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What are some strategies for estimating the variance of a statistic that is an arbitrary function of multiple random variables? I am familiar with the Delta method which uses asymptotics to estimate the variance of a statistic when the function takes a single random variable as its input (e.g. $g(X) = log(X)$). But what are methods for when the statistic involves two or three random variables, such as:

$g(X,Y,Z) = log(X+Z)-log(Y+Z)$

Note in this case the random variables are i.i.d binomials whose mean and variance I have estimated from count data.

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  • $\begingroup$ I'm not very familiar with these, but just earlier today I was reading a paper by Preacher & Selig (2012) that discusses this (albeit perhaps indirectly), available here: quantpsy.org/pubs/preacher_selig_2012.pdf They use two simultaneous regression equations to discuss several methods used to estimate the distribution of the product of two coefficients. $\endgroup$ – Patrick Coulombe Feb 14 '15 at 22:26
  • $\begingroup$ In your example, $g$ has no variance. $\endgroup$ – whuber Feb 14 '15 at 22:30
  • $\begingroup$ If you are okay with not having a closed form solution, simulation should be helpful. See my answer here: stats.stackexchange.com/questions/135665/… $\endgroup$ – TrynnaDoStat Feb 14 '15 at 23:51
  • $\begingroup$ Thanks for the link about using simulation. The total variance will be some function of $p_X$, $p_Y$ and $p_Z$. Would you simulate using observed values $\hat{p}$? I'd be interested to see how such a simulation could be used to derive a CI on the final statistic. $\endgroup$ – agartland Feb 17 '15 at 21:52
  • $\begingroup$ Because both logarithms have positive chances of being applied to arguments that are zero, the modified example has infinite variance and undefined mean. $\endgroup$ – whuber Feb 17 '15 at 22:03

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