Rayleigh scale MLE for wind speeds

Question

Hourly wind speeds (in meters per second) 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 day of $n = 8$ readings has $\sum_i x_i^2 = 64$ .

What is the maximum likelihood estimate of $\sigma$ ?

Accepted Answer

Using \hat\sigma^2 = \dfrac{1}{2n}\sumi xi^2, \hat\sigma^2 = \frac{64}{16} = 4, \qquad \hat\sigma = 2 	ext{ m/s}. Takeaway. The Rayleigh scale estimate needs only the sum of squares and the count.

Rayleigh scale MLE for wind speeds

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