On Stein's lemma, dependent covariates and functional monotonicity in multi-dimensional modeling

Chunming Zhang, Jialiang Li and Jingci Meng

Journal of Multivariate Analysis, 2008, vol. 99, issue 10, 2285-2303

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.

Keywords: primary; 62F30; 62H20 secondary; 62E15; 62H10 Additive model Nonparametric regression Partially monotone function Similarly ordered Stein's Lemma Support vector machine (search for similar items in EconPapers)
Date: 2008
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Citations: View citations in EconPapers (2)

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