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I want to ask if my approach to estimation of the following quantity is correct:

I have $n$ i.i.d. draws $\{(X_i,Z_i) \}_{i=1}^n$ and I want to estimate for a fixed $(i,j)$ pair the quantity: $$ \mathbb{E}[f(X_i,X_j,Z_i,Z_j)|Z_i,Z_j] $$

where $f$ is some function and I have access to the distribution $P_{X|Z}$. My approach is to generate $M$ extra samples of $\{\tilde{X}^{(m)}_i\}_{m=1}^M$ from $P_{X|Z=Z_i}$ and similarly M extra samples of $\{\tilde{X}^{(m)}_j\}_{m=1}^M$ from $P_{X|Z=Z_j}$. I then use these extra samples to build the following estimator:

$$ \mathbb{E}[f(X,X',Z,Z')|Z,Z] \approx \frac{1}{M}\sum_{m=1}^M f(\tilde{X}_j^{(m)}, \tilde{X}_j^{(m)}, Z_i, Z_j). $$

This seems like the intuitive thing to do, but I am wondering if there any issues with it?

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    $\begingroup$ This seems correct! $\endgroup$ – Xi'an Jan 15 at 16:45

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