Example of $g:\mathbb{R}^2 \to \mathbb{R}$ s.t. $E[g(Z)\mid M=\mu]=0, \forall \mu \in C$ where $Z$ is Gaussian and $C$ is an ellipse.

Question

Let $Z \in \mathbb{R}^2$ be an i.i.d. Gaussian vector with mean $M$ where $P_{Z|M}$ is it's distribution.

Question: I am looking for an non-trivla example of function $g: \mathbb{R}^2 \to \mathbb{R}$ such that \begin{align} E[g(Z)\mid M=\mu]=0, \forall \mu \in C \end{align} where $C=\{\mu: \frac{\mu_1^2}{r_1^2}+\frac{\mu_2^2}{r_2^2}=1 \}$. That is, $C$ is an ellipse.

Some Thoughts: I was able to find an example of $g$ when $r_1=r_2=r$ that is when $C$ is a circle. For example, let \begin{align} g(x)= \|x\|^2 -(2-r^2) \end{align} Then, \begin{align} E[g(Z)|M=\mu]&= E[\|Z\|^2 \mid M=\mu]-(2-r^2)\\ &=\operatorname{Var}[Z \mid M=\mu]+ \|\mu\|^2-(2-r^2)\\ &=2 +r -(2-r)=0 \end{align}

However, I was not able to find an example of such a function if $C$ is an ellipse but not a circle.

You say "Where $P_{Z,\mid,M}$ is$,\ldots$", but then you never use that notation is the rest of what you write. Why introduce a notation that you never use? — Michael Hardy, Jun 19 '21 at 18:28

score 0 · Accepted Answer · answered Jun 03 '21 at 01:32

0

Hint

If $\ f(x)=\frac{x_1^2}{r_1^2}+\frac{x_2^2}{r_2^2}\ $, what is $\ E\big[f(x)\,\big|\,M=\mu\big]\ $ for $\ \mu\in C\ $?

answered Jun 03 '21 at 01:32

lonza leggiera

31,984

Ok. I got it. Thanks. – Boby Jun 03 '21 at 12:33
Do you know if this is a unique solution up to scalar multiplication? – Boby Jun 03 '21 at 13:32
No, I don't. After playing around with the question for a few minutes, I couldn't see any obvious simple way of answering it. – lonza leggiera Jun 04 '21 at 01:51
Should I ask this as a separate question? – Boby Jun 04 '21 at 12:21
Ok. I asked this question here: https://math.stackexchange.com/questions/4163003/is-gx-fracx-12r-12-fracx-22r-22-c-a-unique-solution-of-eg – Boby Jun 04 '21 at 12:31

Example of $g:\mathbb{R}^2 \to \mathbb{R}$ s.t. $E[g(Z)\mid M=\mu]=0, \forall \mu \in C$ where $Z$ is Gaussian and $C$ is an ellipse.

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