0
$\begingroup$

Let $\left(y_{i},x_{i}\right):\Omega\rightarrow\mathcal{Y}\times\mathcal{X}$ be random vectors with $\mathcal{Y}\subset\mathbb{R}$ and $\mathcal{X}\subset\mathbb{R}^{p}$. Consider a model $\left\langle \Theta,P\right\rangle$, with $P:\Theta\times\Omega\rightarrow\mathbb{R}_{+}$ a density over $\Omega$ parametrized by $\theta_{0}\in\Theta\subset\mathbb{R}^p$. In other words, $z_i:=\left(y_{i},x_{i}\right)$ follows $P\left(z;\theta_{0}\right)$. In particular assume that $y_i=x_i^\intercal \theta_0+\epsilon$, with $\epsilon$ a zero mean random variable.

Is there any way to characterize the sample space of $\epsilon$ (i.e., the set of outcomes, which the error term maps from, as $\Omega$ constitutes for $y_i,x_i%$)? I assume there is, since not all three variables can be independent.

$\endgroup$
1
  • $\begingroup$ Could you clarify what you mean by "sample space of $\epsilon$"? People seem to use the term "sample space" in various ways, so it's necessary to be more specific concerning what you're asking here. $\endgroup$
    – whuber
    Jan 25 '18 at 14:45
2
$\begingroup$

The sample space of $\epsilon$ is the same as $\Omega$. The reason is simply because $\epsilon_i = f(x_i,y_i)$ hence $\epsilon$ is defined on the same sample space as $(x_i,y_i)$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.