$X_1,\ldots,X_n$ are i.i.d standard random variables.
$a_1,\ldots, a_n$ are constants such that $\min_i a_i > 0$ and $\max_i a_i < \infty$
$\hat c$ is given as the solution to the equation:
$$\sum_{i=1}^n \frac{c -X_i^2}{(c+a_i)^2}=0. $$
Can we prove that $$ \operatorname{Variance} (\hat c)=O(n^{-1}) \text{ as } n \to \infty.$$
Note that the above can be easily proved when all $a_i$' s are equal. What will happen in the general case?