How to compute a conditional expectation

Question

I want to compute a conditionnal expectation, i know that

$Z=(Z_1,\ldots,Z_p)'$ where $ Z_j=\Phi ^{-1}(U_j)$ with $Z \sim N(0,R(\theta))$ and $R(\theta)$ the $p \times p$ positive definite correlation matrix, $\Phi ^{-1}(U_j) $ is the standard normal quantile function and $U\sim[0,1]$.

I want to show $E[Z_i|U_j=u_j]=R_{ij}Z_j$. I tried by using the correlation between $ Z_i $ et $Z_j $

$R_{ij}=cor(Z_i,Z_j)=E[Z_iZ_j]-0 $ so

$R_{ij}=E[Z_iZ_j]=E[\Phi ^{-1}(U_i)\Phi ^{-1}(U_j)]=E[E[\Phi ^{-1}(U_i)\Phi ^{-1}(U_j)|U_j=u_j]]=E[\Phi ^{-1}(U_j)E[\Phi ^{-1}(U_i)|U_j=u_j]] =E[Z_jE[Z_i|U_j=u_j]]. $

But i don't know if it's the right way. Any ideas ?

Thanks

Answer 1

The task is to compute $$ \mathbb E\left[Z_i|U_j \right].$$ Observe that $\Phi$ is invertible hence the $\sigma$-algebra generated by $U_j$ is the same as the $\sigma$-algebra generated by $Z_j$. We are thus reduced to compute $$ \mathbb E\left[Z_i|Z_j \right].$$ Then decompose $Z_i$ as $Z_i-\frac{R_{i,j}}{R_{j,j}}Z_j+ \frac{R_{i,j}}{R_{j,j}}Z_j$ to get $$ \mathbb E\left[Z_i|Z_j \right]=\mathbb E\left[Z_i- R_{i,j} Z_j|Z_j \right]+\mathbb E\left[ R_{i,j} Z_j|Z_j \right]. $$ For the first terms, use and show the independence of $Z_i- R_{i,j} Z_j$ and $Z_j$; the second term is $R_{i,j}Z_j$.