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| '''Pareto interpolation''' is a method of [[estimator|estimating]] the [[median]] and other properties of a population that follows a [[Pareto distribution]]. It is used in [[economics]] when analysing the distribution of incomes in a population, when one must base estimates on a relatively small random sample taken from the population.
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| The family of Pareto distributions is parameterized by
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| * a positive number κ that is the smallest value that a [[random variable]] with a Pareto distribution can take. As applied to distribution of incomes, κ is the lowest income of any person in the population; and
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| * a positive number θ the "Pareto index"; as this increases, the tail of the distribution gets thinner. As applied to distribution of incomes, this means that the larger the value of the Pareto index θ the smaller the proportion of incomes many times as big as the smallest incomes.
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| Pareto interpolation can be used when the available information includes the proportion of the sample that falls below each of two specified numbers ''a'' < ''b''. For example, it may be observed that 45% of individuals in the sample have incomes below ''a'' = $35,000 per year, and 55% have incomes below ''b'' = $40,000 per year.
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| Let
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| :''P''<sub>''a''</sub> = proportion of the sample that lies below ''a''; | |
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| :''P''<sub>''b''</sub> = proportion of the sample that lies below ''b''. | |
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| Then the estimates of κ and θ are
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| :<math>
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| \widehat{\kappa} =
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| \left(
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| \frac{P_b - P_a}
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| { \left(1/a^{\widehat{\theta}}\right) - \left(1/b^{\widehat{\theta}}\right)}
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| \right)^{ 1/\widehat{\theta}}
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| </math>
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| and
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| :<math>
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| \widehat{\theta} \; = \;
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| \frac{\log(1-P_a) - \log(1-P_b)}
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| {\log(b) - \log(a)}.
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| </math> | |
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| The estimate of the median would then be
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| :<math>\mbox{estimated median}=\widehat{\kappa}\cdot 2^{1/\widehat{\theta}},\,</math>
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| since the actual population median is
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| :<math>\mbox{median}=\kappa\,2^{1/\theta}.\,</math>
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| == References ==
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| * [http://www.sipp.census.gov/sipp/sourceac/S&A01_20060323_Long(S&A-3).pdf U.S. Census Bureau, Memorandum on statistical techniques used in 2001 income survey (PDF)]. See Equation 10 on p24.
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| * [http://mumford1.dyndns.org/cen2000/CityProfiles/Profiles/MHHINote.htm Stults, Brian J, ''Deriving median household income'']. Gives a derivation of the equations for Pareto interpolation.
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| [[Category:Estimation for specific distributions]]
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| [[Category:Socioeconomics]]
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