Editing Rayleigh distribution (section)

== Parameter estimation ==

Given a sample of ''N'' [[independent and identically distributed]] Rayleigh random variables <math>x_i</math> with parameter <math>\sigma</math>,

: <math>\widehat{\sigma^2} = \!\,\frac{1}{2N}\sum_{i=1}^N x_i^2</math> is the [[maximum likelihood estimation|maximum likelihood]] estimate and also is [[bias of an estimator|unbiased]].

:<math>\widehat{\sigma}\approx \sqrt{\frac 1 {2N} \sum_{i=1}^N x_i^2}</math> is a biased estimator that can be corrected via the formula

:<math>\sigma = \widehat{\sigma} \frac {\Gamma(N)\sqrt{N}} {\Gamma\left(N + \frac 1 2\right)} = \widehat{\sigma} \frac {4^N N!(N-1)!\sqrt{N}} {(2N)!\sqrt{\pi}}</math><ref>[https://archive.org/details/jresv68Dn9p1005 Siddiqui, M. M. (1964) "Statistical inference for Rayleigh distributions", ''The Journal of Research of the National Bureau of Standards, Sec. D: Radio Science'', Vol. 68D, No. 9, p. 1007]</ref> <math>= \frac{\hat{\sigma}}{c_4(2N+1)}</math>, where [[Standard_deviation#Unbiased_sample_standard_deviation|c<sub>4</sub> is the correction factor used to unbias estimates of standard deviation for normal random variables]].

=== Confidence intervals ===
To find the (1&nbsp;&minus;&nbsp;''α'') confidence interval, first find the bounds <math>[a,b]</math> where:
:&nbsp; <math>P\left(\chi_{2N}^2 \leq a\right) = \alpha/2, \quad P\left(\chi_{2N}^2 \leq b\right) = 1 - \alpha/2</math>
then the scale parameter will fall within the bounds
:&nbsp; <math>\frac{{N}\overline{x^2}}{b} \leq {\widehat{\sigma^2}} \leq \frac{{N}\overline{x^2}}{a}</math><ref>[http://nvlpubs.nist.gov/nistpubs/jres/66D/jresv66Dn2p167_A1b.pdf Siddiqui, M. M. (1961) "Some Problems Connected With Rayleigh Distributions", ''The Journal of Research of the National Bureau of Standards; Sec. D: Radio Propagation'', Vol. 66D, No. 2, p. 169]</ref>