0

i know that its always true only if X,Y are Multivariate normal distribution but i cant find any example that verify that. i mean , i am trying to find any example that show two random variable normal distributed X,Y such that $$E[X|Y]\neq \hat{X}^{lin}_{opt}$$ when $$\hat X_{\mathrm{opt}}^{\mathrm{lin}} = m_{x} + \frac{cov(X,Y)}{var(Y)} (Y-m_y)$$ i want to disprove the claim on the title

Or Elimeleh
  • 1
  • Try to look at this post https://stats.stackexchange.com/questions/30588/deriving-the-conditional-distributions-of-a-multivariate-normal-distribution – Jesper for President Jan 29 '20 at 16:03
  • thank you for the answer but what i am searching for is different, i know that the sentence is true only if X,Y are Multivariate normal distribution. but i am trying to find example that shows X~N,Y~N and $$E[X|Y]\neq \hat{X}^{lin}_{opt}$$ – Or Elimeleh Jan 29 '20 at 16:37
  • Do you mean examples where $X,Y$ are marginally Normal but not jointly Normal? – Alecos Papadopoulos Jan 29 '20 at 18:05
  • i don't understand what its mean marginally normal but i guess that its what i mean for. i will try to explain myself again, i know sentence that says • If X, Y are jointly Gaussian, then the linear MMSE estimator and the optimal MMSE estimator coincide. and i want to find example about two Gaussian random variable that aren't jointly Gaussian and the optimal MMSE estimator is not equal to the linear MMSE estimator – Or Elimeleh Jan 29 '20 at 19:15

1 Answers1

1

Based on the clarifications provided in the comments by the OP, it appears that what we are searching here is an example where
a) we have two random variables $X,Y$ that are dependent
b) the marginal distribution of each random variable is Normal
c) their joint distribution is such that the conditional expectation $E(X\mid Y)$ is not equal to the best linear predictor.

The OP should look into Copulas, since this is the method that allows us to take two random variables with arbitrary marginal distributions, and endow them with a joint distribution that is not the multivariate extension of their marginal distributions.

If $X,Y$ are maginally Normally distributed, their distribution functions are $\Phi((x-\mu_x)/\sigma_x)$, $\Phi((y-\mu_y)/\sigma_y)$, and we can use a Copula to assume that their joint distribution function is

$$C_{X,Y} = C\left [\Phi((x-\mu_x)/\sigma_x), \Phi((y-\mu_y)/\sigma_y)\right]$$

If we take $C$ to be the "Gaussian Copula", we would be saying that their joint distribution is bivariate Normal. But if we consider any other Copula, this won't be the case, and it is certain that at least for some of the existing Copulas, the resulting $E(X\mid Y)$ will not equal the best linear predictor.

Alecos Papadopoulos
  • 52,923
  • 5
  • 131
  • 241