I wonder whether it is possible to derive influence function (explained for example here: Influence functions and OLS) for an inequality index designed for ordinal (non-continuous) data in this paper: http://eprints.lse.ac.uk/6538/1/Inequality_Measurement_for_Ordered_Response_Health_Data.pdf

The index ($I_{1,1}$ in the paper) is built as follows. The data is ordinal - for example is is self-rated health status measured in $n= 5$ categories (very bad, bad, fair, good, very good). To categories we assign an ordered increasing scale, for example $c=(1,2,3,4,5)$. Let $p_i$ denote proportions of individuals in the class $c_i$. A cumulative distribution is $(P_1, ..., P_n)$ with $P_i=\sum_{k=1}^ip_k$. Category $m$ is the median $(P_{m-1}\leq0.5$ and $P_{m}\geq0.5)$.

The $I_{1,1}$ index is defined as: $$\frac{\sum_{i<m}{P_i}-\sum_{i\geq m}{P_i}+(n+1-m)}{(n-1)/2}$$

It is possible to calculate the influence function for $I_{1,1}$ and if so can you give me some hints? Thank you very much.


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