## 1.5 The Variance Estimator

Using the notation in (Osier 2009Osier, Guillaume. 2009. “Variance Estimation for Complex Indicators of Poverty and Inequality.” Journal of the European Survey Research Association 3 (3): 167–95. http://ojs.ub.uni-konstanz.de/srm/article/view/369.), the variance of the estimator $$T(\hat{M})$$ can approximated by:

$$$Var\left[T(\hat{M})\right]\cong var\left[\sum_s w_i z_i\right] \tag{1.1}$$$

The linearized variable $$z$$ is given by the derivative of the functional:

$$$z_k=lim_{t\rightarrow0}\frac{T(M+t\delta_k)-T(M)}{t}=IT_k(M) \tag{1.2}$$$

where, $$\delta_k$$ is the Dirac measure in $$k$$: $$\delta_k(i)=1$$ if and only if $$i=k$$.

This derivative is called Influence Function and was introduced in the area of Robust Statistics.