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Hello,

I am conducting some simulations where I need to model E[ X | Y ] (the conditional expectation of X given Y) for X and Y two random variables. I already know that the random variable

E[ X ]  + Cov( X, Y ) / Var( Y ) ( Y - E[ Y ]), 

produces the best linear approximation for E[ X | Y ]. However, I could not find yet any method to compute empirically the conditional expectation for some sample (X_i, Y_i), i = 1,...,n.

Does anyone have any suggestion to solve this?

Best regards,

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Maikol Solís
University of Costa Rica
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1) When Y is discrete, the empirical estimator of E(X|Y) is the mean of X at each level of Y, assuming your sample is either iid or Y-stratified.
2) When Y is continuous, you would need to make some assumptions on (i.e. model) the marginal distribution of X, the marginal distribution of Y, the conditional distribution of Y given X et al. Then apply semi-parametric likelihood method to obtain the empirical distribution of X given Y. This will yield an empirical estimator of E(X|Y).        

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Qing Kang
Chief Scientist
Statistical Intelligence Group, LLC
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I quite agree.  Nicely done Qing
Ray
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Raymond Hoffmann
Professor of Biostatistics
Medical College of Wisconsin
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Original Message:
Sent: 11-05-2014 09:34
From: Qing Kang
Subject: Empirical estimator for the conditional expectation


1) When Y is discrete, the empirical estimator of E(X|Y) is the mean of X at each level of Y, assuming your sample is either iid or Y-stratified.
2) When Y is continuous, you would need to make some assumptions on (i.e. model) the marginal distribution of X, the marginal distribution of Y, the conditional distribution of Y given X et al. Then apply semi-parametric likelihood method to obtain the empirical distribution of X given Y. This will yield an empirical estimator of E(X|Y).        

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Qing Kang
Chief Scientist
Statistical Intelligence Group, LLC
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