Combinatorics of the free Baxter algebra

We study the free (associative, non-commutative) Baxter algebra on one generator. The first explicit description of this object is due to Ebrahimi-Fard and Guo. We provide an alternative description in terms of a certain class of trees, which form a linear basis for this algebra. We use this to treat other related cases, particularly that in which the Baxter map is required to be quasi-idempotent, in a unified manner. Each case corresponds to a different class of trees. Our main focus is on the underlying combinatorics. In several cases, we provide bijections between our various classes of trees and more familiar combinatorial objects including certain Schroeder paths and Motzkin paths. We calculate the dimensions of the homogeneous components of these algebras (with respect to a bidegree related to the number of nodes and the number of angles in the trees) and the corresponding generating series. An important feature is that the combinatorics is captured by the idempotent case; the others are obtained from this case by various binomial transforms. We also relate free Baxter algebras to Loday's dendriform trialgebras and dialgebras. We show that the free dendriform trialgebra (respectively, dialgebra) on one generator embeds in the free Baxter algebra with a quasi-idempotent map (respectively, with a quasi-idempotent map and an idempotent generator). This refines results of Ebrahimi-Fard and Guo.Comment: Fixed errata about grading in the idempotent cas

Optimal Inference in Regression Models with Nearly Integrated Regressors

This paper considers the problem of conducting inference on the regression coefficient in a bivariate regression model with a highly persistent regressor. Gaussian power envelopes are obtained for a class of testing procedures satisfying a conditionality restriction. In addition, the paper proposes feasible testing procedures that attain these Gaussian power envelopes whether or not the innovations of the regression model are normally distributed.

Optimal Inference in Regression Models with Nearly Integrated Regressors

This paper considers the problem of conducting inference on the regression coeffcient in a bivariate regression model with a highly persistent regressor. Gaussian power envelopes are obtained for a class of testing procedures satisfying a conditionality restriction. In addition, the paper proposes feasible testing procedures that attain these Gaussian power envelopes whether or not the innovations of the regression model are normally distributed.

Asymptotic power of sphericity tests for high-dimensional data

This paper studies the asymptotic power of tests of sphericity against perturbations in a single unknown direction as both the dimensionality of the data and the number of observations go to infinity. We establish the convergence, under the null hypothesis and contiguous alternatives, of the log ratio of the joint densities of the sample covariance eigenvalues to a Gaussian process indexed by the norm of the perturbation. When the perturbation norm is larger than the phase transition threshold studied in Baik, Ben Arous and Peche [Ann. Probab. 33 (2005) 1643-1697] the limiting process is degenerate, and discrimination between the null and the alternative is asymptotically certain. When the norm is below the threshold, the limiting process is nondegenerate, and the joint eigenvalue densities under the null and alternative hypotheses are mutually contiguous. Using the asymptotic theory of statistical experiments, we obtain asymptotic power envelopes and derive the asymptotic power for various sphericity tests in the contiguity region. In particular, we show that the asymptotic power of the Tracy-Widom-type tests is trivial (i.e., equals the asymptotic size), whereas that of the eigenvalue-based likelihood ratio test is strictly larger than the size, and close to the power envelope.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1100 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org