Rayleigh distribution sufficiency

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March 27th 2010, 08:30 AM
redwings6

Rayleigh distribution sufficiency

Let Y1,Y2,...Yn denote a random sample from a Rayleigh distribution with parameter θ. Show ∑ Yi^2 is sufficient for θ.

I know what the Rayleigh distribution is but I am unsure of how to answer it.
March 27th 2010, 08:56 AM
Moo

Hello,

The pdf of the random sample is $f_{\sigma}(y_1,\dots,y_n)=\prod_{i=1}^n f_{\sigma}(y_i)=\frac{\prod_{i=1}^n y_i}{\sigma^{2n}}\cdot\exp\left(-\frac{\sum_{i=1}^n y_i^2}{2\sigma^2}\right)$

Then use this : Sufficient statistic - Wikipedia, the free encyclopedia

with $h(y_1,\dots,y_n)=\prod_{i=1}^n y_i$ and $g_{\sigma}(T(\bold{y}))=\frac{1}{\sigma^{2n}}\cdot \exp\left(-\frac{T(\bold{y})}{2\sigma^2}\right)$

Hence $T(\bold{y})=\sum_{i=1}^n y_i^2$ is a sufficient statistic for $\sigma$ .

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