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In statistics, the Box–Cox distribution (also known as the power-normal distribution) is the distribution of a random variable X for which the Box–Cox transformation on X follows a truncated normal distribution. It is a continuous probability distribution having probability density function (pdf) given by

f (y) = \frac{1}{(1 - I (f < 0) - sgn (f) Φ (0, m, \sqrt{s})) \sqrt{2 π s^{2}}} \exp {- \frac{1}{2 s^{2}} {(\frac{y^{f}}{f} - m)}^{2}}

for y > 0, where m is the location parameter of the distribution, s is the dispersion, ƒ is the family parameter, I is the indicator function, Φ is the cumulative distribution function of the standard normal distribution, and sgn is the sign function.

Special cases

ƒ = 1 gives a truncated normal distribution.

References

Template:Cite paper

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+In [[statistics]], the '''Box–Cox distribution''' (also known as the '''power-normal distribution''') is the distribution of a [[random variable]] ''X'' for which the [[Box–Cox transformation]] on ''X'' follows a [[truncated normal distribution]]. It is a continuous [[probability distribution]] having [[probability density function]] (pdf) given by
+:<math>
+f(y) = \frac{1}{\left(1-I(f<0)-\sgn(f)\Phi(0,m,\sqrt{s})\right)\sqrt{2 \pi s^2}} \exp\left\{-\frac{1}{2s^2}\left(\frac{y^f}{f} - m\right)^2\right\}
+</math>
+for ''y'' > 0, where ''m'' is the [[location parameter]] of the distribution, ''s'' is the dispersion, ''&fnof;'' is the family parameter, ''I'' is the [[indicator function]], &Phi; is the [[cumulative distribution function]] of the [[standard normal distribution]], and sgn is the [[sign function]].
+==Special cases==
+* ''&fnof;'' = 1 gives a [[truncated normal distribution]].
+==References==
+*{{cite paper
+|title = Properties of the Power-Normal Distribution
+|last = Freeman
+|first = Jade
+|coauthors = Reza Modarres
+|publisher = U.S. Environmental Protection Agency
+|url = http://www.udc.edu/docs/dc_water_resources/technical_reports/report_n_190.pdf
+}}
+{{DEFAULTSORT:Box-Cox distribution}}
+{{ProbDistributions|continuous-semi-infinite}}
+[[Category:Continuous distributions]]
+[[Category:Probability distributions]]
+{{statistics-stub}}

Image functors for sheaves: Difference between revisions

Revision as of 22:46, 22 March 2013

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