I am currently studying "Plane Answers to Complex Questions: The Theory of Linear Models" by Ronald Christensen, and I have encountered Proposition 6.3.2 on page 133. The proposition is stated as follows:
I am having difficulty understanding the notation used in this proof, particularly the expressions involving conditional expectations, such as $E_{yx}$, $ E_x(E_y|x) $ and $E_x$. If someone could help break down and explain these notations, as well as provide insights into how they are applied in the proof, it would greatly assist in my understanding.
Thank you for your assistance!
My attempt to understand the proof:
First, the definition of covariance was used: $Cov[y,f(x)] = E[(y-E(y)(f(x)-E(f(x))]$ , then I assume $E(f(x)) = 0$, because we would get instantly the form in the proof $Cov[y,f(x)] = E[(y-E(y)f(x)]$. Now comes the confusing part, how $E_{yx}(...)$ is interpreted? Afterward $m(x)$ is smuggled in and linearity is used. My guess is, that the tower property is used, but again the notation is confusing me, how it is rightly interpreted?
Here $x$ and $y$ are random variables. The quantity $E_{yx}[g(x,y)]$ is nothing but $E[g(x,y)]$; the role of the subscript "${}_{yx}$" is to indicate that the random variables in $g(x,y)$ are $x$ and $y$. The notation $E_{y\mid x}[g(x,y)]$ denotes the conditional expectation $E[g(x,y)\mid x]$. This conditional expectation is a function of $x$, hence the author uses the notation $E_x[E[g(x,y)\mid x]]$ to denote nothing but $E[E[g(x,y)\mid x]]$. Again, the subscript "${}_x$" just indicates that we take the expectation of a function of $x$. Indeed the tower property is used.