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.     

Preface: An Appreciation of Professor Jose B. Cruz,Jr.

Preface

Preface: An Appreciation of Professor Jose B. Cruz, Jr.     

Dynamic Sampling Applied to Problems in Optimal Control

Dynamic sampling utilizes the option of varying the sampling rates according to the situation of the systems, thus obtaining procedures with improved efficiencies. In this paper, the technique is applied to a typical problem in optimal control theory, that of tracking and controlling the position of an object. It is shown that the dynamic sampling results in a significantly improved procedure for this case, even when applying a suboptimal policy which can be analyzed in closed form.     

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.

     

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.     

Hidden Markov model likelihoods and their derivatives behave like i.i.d. ones

We consider the log-likelihood function of hidden Markov models, its derivatives and expectations of these (such as different information functions). We give explicit expressions for these functions and bound them as the size of the chain increases. We apply our bounds to obtain partial second order asymptotics and some qualitative properties of a special model as well as to extend some results of [19].     

A note on stable sets,groups, and theories with NIP

Let M be an arbitrary structure. Then we say that an M ‐formula φ (x) defines a stable set in M if every formula φ (x) ∧ α (x, y) is stable. We prove: If G is an M ‐definable group and every definable stable subset of G has U ‐rank at most n (the same n for all sets), then G has a maximal connected stable normal subgroup H such that G /H is purely unstable. The assumptions hold for example if M is interpretable in an o‐minimal structure. More generally, an M ‐definable set X is weakly stable if the M ‐induced structure on X is stable. We observe that, by results of Shelah, every weakly stable set in theories with NIP is stable. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)