preprint · arXiv (Cornell University) · 2017

Conditional Independence, Conditional Mean Independence, and Zero\n Conditional Covariance

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Abstract

Investigation of the reversibility of the directional hierarchy in the\ninterdependency among the notions of conditional independence, conditional mean\nindependence, and zero conditional covariance, for two random variables X and Y\ngiven a conditioning element Z which is not constrained by any topological\nrestriction on its range, reveals that if the first moments of X, Y, and XY\nexist, then conditional independence implies conditional mean independence and\nconditional mean independence implies zero conditional covariance, but the\ndirection of the hierarchy is not reversible in general. If the conditional\nexpectation of Y given X and Z is "affine in X," which happens when X is\nBernoulli, then the "intercept" and "slope" of the conditional expectation\n(that is, the nonparametric regression function) equal the "intercept" and\n"slope" of the "least-squares linear regression function", as a result of which\nzero conditional covariance implies conditional mean independence.\n

Cite this paper

APA 7

Majumdar, R. (2017). Conditional Independence, Conditional Mean Independence, and Zero\n Conditional Covariance. arXiv (Cornell University). https://doi.org/10.48550/arxiv.1707.04802

MLA 9

Majumdar, Rajeshwari. “Conditional Independence, Conditional Mean Independence, and Zero\n Conditional Covariance.” arXiv (Cornell University), 2017, https://doi.org/10.48550/arxiv.1707.04802.

Chicago

Majumdar, Rajeshwari. 2017. “Conditional Independence, Conditional Mean Independence, and Zero\n Conditional Covariance.” arXiv (Cornell University). https://doi.org/10.48550/arxiv.1707.04802.

Harvard

Majumdar, R. (2017) ‘Conditional Independence, Conditional Mean Independence, and Zero\n Conditional Covariance’, arXiv (Cornell University). Available at: https://doi.org/10.48550/arxiv.1707.04802.

BibTeX

@misc{majumdar2017conditional,
  title = {{Conditional Independence, Conditional Mean Independence, and Zero\n Conditional Covariance}},
  author = {Majumdar, Rajeshwari},
  journal = {arXiv (Cornell University)},
  year = {2017},
  doi = {10.48550/arxiv.1707.04802}
}

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