This article surveys the development of nonparametric models and methods for estimation of choice models with nonlinear budget sets. The discussion focuses on the budget set regression, that is, the conditional expectation of a choice variable given the budget set. Utility maximization in a nonparametric model with general heterogeneity reduces the curse of dimensionality in this regression. Empirical results using this regression are different from maximum likelihood and give informative inference. The article also considers the information provided by kink probabilities for nonparametric utility with general heterogeneity. Instrumental variable estimation and the evidence it provides of heterogeneity in preferences are also discussed.
more »
« less
On the Convergence of the Backward Reachable Sets of Robust Controlled Invariant Sets For Discretetime Linear Systems
 Award ID(s):
 1918123
 NSFPAR ID:
 10472554
 Publisher / Repository:
 IEEE
 Date Published:
 ISBN:
 9781665467612
 Page Range / eLocation ID:
 4270 to 4275
 Format(s):
 Medium: X
 Location:
 Cancun, Mexico
 Sponsoring Org:
 National Science Foundation
More Like this


A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. π²πΎπΊπ πππ is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by set forcings. The π«πΊπππΎππ π²πππ ππ π ππππ ( π«π²π ) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable surjections. Let π«π²π  πππΎπ  ππ‘ be the statement that in all (set) generic extensions there is a model of π«π²π whose Suslin, coSuslin sets are the universally Baire sets. We outline the proof that over some mild large cardinal theory, π²πΎπΊπ πππ is equiconsistent with π«π²π  πππΎπ  ππ‘ . In fact, we isolate an exact theory (in the hierarchy of strategy mice) that is equiconsistent with both (see Definition 3.1). As a consequence, we obtain that π²πΎπΊπ πππ is weaker than the theory β πΉπ₯π’ + there is a Woodin cardinal which is a limit of Woodin cardinals.β This significantly improves upon the earlier consistency proof of π²πΎπΊπ πππ by Woodin. A variation of π²πΎπΊπ πππ , called π³πππΎπ π²πΎπΊπ πππ , is also shown to be equiconsistent with π²πΎπΊπ πππ over the same large cardinal theory. We also outline the proof that if V has a proper class of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, then π²πΎπΊπ πππ holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to the aforementioned equiconsistency result, where it is shown that π²πΎπΊπ πππ holds in a generic extension of a certain minimal universe. This theorem is more general in that no minimal assumption is needed. A corollary of this is that π«π²π  πππΎπ  ππ‘ is not equivalent to π²πΎπΊπ πππ .more » « less

The shape and orientation of data clouds reflect variability in observations that can confound pattern recognition systems. Subspace methods, utilizing Grassmann manifolds, have been a great aid in dealing with such variability. However, this usefulness begins to falter when the data cloud contains sufficiently many outliers corresponding to stray elements from another class or when the number of data points is larger than the number of features. We illustrate how nested subspace methods, utilizing flag manifolds, can help to deal with such additional confounding factors. Flag manifolds, which are parameter spaces for nested sequences of subspaces, are a natural geometric generalization of Grassmann manifolds. We utilize and extend known algorithms for determining the minimal length geodesic, the initial direction generating the minimal length geodesic, and the distance between any pair of points on a flag manifold. The approach is illustrated in the context of (hyper) spectral imagery showing the impact of ambient dimension, sample dimension, and flag structure.more » « less