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Suppose that $X$ and $Y$ have an uknown joint distribution $f_{XY}$.

How can I formally demostrate that it always exists a unique decomposition of the form : $$ Y = E[Y|X] +\epsilon $$

without assuming any explicity form of $f_{XY}$?

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This follows from the linearity of expectation and law of total expectation

$X$ and $Y$ have an unknown joint distribution $F_{XY}$, and the distribution of $Y\mid X$ is some unknown $F_{Y|X}$. Suppose the mean of $Y|X$ is $\mu(X)$.

Consider the random variable $\epsilon = Y - \mu(X)$. Then $\epsilon$ is a mean 0 random variable. To see this, \begin{align*} E(\epsilon)& = E(Y - \mu(X))\\ & = E(E(Y - \mu(X) \mid X))\\ & = E\left[ E(Y\mid X) - \mu(X) \right]\\ & = E(0)\\ & = 0\,. \end{align*}

Thus, we can always write

$$Y = \mu(X) + \epsilon = E(Y\mid X) + \epsilon \,.$$


Uniqueness:

Assume that there exists a $\delta(X)$ and $\eta$, such that

$$Y = \delta(x) + \eta. $$

Uniqueness holds under two constraints.

  1. $\eta$ is independent of $X$
  2. $E(\eta) = 0$

Taking expectation with respect to $F_{Y|X}$, \begin{align*} E(Y\mid X) & = E( \delta(X) \mid X) + E(\eta \mid X)\\ \Rightarrow \mu(X) & = \delta(X) + 0\,. \end{align*}

Thus, we obtain that $\delta(X) = \mu(X)$, which implies $\eta = \epsilon$.

Note that the two constraints are crucial.

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  • $\begingroup$ Thanks for the help. I'm ok with the demostration that if you define $\epsilon$ as $ Y - \mu (X)$ , the mean of this random variable is $0$ but how does it implies that there's an unique decomposition of $Y$ ? . If you define $\epsilon = \lambda (Y - \mu (X) ) $ you can actually show likewise that $E [\epsilon] =0$ . $\endgroup$ – R.Lac Apr 24 '18 at 9:27
  • $\begingroup$ @R.Lac, yes but then you obtain the equation $\lambda Y = \lambda \mu(X) + \epsilon$, which is not what you need. $\endgroup$ – Greenparker Apr 24 '18 at 9:30
  • $\begingroup$ Sorry, maybe i'm overthinking on it but I miss the point. If I obtain the equation as you shown, is possible to write a decomposition of $Y$ of the form $ Y = E[Y|X] + \frac{\epsilon}{\lambda} $ so if you recall $\frac{\epsilon}{\lambda} = \eta $ , $Y$ has infinite many decomposition of that form. $\endgroup$ – R.Lac Apr 24 '18 at 9:37
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    $\begingroup$ @R.Lac I realized this needed more explanation. I have added a discussion on uniqueness of such a decomposition. The key aspect is the constraints imposed on the decomposition. $\endgroup$ – Greenparker Apr 24 '18 at 9:56

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