False coverage rate: Difference between revisions

Latest revision as of 21:32, 31 July 2014

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-In [[probability theory]], '''Cantelli's inequality''', named after [[Francesco Paolo Cantelli]], is a generalization of [[Chebyshev's inequality]] in the case of a single "tail".<ref>''Research and practice in multiple criteria decision making: proceedings of the XIVth International Conference on Multiple Criteria Decision Making (MCDM), Charlottesville, Virginia, USA, June 8–12, 1998'', edited by Y.Y. Haimes and R.E. Steuer, [[Springer Verlag|Springer]], 2000, ISBN 3540672664.</ref><ref>[http://www.cse.buffalo.edu/~hungngo/classes/2011/Spring-694/lectures/l4.pdf "Tail and Concentration Inequalities" by Hung Q. Ngo]</ref><ref>[http://www.econ.upf.edu/~lugosi/anu.pdf "Concentration-of-measure inequalities" by Gábor Lugois]</ref> The inequality states that
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-: <math>
-\Pr(X-\mu\ge\lambda)\quad\begin{cases}
-\le \frac{\sigma^2}{\sigma^2 + \lambda^2} & \text{if } \lambda > 0, \\[8pt]
-\ge 1 - \frac{\sigma^2}{\sigma^2 + \lambda^2} & \text{if }\lambda < 0.
-\end{cases}
-</math>
-where
-:<math>X</math> is a real-valued [[random variable]],
-:<math>\Pr</math> is the [[probability measure]],
-:<math>\mu</math> is the [[expected value]] of <math>X</math>,
-:<math>\sigma^2</math> is the [[variance]] of <math>X</math>.
-With the same methods one can prove the following variant:
-: <math>
-\Pr(|X-\mu|\ge\lambda)\le \frac{2\sigma^2}{\sigma^2+\lambda^2}.
-</math>
-The inequality is due to [[Francesco Paolo Cantelli]]. The Chebyshev inequality implies that in any [[sample (statistics)|data sample]] or [[probability distribution]], "nearly all" values are close to the [[expected value|mean]] in terms of the [[absolute value]] of the difference between the points of the data sample and the weighted average of the data sample. The Cantelli inequality (sometimes called the "Chebyshev–Cantelli inequality" or the "one-sided Chebyshev inequality") gives a way of estimating how the points of the data sample are bigger than or smaller than their weighted average without the two tails of the absolute value estimate. The Chebyshev inequality has [[Chebyshev's inequality#Higher moments|"higher moments versions"]] and [[Chebyshev's inequality#Vector version|"vector versions"]], and so does the Cantelli inequality.
-==References==
-{{reflist}}
-{{probability-stub}}
-[[Category:Probability theory]]

False coverage rate: Difference between revisions

Latest revision as of 21:32, 31 July 2014

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