Order statistic - Revision history

en>Cherkash at 23:30, 4 September 2014

2014-09-04T23:30:42Z

← Older revision		Revision as of 01:30, 5 September 2014
Line 1:		Line 1:
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en>David Eppstein: /* Order statistics sampled from a uniform distribution */ add sources

2014-02-04T23:46:18Z

Order statistics sampled from a uniform distribution: add sources

← Older revision		Revision as of 01:46, 5 February 2014
Line 1:		Line 1:
	~~'''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~~.		Individual who wrote the is called Eusebio. South Carolina is his birth place. The favorite hobby for him and as well , his kids is when you need to fish and he's been really doing it for quite a while. Filing has been his profession although. Go to his website locate out more: http://prometeu.net<br><br>Here is my site - [http://prometeu.net clash of clans unlimited gems]

	~~The family of Pareto distributions is parameterized by~~

	* 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
	* 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.~~

	~~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~~.

	~~Let~~

	:~~''P''<sub>''a''</sub> = proportion of the sample that lies below ''a'';~~

	:~~''P''<sub>''b''<~~/~~sub> = proportion of the sample that lies below ''b''.~~

	~~Then the estimates of κ and θ are~~

	~~:<math>~~
	~~\widehat{\kappa} =~~
	~~\left(~~
	~~\frac{P_b - P_a}~~
	~~{ \left(1~~/~~a^{\widehat{\theta}}\right) - \left(1/b^{\widehat{\theta}}\right)}~~
	~~\right)^{ 1/\widehat{\theta}}~~
	~~</math>~~

	~~and~~

	~~:<math>~~
	~~\widehat{\theta} \; = \;~~
	~~\frac{\log(1-P_a) - \log(1-P_b)}~~
	~~{\log(b) - \log(a)}~~.
	<~~/math~~>

	~~The estimate of the median would then be~~

	:<~~math~~>~~\mbox{estimated median}=\widehat{\kappa}\cdot 2^{1/\widehat{\theta}},\,</math>~~

	~~since the actual population median~~ is

	~~:<math>\mbox{median}=\kappa\,2^{1/\theta}.\,</math>~~

	~~== References ==~~
	* [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.~~
	* [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.~~

	~~[[Category:Estimation for specific distributions]]~~
	~~[[Category:Socioeconomics]~~]

en>Melchoir: /* Computing order statistics */ link to new stub, Sampling in order

2014-01-06T00:06:08Z

Computing order statistics: link to new stub, Sampling in order

← Older revision		Revision as of 02:06, 6 January 2014
Line 1:		Line 1:
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			The family of Pareto distributions is parameterized by

			* 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
			* 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.

			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.

			Let

			:''P''<sub>''a''</sub> = proportion of the sample that lies below ''a'';

			:''P''<sub>''b''</sub> = proportion of the sample that lies below ''b''.

			Then the estimates of κ and θ are

			:<math>
			\widehat{\kappa} =
			\left(
			\frac{P_b - P_a}
			{ \left(1/a^{\widehat{\theta}}\right) - \left(1/b^{\widehat{\theta}}\right)}
			\right)^{ 1/\widehat{\theta}}
			</math>

			and

			:<math>
			\widehat{\theta} \; = \;
			\frac{\log(1-P_a) - \log(1-P_b)}
			{\log(b) - \log(a)}.
			</math>

			The estimate of the median would then be

			:<math>\mbox{estimated median}=\widehat{\kappa}\cdot 2^{1/\widehat{\theta}},\,</math>

			since the actual population median is

			:<math>\mbox{median}=\kappa\,2^{1/\theta}.\,</math>

			== References ==
			* [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.
			* [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.

			[[Category:Estimation for specific distributions]]
			[[Category:Socioeconomics]]

en>Gareth Jones: /* Probabilistic analysis */ rm unnecessary $tags$

2012-08-10T20:32:52Z

Probabilistic analysis: rm unnecessary <math> tags

New page

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Order statistic - Revision history

en>Cherkash at 23:30, 4 September 2014

en>David Eppstein: /* Order statistics sampled from a uniform distribution */ add sources

en>Melchoir: /* Computing order statistics */ link to new stub, Sampling in order

en>Gareth Jones: /* Probabilistic analysis */ rm unnecessary tags

en>Gareth Jones: /* Probabilistic analysis */ rm unnecessary $tags$