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

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  • Zhang, Chunming
  • Li, Jialiang
  • Meng, Jingci

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.

Suggested Citation

  • Zhang, Chunming & Li, Jialiang & Meng, Jingci, 2008. "On Stein's lemma, dependent covariates and functional monotonicity in multi-dimensional modeling," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2285-2303, November.
  • Handle: RePEc:eee:jmvana:v:99:y:2008:i:10:p:2285-2303
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    References listed on IDEAS

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    1. Hardle, Wolfgang & LIang, Hua & Gao, Jiti, 2000. "Partially linear models," MPRA Paper 39562, University Library of Munich, Germany, revised 01 Sep 2000.
    2. Langford E. & Schwertman N. & Owens M., 2001. "Is the Property of Being Positively Correlated Transitive?," The American Statistician, American Statistical Association, vol. 55, pages 322-325, November.
    3. Wand, M. P., 1999. "A Central Limit Theorem for Local Polynomial Backfitting Estimators," Journal of Multivariate Analysis, Elsevier, vol. 70(1), pages 57-65, July.
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    Cited by:

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    2. M. Arashi & Mahdi Roozbeh, 2019. "Some improved estimation strategies in high-dimensional semiparametric regression models with application to riboflavin production data," Statistical Papers, Springer, vol. 60(3), pages 667-686, June.

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