Base Size Sets and Determining Sets

Bridging the work of Cameron, Harary, and others, we examine the base size set B(G) and determining set D(G) of several families of groups. The base size set is the set of base sizes of all faithful actions of the group G on finite sets. The determining set is the subset of B(G) obtained by restricting the actions of G to automorphism groups of finite graphs. We show that for finite abelian groups, B(G)=D(G)={1,2,...,k} where k is the number of elementary divisors of G. We then characterize B(G) and D(G) for dihedral groups of the form D_{p^k} and D_{2p^k}. Finally, we prove B(G) is not equal to D(G) for dihedral groups of the form D_{pq} where p and q are distinct odd primes.Comment: 10 pages, 1 figur

Monte Carlo Confidence Sets for Identified Sets

In complicated/nonlinear parametric models, it is generally hard to know whether the model parameters are point identified. We provide computationally attractive procedures to construct confidence sets (CSs) for identified sets of full parameters and of subvectors in models defined through a likelihood or a vector of moment equalities or inequalities. These CSs are based on level sets of optimal sample criterion functions (such as likelihood or optimally-weighted or continuously-updated GMM criterions). The level sets are constructed using cutoffs that are computed via Monte Carlo (MC) simulations directly from the quasi-posterior distributions of the criterions. We establish new Bernstein-von Mises (or Bayesian Wilks) type theorems for the quasi-posterior distributions of the quasi-likelihood ratio (QLR) and profile QLR in partially-identified regular models and some non-regular models. These results imply that our MC CSs have exact asymptotic frequentist coverage for identified sets of full parameters and of subvectors in partially-identified regular models, and have valid but potentially conservative coverage in models with reduced-form parameters on the boundary. Our MC CSs for identified sets of subvectors are shown to have exact asymptotic coverage in models with singularities. We also provide results on uniform validity of our CSs over classes of DGPs that include point and partially identified models. We demonstrate good finite-sample coverage properties of our procedures in two simulation experiments. Finally, our procedures are applied to two non-trivial empirical examples: an airline entry game and a model of trade flows

Local set approximation: Mattila-Vuorinen type sets, Reifenberg type sets, and tangent sets

We investigate the interplay between the local and asymptotic geometry of a set $A \subseteq \mathbb{R}^n$ and the geometry of model sets $\mathcal{S} \subset \mathcal{P}(\mathbb{R}^n)$, which approximate $A$ locally uniformly on small scales. The framework for local set approximation developed in this paper unifies and extends ideas of Jones, Mattila and Vuorinen, Reifenberg, and Preiss. We indicate several applications of this framework to variational problems that arise in geometric measure theory and partial differential equations. For instance, we show that the singular part of the support of an $(n-1)$-dimensional asymptotically optimally doubling measure in $\mathbb{R}^n$ ($n\geq 4$) has upper Minkowski dimension at most $n-4$.Comment: 52 pages, 5 figure