1.8 Examples of Linearization Using the Influence Function

  • At-risk-of-poverty threshold: \[ arpt = 0.6\times m \] where \(m\) is the median income.

\[ z_k= -\frac{0.6}{f(m)}\times\frac{1}{N}\times\left[I(y_k\leq m-0.5) \right] \]

  • At-risk-of-poverty rate:

\[ arpr=\frac{\sum_U I(y_i \leq t)}{\sum_U w_i}.100 \] \[ z_k=\frac{1}{N}\left[I(y_k\leq t)-t\right]-\frac{0.6}{N}\times\frac{f(t)}{f(m)}\left[I(y_k\leq m)-0.5\right] \]

where:

\(N\) - population size;

\(t\) - at-risk-of-poverty threshold;

\(y_k\) - income of person \(k\);

\(m\) - median income;

\(f\) - income density function;