Liliana Forzani

Likelihood-Based Sufficient Dimension Reduction

We obtain the maximum likelihood estimator of the central subspace under conditional normality of the predictors given the response. Analytically and in simulations we found that our new estimator can preform much better than sliced inverse regression, sliced average variance estimation and directional regression, and that it seems quite robust to deviations from normality.

Principal Fitted Components for Dimension Reduction in Regression.

We provide a remedy for two concerns that have dogged the use of principal components in regression: (i) principal components are computed from the predictors alone and do not make apparent use of the response, and (ii) principal components are not invariant or equivariant under full rank linear transformation of the predictors. The development begins with principal fitted components (Cook, 2007) and uses normal models for the inverse regression of the predictors on the response to gain reductive information for the forward regression of interest. This approach includes methodology for testing hypotheses about the number of components and about conditional independencies among

On weighted inequalities for singular integrals

In this note we consider singular integrals associated to CalderonZygmund kernels. We prove that if the kernel is supported in (0, oo) then the one-sided Ap condition, A-, is a sufficient condition for the singular integral to be bounded in LP(w), 1 < p < oo, or from Ll(wdx) into weak-Ll(wdx) if p = 1. This one-sided Ap condition becomes also necessary when we require the uniform boundedness of the singular integrals associated to the dilations of a kernel which is not identically zero in (0, oo). The two-sided version of this result is also obtained: Muckenhoupts Ap condition is necessary for the uniform boundedness of the singular integrals associated to the dilations of a general Calder6n-Zygmund kernel which is not the function zero either in (-oo,cO) or in (0, oo). INTRODUCTION It is a classical result in the theory of weighted inequalities the fact that the Ap condition of B. Muckenhoupt on a weight w is equivalent to the LP(wdx) boundedness of the Hilbert transform. This result was proved in 1973 by Hunt, Muckenhoupt and Wheeden [HMW]. In 1974 Coifman and Fefferman [CF] gave a different proof which relies on a good-A inequality, producing an integral estimate of the singular integral in terms of the Hardy-Littlewood maximal operator. Since 1986 the work by E. Sawyer [S], Andersen and Sawyer [AS], Martin Reyes, Ortega Salvador and de la Torre [MOT], [MT] has shown that many positive operators of real analysis have one-sided versions for which the classes of weights are larger than Muckenhoupts ones. Our purpose here is to study the corresponding problems for singular integrals. The situation for one-sided singular integrals is different. The symmetry properties of the Hilbert kernel produce the necessary cancellation properties of a singular integral, so that, no one-sided truncation of l/x is expected to produce a one-sided singular integral. Nevertheless, as we show in Lemma (1.5), the class of general singular integral Calderon-Zygmund kernels supported on a half line is nontrivial. We ask for the more general class of weights w for which such singular integral operators are bounded in LP(wdx). It turns out (Theorem (2.1)) that the one-sided Ap condition is a sufficient condition which becomes also necessary when we require Received by the editors March 15, 1995 and, in revised form, January 30, 1996. 1991 Mathematics Subject Classification. Primary 42B25.

Sufficient dimension reduction for longitudinally measured predictors

We propose a method to combine several predictors (markers) that are measured repeatedly over time into a composite marker score without assuming a model and only requiring a mild condition on the predictor distribution. Assuming that the first and second moments of the predictors can be decomposed into a time and a marker component via a Kronecker product structure that accommodates the longitudinal nature of the predictors, we develop first-moment sufficient dimension reduction techniques to replace the original markers with linear transformations that contain sufficient information for the regression of the predictors on the outcome. These linear combinations can then be combined into a score that has better predictive performance than a score built under a general model that ignores the longitudinal structure of the data. Our methods can be applied to either continuous or categorical outcome measures. In simulations, we focus on binary outcomes and show that our method outperforms existing alternatives by using the AUC, the area under the receiver-operator characteristics (ROC) curve, as a summary measure of the discriminatory ability of a single continuous diagnostic marker for binary disease outcomes.

ESTIMATING SUFFICIENT REDUCTIONS OF THE PREDICTORS IN ABUNDANT HIGH-DIMENSIONAL REGRESSIONS

We study the asymptotic behavior of a class of methods for sufficient dimension reduction in high-dimension regressions, as the sample size and number of predictors grow in various alignments. It is demonstrated that these methods are consistent in a variety of settings, particularly in abundant regressions where most predictors contribute some information on the response, and oracle rates are possible. Simulation results are presented to support the theoretical conclusion.

On geometric characterizations for Monge-Ampère doubling measures

Abstract In this article we prove a theorem on the size of the image of sections of a convex function under its normal mapping when the sections satisfy a geometric property. We apply this result to get new geometric characterizations for Monge–Ampere doubling measures.

A FAMILY OF MODELS EXPLAINING THE LEVEL-SLOPE-CURVATURE EFFECT

One of the most widely used methods to build yield curve models is to use principal components analysis on the correlation matrix of the innovations. R. Litterman and J. Scheinkman found that three factors are enough to explain most of the moves in the case of the US treasury curve. These factors are level, steepness and curvature. Working in the context of commodity futures, G. Cortazar and E. Schwartz found that the spectral structure of the correlation matrices is strikingly similar to those found by R. Litterman and J. Scheinkman. We observe that in both cases the correlation between two different contracts maturing at timestandsis roughly of the formρ|t-s|, for a certain (fixed)0 ≤ ρ ≤ 1. Assuming this correlation structure we prove that the observed factors are perturbations of cosine waves and we extend the analysis to multiple curves.

On the L^p boundedness of the non-centered Gaussian Hardy-Littlewood maximal function

The purpose of this paper is to prove the L p (R n , dγ) boundedness, for p > 1, of the non-centered Hardy-Littlewood maximal operator associated with the Gaussian measure dγ = e -|x|2 dx.

On the mean and variance of the generalized inverse of a singular wishart matrix

We derive the first and the second moments of the MoorePenrose generalized inverse of a singular standard Wishart matrix without relying on a density. Instead, we use the moments of an inverse Wishart distribution and an invariance argument which is related to the literature on tensor functions. We also find the order of the spectral norm of the generalized inverse of a Wishart matrix as its dimension and degrees of freedom diverge. AMS 2000 subject classifications: Primary 62H05, secondary 62E15.

On Riesz transforms and maximal functions in the context of Gaussian Harmonic Analysis

The purpose of this paper is twofold. We introduce a general maximal function on the Gaussian setting which dominates the Ornstein-Uhlenbeck maximal operator and prove its weak type (1,1) by using a covering lemma which is halfway between Besicovitch and Wiener. On the other hand, by taking as a starting point the generalized Cauchy-Riemann equations, we introduce a new class of Gaussian Riesz Transforms. We prove, using the maximal function defined in the first part of the paper, that unlike the ones already studied, these new Riesz Transforms are weak type (1,1) independently of their orders.