at Random Times under Deteriorating Waiting Conditions

We consider a memoryless first-come first-served queue in which customers' waiting costs are increasing and convex with time. Hence, customers may opt to renege if service has not commenced after waiting for some time. We assume a homogeneous population of customers and we look for their symmetric Nash equilibrium reneging strategy. Besides the model parameters, customers are aware only, if they are in service or not, and they recall for how long they are have been waiting. They are informed of nothing else. We show that under some assumptions on customers' utility function, Nash equilibrium prescribes reneging after random times. We give a closed form expression for the resulting distribution. In particular, its support is an interval (in which it has a density) and it has at most two atoms (at the edges of the interval). Moreover, this equilibrium is unique. Finally, we indicate a case in which Nash equilibrium prescribes a deterministic reneging time.     

Bounds on the error of an approximate invariant subspace for non-self-adjoint matrices

Summary. Suppose one approximates an invariant subspace of an matrix in which in not necessarily self--adjoint. Suppose that one also has an approximation for the corresponding eigenvalues. We consider the question of how good the approximations are. Specifically, we develop bounds on the angle between the approximating subspace and the invariant subspace itself. These bounds are functions of the following three terms: (1) the residual of the approximations; (2) singular--value separation in an associated matrix; and (3) the goodness of the approximations to the eigenvalues. Received December 1, 1992 / Revised version received October 20, 1993     

Groups, measures, and the NIP

We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups in saturated -minimal structures to compact Lie groups. We also prove some other structural results about such , for example the existence of a left invariant finitely additive probability measure on definable subsets of . We finally introduce the new notion of ``compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the -minimal case.

     

Linear O-minimal structures

A linearly ordered structure is called o-minimal if every definable subset ofM is a finite union of points and intervals. Such an is aCF structure if, roughly said, every definable family of curves is locally a one-parameter family. We prove that if is aCF structure which expands an (interval in an) ordered group, then it is elementary equivalent to a reduct of an (interval in an) ordered vector space. Along the way we prove several quantifier-elimination results for expansions and reducts of ordered vector spaces. The research for this article was begun when the authors were at Berkeley during the logic year at the Mathematical Science Research Institute. It was completed at McGill University. The research was supported by grants from NSERC and FCAR. JL would, as always, like to thank Alistair.     

Computing o-minimal topological invariants using differential topology

We work in an o-minimal expansion of a real closed field. Using piecewise smoothness of definable functions we define the topological degree for definable continuous functions. Using this notion of the degree we obtain a new proof for the existence of torsion points in a definably compact group, and also a new proof of an o-minimal analogue of the Brouwer fixed point theorem.

     

Definable Compactness and Definable Subgroups of o-Minimal Groups

The paper introduces the notion of definable compactness andwithin the context of o-minimal structures proves several topologicalproperties of definably compact spaces. In particular a definableset in an o-minimal structure is definably compact (with respectto the subspace topology) if and only if it is closed and bounded.Definable compactness is then applied to the study of groupsand rings in o-minimal structures. The main result proved isthat any infinite definable group in an o-minimal structurethat is not definably compact contains a definable torsion-freesubgroup of dimension 1. With this theorem, a complete characterizationis given of all rings without zero divisors that are definablein o-minimal structures. The paper concludes with several examplesillustrating some limitations on extending the theorem.     

Mild manifolds and a non-standard Riemann existence theorem

Let be an o-minimal expansion of a real closed field R, and K be the algebraic closure of R. In earlier papers we investigated the notions of -definable K-holomorphic maps, K-analytic manifolds and their K-analytic subsets. We call such a K-manifold mild if it eliminates quantifers after endowing it with all it K-analytic subsets. Examples are compact complex manifolds and non-singular algebraic curves over K. We examine here basic properties of mild manifolds and prove that when a mild manifold M is strongly minimal and not locally modular then it is biholomorphic to a non-singular algebraic curve over K.