Pareto interpolation
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Pareto interpolation is a nonlinear method of interpolation to find the median of a set of data. It is used in economics when analysing income figures. It assumes that the data fits a curve known as the Pareto distribution.
The median is given by
- <math>{\rm median}=\kappa\,2^{1/\theta},<math>
where parameters κ and θ are given by:
- <math>
K = \left( \frac{P_b - P_a} { \frac{1}{a^{\theta}} - \frac{1}{b^{\theta}}} \right) ^{ \frac{1} {\theta}} <math>
and
- <math>
\theta \; = \; \frac{\log(1-P_a) - \log(1-P_b)} {\log(b) - \log(a)} <math>
where
- a = lower limit of the category containing the median
- b = upper limit of the category containing the median
- Pa = proportion of the distribution that lies below the lower limit
- Pb = proportion of the distribution that lies below the upper limitzh:帕累托插值