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Math Help - Rayleigh distribution sufficiency

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    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.
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    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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