Using the Delta Method when Data is Missing

The idea of the delta-method is that if you have a "nice" function $f$ and a consistent estimator $B = (B_1, B_2)$ of $\beta$, then:

$$\sqrt{n} \big( f(B) - f(\beta) \big) \dot{\sim} N\big(0, \nabla{f(\beta)^T} \Sigma \nabla{f(\beta)} \big),$$

where $\dot{\sim}$ denotes "asymptotically distributed". This presentation of the delta method assumes you have $n$ observations of both variables in $B$. In real life things are not so pretty... You might have $n_1$ observations of $B_1$, and $n_2$ observations of $B_2$ of which only $m <= n_1,n_2$ observations of $B$ have non-missing data both for $B_1$ and $B_2$

If you wanted to use the delta method you could keep the subset of those $B$s that have all measurements for both variables ($m$ variables only). But that seems such a terrible waste!

Is there a way in which you can include all the observations of your sample in the delta method's $n$?

• The $\Delta$ method is an asymptotic result. You would apply whatever estimation routine is appropriate for missing data to estimate $\beta$ and $\Sigma$ then use whatever analytic result you obtain for $\nabla$f. – AdamO Aug 4 '16 at 21:17