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We propose a notion of central mean dimension reduction subspace for time series {x1} which does not require specification of a model but seeks to find a p x d matrix Φd, d ≤ p, so that the d x 1 vector where ΦdTXt-1 where Xt-1= (xt-1,…Xt-p)T for some p ≥ 1, includes all the information about xt that is available from E(xt∣Xt-1). For known p and d, we estimate the mean central subspace through the Nadaraya-Watson kernel smoother and establish the strong consistency of our estimator. In addition, we propose estimation of d and p using a modified Schwarz Bayesian criterion, if either of d and p is unknown. Finally, we examine the performance of all the estimators extensively through a variety of simulations and provide a new analysis of the well-known Canadian lynx data. Supplemental materials for this article are available online. [ABSTRACT FROM AUTHOR]

In this article we propose a new general index that measures relationships between multiple sets of random vectors. This index is based on Kullback-Leibler (KL) information, which measures linear or nonlinear dependence between multiple sets using joint and marginal densities of affine transformations of the random vectors. Estimates of the matrixes are obtained by maximizing the KL information and are shown to be consistent. The motivation for introducing such an index comes from morphological integration studies, a topic in biological science. As a special case of this index, we define an overall measure of association and two other measures for dimension reduction. The use of these measures is illustrated through real data analysis in morphometric studies and extensive simulations, and their performance is compared with that of approaches based on canonical correlation analysis. Extensions of the aforementioned measures to multiple groups are also discussed. In contrast to canonical correlation analysis, our general index not only provides an overall measure of association, but also determines nonlinear relationships, thereby making it useful in many other applications. [ABSTRACT FROM AUTHOR]

This article develops an estimation method for generalized partially linear single-index models that can be extended to multi-index models that deal with two sets of predictors. We provide theoretical justification for the method. Illustrative examples are also presented. [ABSTRACT FROM AUTHOR]

Sufficient dimension-reduction methods provide effective ways to visualize discriminant analysis problems. For example, Cook and Yin showed that the dimension-reduction method of sliced average variance estimation (SAVE) identifies variates that are equivalent to a quadratic discriminant analysis (QDA) solution. This article makes this connection explicit to motivate the use of SAVE variates in exploratory graphics for discriminant analysis. Classification can then be based on the SAVE variates using a suitable distance measure, If the chosen measure is Mahalanobis distance, then classification is identical to QDA using the original variables. Just as canonical variates provide a useful way to visualize linear discriminant analysis (LDA), so do SAVE variates help visualize QDA. This would appear to be particularly useful given the lack of graphical tools for QDA in current software. Furthermore, whereas LDA and QDA can be sensitive to nonnormality, SAVE is more robust. [ABSTRACT FROM AUTHOR]

In this article we develop the asymptotic distribution for the eigenvalues of a quadratic response sur- face by using the result of Eaton and Tyler to extend the results of Carter, Chinchilli, and Campbell and Bisgaard and Ankenman to the case including nondistinct eigenvalues. In particular, we suggest a method for estimating the confidence regions for multiple roots. [ABSTRACT FROM AUTHOR]

The conditional mean of the response given the predictors is often of interest in regression problems. The central mean subspace, recently introduced by Cook and Li, allows inference about aspects of the mean function in a largely nonparametric context. We propose a marginal fourth moments method for estimating directions in the central mean subspace that might be missed by existing methods such as ordinary least squares (OLS) and principal Hessian directions (pHd). Our method, targeting higher order trends, particularly cubics, complements OLS and pHd because there is no inclusion among them. Theory, estimation and inferences as well as illustrative examples are presented. [ABSTRACT FROM AUTHOR]