Rayleigh scale MLE from a sum of squares

Question

Fading amplitudes are modeled as Rayleigh with unknown scale $\sigma$ , density $f(x \mid \sigma) = \dfrac{x}{\sigma^2} e^{-x^2/(2\sigma^2)}$ for $x > 0$ . A run records $n = 50$ amplitudes with $\sum_i x_i^2 = 2500$ .

Find the maximum likelihood estimate of $\sigma$ .

Accepted Answer

The Rayleigh scale MLE depends on the data only through the sum of squares: \hat\sigma^2 = \frac{1}{2n}\sumi xi^2 = \frac{2500}{100} = 25, \qquad \hat\sigma = 5. Takeaway. The sum of squared amplitudes is the sufficient statistic; halve its average per observation to get \hat\sigma^2.

Rayleigh scale MLE from a sum of squares

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