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Consider a 3-variate random vector $(\epsilon_0, \epsilon_1, \epsilon_2)$ which is distributed as a Gaussian mixture: (with some abuse of notation) $$ f(\epsilon_0, \epsilon_1, \epsilon_2)=\underbrace{w_a \mathcal{N}\Big(\begin{pmatrix} \mu_a\\ \mu_a\\ \mu_a \end{pmatrix}, \begin{pmatrix} \sigma^2_a & \rho_a & \rho_a\\ \rho_a &\sigma^2_a & \rho_a\\ \rho_a & \rho_a &\sigma^2_a \\ \end{pmatrix}\Big)}_{\text{component (a)}} +\underbrace{w_b\mathcal{N}\Big(\begin{pmatrix} \mu_b\\ \mu_b\\ \mu_b \end{pmatrix}, \begin{pmatrix} \sigma^2_b & \rho_b & \rho_b\\ \rho_b &\sigma^2_b & \rho_b\\ \rho_b & \rho_b &\sigma^2_b \\ \end{pmatrix}\Big)}_{\text{component (b)}} $$ where $f$ denotes the PDF, $(w_a, w_b)$ are the mixture weights, $\mathcal{N}(\mu, \Sigma)$ stays for 3-variate normal distribution with mean $\mu$ and var-cov matrix $\Sigma$.

Is it possible to derive $f(\epsilon_1-\epsilon_0, \epsilon_1-\epsilon_2)$? If yes, could you walk me through the derivation?

I'm confused on how to proceed:

  • Deriving the (marginal) PDF of $(\epsilon_1-\epsilon_0)$ and $(\epsilon_1-\epsilon_2)$ when $\{\epsilon_0, \epsilon_1, \epsilon_2\}$ are mutually independent is explained here for example.

  • Here, however, I don't have mutual independence. Moreover, I want the joint of pairs of differences.

Any hint would be extremely appreciated.

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  • $\begingroup$ Sorry when I try to answer you question it was mixed up. Could you try to recover to your original question? $\endgroup$
    – user158565
    Nov 1, 2018 at 20:55

1 Answer 1

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Let $\left (\begin{matrix} \epsilon_1-\epsilon_0 \\ \epsilon_1-\epsilon_2\end{matrix}\right) = \left (\begin{matrix} 1 & -1 & 0 \\ 1& 0 & -1 \end{matrix}\right)\left (\begin{matrix} \epsilon_0 \\ \epsilon_1\\\epsilon_2\end{matrix}\right) = A\epsilon$

Then $$f(\epsilon_0-\epsilon_1, \epsilon_0-\epsilon_2)=\underbrace{w_a \mathcal{N}\Big(A\begin{pmatrix} \mu_a\\ \mu_a\\ \mu_a \end{pmatrix}, A\begin{pmatrix} \sigma^2_a & \rho_a & \rho_a\\ \rho_a &\sigma^2_a & \rho_a\\ \rho_a & \rho_a &\sigma^2_a \\ \end{pmatrix}A'\Big)}_{\text{component (a)}} +\underbrace{w_b\mathcal{N}\Big(A\begin{pmatrix} \mu_b\\ \mu_b\\ \mu_b \end{pmatrix}, A\begin{pmatrix} \sigma^2_b & \rho_b & \rho_b\\ \rho_b &\sigma^2_b & \rho_b\\ \rho_b & \rho_b &\sigma^2_b \\ \end{pmatrix}A'\Big)}_{\text{component (b)}} $$

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  • $\begingroup$ is $A'$ inside the brackets? $\endgroup$
    – Star
    Nov 1, 2018 at 21:13
  • $\begingroup$ I have another question on the Gaussian mixture in case you may have time stats.stackexchange.com/questions/374834/… $\endgroup$
    – Star
    Nov 1, 2018 at 21:15
  • $\begingroup$ Yes, edited, put A inside. $\endgroup$
    – user158565
    Nov 1, 2018 at 21:15
  • $\begingroup$ I am not sure I understand your mixture correctly. If you think my understanding for this question is correct, I will go to your other similar questions. $\endgroup$
    – user158565
    Nov 1, 2018 at 21:18
  • $\begingroup$ Thanks. 1) I think to have $A$ as you define the vector of epsilons should be ordered $(\epsilon_1, \epsilon_0, \epsilon_2)$. 2) I'm confused about the way you work with mixtures: according to your methodology, for example, $f(\epsilon_1-\epsilon_0)=w_a \mathcal{N}(0, 2(\sigma^2_a-\rho_a))+w_b \mathcal{N}(0, 2(\sigma^2_b-\rho_b))$. $\endgroup$
    – Star
    Nov 2, 2018 at 9:45

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