Saddlepoint approximations for likelihood ratio like statistics with applications to permutation tests

We obtain two theorems extending the use of a saddlepoint approximation to multiparameter problems for likelihood ratio-like statistics which allow their use in permutation and rank tests and could be used in bootstrap approximations. In the first, we show that in some cases when no density exists, the integral of the formal saddlepoint density over the set corresponding to large values of the likelihood ratio-like statistic approximates the true probability with relative error of order $1/n$. In the second, we give multivariate generalizations of the Lugannani--Rice and Barndorff-Nielsen or $r^*$ formulas for the approximations. These theorems are applied to obtain permutation tests based on the likelihood ratio-like statistics for the $k$ sample and the multivariate two-sample cases. Numerical examples are given to illustrate the high degree of accuracy, and these statistics are compared to the classical statistics in both cases.Comment: Published in at http://dx.doi.org/10.1214/11-AOS945 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Vertex Ramsey problems in the hypercube

If we 2-color the vertices of a large hypercube what monochromatic substructures are we guaranteed to find? Call a set S of vertices from Q_d, the d-dimensional hypercube, Ramsey if any 2-coloring of the vertices of Q_n, for n sufficiently large, contains a monochromatic copy of S. Ramsey's theorem tells us that for any r \geq 1 every 2-coloring of a sufficiently large r-uniform hypergraph will contain a large monochromatic clique (a complete subhypergraph): hence any set of vertices from Q_d that all have the same weight is Ramsey. A natural question to ask is: which sets S corresponding to unions of cliques of different weights from Q_d are Ramsey? The answer to this question depends on the number of cliques involved. In particular we determine which unions of 2 or 3 cliques are Ramsey and then show, using a probabilistic argument, that any non-trivial union of 39 or more cliques of different weights cannot be Ramsey. A key tool is a lemma which reduces questions concerning monochromatic configurations in the hypercube to questions about monochromatic translates of sets of integers.Comment: 26 pages, 3 figure

Do marital prospects dissuade unmarried fertility?

Unmarried fertility was a lot lower in the 1970s than in the 1990s. It was also the case that unmarried mothers had much lower marriage rates than non-mothers, a differential that has largely vanished over time. Could this marriage-market penalty have been strong enough to explain why unmarried fertility rates were lower then? To explore this issue, we introduce a new model of fertility and marriage, based on directed search. Relative to the existing literature, the essential contributions of the model are to allow for accumulation of children over the lifecycle and for the marriage of single mothers. We use the model, in conjunction with US survey data, to explore the impact of marital prospects on the fertility decisions of unmarried women. We find that the decline, from the 1970s to 1995, in marriage rates of unmarried women with no children, can account for the dramatic rise in unmarried women’s share of births over that period

The Algebra of Grand Unified Theories

The Standard Model of particle physics may seem complicated and arbitrary, but it has hidden patterns that are revealed by the relationship between three "grand unified theories": theories that unify forces and particles by extending the Standard Model symmetry group U(1) x SU(2) x SU(3) to a larger group. These three theories are Georgi and Glashow's SU(5) theory, Georgi's theory based on the group Spin(10), and the Pati-Salam model based on the group SU(2) x SU(2) x SU(4). In this expository account for mathematicians, we explain only the portion of these theories that involves finite-dimensional group representations. This allows us to reduce the prerequisites to a bare minimum while still giving a taste of the profound puzzles that physicists are struggling to solve.Comment: 73 pages, 20 ps figure

Fat Cats and Thin Kittens: Are People Who Make Large Campaign Contributions Different?

Critics of campaign finance in the United States often direct their fire toward contributors who make large donations. Critics charge that large contributions are unfair, unrepresentative, and undemocratic. Accordingly, they push for "reforms" that would favor small contributions over large, and public money over private donations. Survey data on contributors contradict that stereotype of contributors of large amounts and their effects on American politics. Overall, "fat cats" differ less from contributors of smaller amounts than critics have alleged. The differences that do exist are mostly unsurprising and generally small in magnitude. Survey results show that both policy liberalism and Democratic partisanship are well represented among contributors of large sums.The supporters of McCain-Feingold argue that new restrictions on large contributions will profoundly alter American politics for the better. Their claims have no basis in fact. New laws aimed at restricting large donations in favor of smaller ones will have little effect on practical politics