delta method with higher order terms to improve variance estimation accuracy I need to apply the delta method principle using a Taylor expansion that retains higher order terms (i.e. to second or third order) in order to improve the accuracy of variance estimation.  The literature on the higher order delta method seems to focus on the special case where the first order term is zero leaving only a second order terms.  In contrast, I would like to retain all terms for the purpose of improving estimation accuracy.  Does anyone know of or has derived a delta method equation that includes higher order terms?
Thanks very much.
 A: It's not even as tedious as all that:
$g(\mu+X-\mu) = g(\mu)+(X-\mu)g'(\mu)+(X-\mu)^2/2 g''(\mu)+(X-\mu)^3/3 g'''(\mu)+...$
\begin{eqnarray}\text{Var}(g(\mu+X-\mu)) &=& \text{Var}((X-\mu)g'(\mu))+\text{Var}(\frac{(X-\mu)^2}{2} g''(\mu))\\& &+2\text{Cov}((X-\mu)g'(\mu),\frac{(X-\mu)^2}{2} g''(\mu))\\& &+\text{Var}(\frac{(X-\mu)^3}{3!} g'''(\mu))\\& &+2\text{Cov}((X-\mu)g'(\mu),\frac{(X-\mu)^3}{3!} g'''(\mu))\\& &+2\text{Cov}(\frac{(X-\mu)^2}{2}g''(\mu),\frac{(X-\mu)^3}{3!} g'''(\mu))+...\end{eqnarray}
\begin{eqnarray}              &=& g'(\mu)^2\text{Var}(X)
                            \\& &+2g'(\mu)\frac{g''(\mu)}{2}\text{E}((X-\mu)^3 )
                            \\& &+\frac{g''(\mu)^2}{4}\text{E}((X-\mu)^4)
                            \\& &+2g'(\mu)\frac{g'''(\mu)}{3!}\text{Cov}((X-\mu),(X-\mu)^3 )
                            \\& &+2\frac{g''(\mu)}{2} \frac{g'''(\mu)}{3!}\text{Cov}((X-\mu)^2,(X-\mu)^3)
                            \\& &+(\frac{g'''(\mu)}{3!})^2\text{Var}((X-\mu)^3 )
+...\end{eqnarray}
\begin{eqnarray}              &=& g'(\mu)^2\,\text{Var}(X)
                            \\& &+2g'(\mu)\frac{g''(\mu)}{2}\,\text{E}((X-\mu)^3 )
                            \\& &+[\frac{g''(\mu)^2}{4}+2g'(\mu)\frac{g'''(\mu)}{3!}]\,\text{E}((X-\mu)^4)
                            \\& &+[2g'(\mu) \frac{g^{(4)}(\mu)}{4!}+2\frac{g''(\mu)}{2} \frac{g'''(\mu)}{3!}]\,\text{E}((X-\mu)^5)
+...\end{eqnarray}
... and so on
Then you just make sure that at whatever point you stop, say to terms in  $(X-\mu)^k$, you have all the terms to that order.
