Search

Abstract Principal Hessian directions (pHd) based on the Hessian matrix is a moment-based method and a promising methodology for sufficient dimension reduction because of its easy implementation. However, it requires strong conditions on the distribution of the predictors, which must be nearly Gaussian. We investigate here whether and how this method is applicable when the distribution is a mixture multivariate skew elliptical (MMSE) distribution, and if not, how to adapt the technique. Further, we propose two new estimation algorithms for an extended version of pHd. The theoretical results also serve as a reminder for researchers and users to pay attention to the theoretical conditions on which pHd critically relies. Numerical studies are conducted to examine its performance in finite-sample cases. [ABSTRACT FROM AUTHOR]

Abstract: We study the degrees of freedom in shrinkage estimation of regression coefficients. Generalizing the idea of the Lasso, we consider the problem of estimating the coefficients by minimizing the sum of squares with the constraint that the coefficients belong to a closed convex set. Based on a differential geometric approach, we derive an unbiased estimator of the degrees of freedom for this estimation method, under a smoothness assumption on the boundary of the closed convex set. The result presented in this paper is applicable to estimation with a wide class of constraints. As an application, we obtain a type criterion and AIC for selecting tuning parameters. [Copyright &y& Elsevier]

Abstract: Tracking the correct directions of monotonicity in multi-dimensional modeling plays an important role in interpreting functional associations. In the presence of multiple predictors, we provide empirical evidence that the observed monotone directions via parametric, nonparametric or semiparametric fit of commonly used multi-dimensional models may entirely violate the actual directions of monotonicity. This breakdown is caused primarily by the dependence structure of covariates, with negligible influence from the bias of function estimation. To examine the linkage between the dependent covariates and monotone directions, we first generalize Stein’s Lemma for random variables which are mutually independent Gaussian to two important cases: dependent Gaussian, and independent non-Gaussian. We show that in both two cases, there is an explicit one-to-one correspondence between the monotone directions of a multi-dimensional function and the signs of a deterministic surrogate vector. Moreover, we demonstrate that the second case can be extended to accommodate a class of dependent covariates. This generalization further enables us to develop a de-correlation transform for arbitrarily dependent covariates. The transformed covariates preserve modeling interpretability with little loss in modeling efficiency. The simplicity and effectiveness of the proposed method are illustrated via simulation studies and real data application. [Copyright &y& Elsevier]

Abstract: For the family of multivariate normal distribution functions, Stein''s Lemma presents a useful tool for calculating covariances between functions of the component random variables. Motivated by applications to corporate finance, we prove a generalization of Stein''s Lemma to the family of elliptical distributions. [Copyright &y& Elsevier]