1.5 The Variance Estimator

Using the notation in Osier (2009)Osier, 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:

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

The linearized variable \(z\) is given by the derivative of the functional:

\[\begin{equation} z_k=lim_{t\rightarrow0}\frac{T(M+t\delta_k)-T(M)}{t}=IT_k(M) \tag{1.2} \end{equation}\]

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.