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Let $g:\mathbb{R}^2\rightarrow \mathbb{R}$ be a smooth, but not necessarily bounded function and $X$ and $Y$ two random variables that are not independent. (assuming they yield sufficiently many moments for the following question)

I'm wanting to approximate the mean $\mathbb{E}[g(X,Y)]$ with a multivariate Taylor expansion around the mean values of $X$ and $Y$ (delta method) and would like to have an estimate for the order of the error in terms of the sample size $n$ of an i.i.d. sample of $X,Y$.

Oehlert [1] has given a result for a one-variable function and I am looking for a similar result for my two-variable function $g$.

Does anyone know whether it is possible to establish a result of this type for my problem? If so, under which reference can it be found?

[1] Oehlert (1992), A Note on the Delta Method, The American Statistician, Vol. 46, No. 1, p. 27-29.

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I am not sure to understand the question but concerning the multivariate Delta method see the book "Testing statistical hypotheses" by Lehmann & Romano –  Stéphane Laurent Feb 21 '12 at 19:14
    
Serfling's book, "Approximation Theorems of Mathematical Statistics", contains a number of results concerning the second-order multivariate delta method, but it is rather a slog. Start around chapter 6. –  Steven Pav Apr 21 at 19:14

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