Prediction Weighted Maximum Frequency Selection

Shrinkage estimators that possess the ability to produce sparse solutions have become increasingly important to the analysis of today's complex datasets. Examples include the LASSO, the Elastic-Net and their adaptive counterparts. Estimation of penalty parameters still presents difficulties however. While variable selection consistent procedures have been developed, their finite sample performance can often be less than satisfactory. We develop a new strategy for variable selection using the adaptive LASSO and adaptive Elastic-Net estimators with $p_n$ diverging. The basic idea first involves using the trace paths of their LARS solutions to bootstrap estimates of maximum frequency (MF) models conditioned on dimension. Conditioning on dimension effectively mitigates overfitting, however to deal with underfitting, these MFs are then prediction-weighted, and it is shown that not only can consistent model selection be achieved, but that attractive convergence rates can as well, leading to excellent finite sample performance. Detailed numerical studies are carried out on both simulated and real datasets. Extensions to the class of generalized linear models are also detailed.Comment: This manuscript contains 41 pages and 14 figure

Spike and slab variable selection: Frequentist and Bayesian strategies

Variable selection in the linear regression model takes many apparent faces from both frequentist and Bayesian standpoints. In this paper we introduce a variable selection method referred to as a rescaled spike and slab model. We study the importance of prior hierarchical specifications and draw connections to frequentist generalized ridge regression estimation. Specifically, we study the usefulness of continuous bimodal priors to model hypervariance parameters, and the effect scaling has on the posterior mean through its relationship to penalization. Several model selection strategies, some frequentist and some Bayesian in nature, are developed and studied theoretically. We demonstrate the importance of selective shrinkage for effective variable selection in terms of risk misclassification, and show this is achieved using the posterior from a rescaled spike and slab model. We also show how to verify a procedure's ability to reduce model uncertainty in finite samples using a specialized forward selection strategy. Using this tool, we illustrate the effectiveness of rescaled spike and slab models in reducing model uncertainty.Comment: Published at http://dx.doi.org/10.1214/009053604000001147 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Strategy for the Design of Flame Retardants: Cross-linking Processes

Cross-linking is identified as an effective means for flame retardation of polymers and schemes for the cross-linking of poly(ethylene terephthalate) and poly(methyl methacrylate) are presented. For poly(ethylene terephthalate) the scheme involves polymerization of the initially produced vinyl ester. This is followed by chain-stripping, producing a polyene, and cyclization of this polyene. For poly(methyl methacrylate) the scheme entails the formation of anhydride linkages between adjacent polymer strands. Evidence is presented to show the efficacy of these processes and information is produced to aid in the identification of new flame retardants

Groups which do not admit ghosts

A ghost in the stable module category of a group G is a map between representations of G that is invisible to Tate cohomology. We show that the only non-trivial finite p-groups whose stable module categories have no non-trivial ghosts are the cyclic groups of order 2 and 3. We compare this to the situation in the derived category of a commutative ring. We also determine for which groups G the second power of the Jacobson radical of kG is stably isomorphic to a suspension of k.Comment: 9 pages, improved exposition and fixed several typos, to appear in the Proceedings of the AM

Boxicity of Series Parallel Graphs

The three well-known graph classes, planar graphs (P), series-parallel graphs(SP) and outer planar graphs(OP) satisfy the following proper inclusion relation: OP C SP C P. It is known that box(G) <= 3 if G belongs to P and box(G) <= 2 if G belongs to OP. Thus it is interesting to decide whether the maximum possible value of the boxicity of series-parallel graphs is 2 or 3. In this paper we construct a series-parallel graph with boxicity 3, thus resolving this question. Recently Chandran and Sivadasan showed that for any G, box(G) <= treewidth(G)+2. They conjecture that for any k, there exists a k-tree with boxicity k+1. (This would show that their upper bound is tight but for an additive factor of 1, since the treewidth of any k-tree equals k.) The series-parallel graph we construct in this paper is a 2-tree with boxicity 3 and is thus a first step towards proving their conjecture.Comment: 10 pages, 0 figure

Hadwiger Number and the Cartesian Product Of Graphs

The Hadwiger number mr(G) of a graph G is the largest integer n for which the complete graph K_n on n vertices is a minor of G. Hadwiger conjectured that for every graph G, mr(G) >= chi(G), where chi(G) is the chromatic number of G. In this paper, we study the Hadwiger number of the Cartesian product G [] H of graphs. As the main result of this paper, we prove that mr(G_1 [] G_2) >= h\sqrt{l}(1 - o(1)) for any two graphs G_1 and G_2 with mr(G_1) = h and mr(G_2) = l. We show that the above lower bound is asymptotically best possible. This asymptotically settles a question of Z. Miller (1978). As consequences of our main result, we show the following: 1. Let G be a connected graph. Let the (unique) prime factorization of G be given by G_1 [] G_2 [] ... [] G_k. Then G satisfies Hadwiger's conjecture if k >= 2.log(log(chi(G))) + c', where c' is a constant. This improves the 2.log(chi(G))+3 bound of Chandran and Sivadasan. 2. Let G_1 and G_2 be two graphs such that chi(G_1) >= chi(G_2) >= c.log^{1.5}(chi(G_1)), where c is a constant. Then G_1 [] G_2 satisfies Hadwiger's conjecture. 3. Hadwiger's conjecture is true for G^d (Cartesian product of G taken d times) for every graph G and every d >= 2. This settles a question by Chandran and Sivadasan (They had shown that the Hadiwger's conjecture is true for G^d if d >= 3.)Comment: 10 pages, 2 figures, major update: lower and upper bound proofs have been revised. The bounds are now asymptotically tigh

The generating hypothesis for the stable module category of a pp-group

Freyd's generating hypothesis, interpreted in the stable module category of a finite p-group G, is the statement that a map between finite-dimensional kG-modules factors through a projective if the induced map on Tate cohomology is trivial. We show that Freyd's generating hypothesis holds for a non-trivial finite p-group G if and only if G is either C_2 or C_3. We also give various conditions which are equivalent to the generating hypothesis.Comment: 6 pages, fixed minor typos, to appear in J. Algebr

Fence methods for mixed model selection

Many model search strategies involve trading off model fit with model complexity in a penalized goodness of fit measure. Asymptotic properties for these types of procedures in settings like linear regression and ARMA time series have been studied, but these do not naturally extend to nonstandard situations such as mixed effects models, where simple definition of the sample size is not meaningful. This paper introduces a new class of strategies, known as fence methods, for mixed model selection, which includes linear and generalized linear mixed models. The idea involves a procedure to isolate a subgroup of what are known as correct models (of which the optimal model is a member). This is accomplished by constructing a statistical fence, or barrier, to carefully eliminate incorrect models. Once the fence is constructed, the optimal model is selected from among those within the fence according to a criterion which can be made flexible. In addition, we propose two variations of the fence. The first is a stepwise procedure to handle situations of many predictors; the second is an adaptive approach for choosing a tuning constant. We give sufficient conditions for consistency of fence and its variations, a desirable property for a good model selection procedure. The methods are illustrated through simulation studies and real data analysis.Comment: Published in at http://dx.doi.org/10.1214/07-AOS517 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org