In his book on adaptive filtering, Sayed mentions a subclass of affine estimators in which not only the predictions y are linearly dependent on the observations x, but x and y are jointly Gaussian. From that, I wonder: in case two random variables are jointly Gaussian, then the optimal estimator E(y|x) (the so-called "least mean square estimator") is necessarily affine? If yes, are there any cases where two random variables are linearly related but are not jointly Gaussian?
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1$\begingroup$ With respect to your last question, there are infinitely many: $y = 2x + e$, where $x \sim U(0,1)$ and $e \sim U(-1,1)$, for example. $\endgroup$– jbowmanCommented Apr 10, 2022 at 14:17
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1$\begingroup$ Your second question is answered with an explicit example at stats.stackexchange.com/questions/257779/…. Your first one is answered by examining any formula you like for the least squares solution. $\endgroup$– whuber ♦Commented Apr 10, 2022 at 15:15
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