Rayleigh scale MLE for dart distances

Question

The distance from the bullseye (in centimeters) where a dart lands is modeled as Rayleigh with unknown scale $\sigma$ , density $f(x \mid \sigma) = \dfrac{x}{\sigma^2} e^{-x^2/(2\sigma^2)}$ for $x > 0$ . Four throws land at distances 1, 2, 2, 3.

Estimate $\sigma$ by maximum likelihood.

Accepted Answer

The scale MLE is \hat\sigma^2 = \dfrac{1}{2n}\sumi xi^2. With \sumi xi^2 = 1 + 4 + 4 + 9 = 18 and n = 4, \hat\sigma^2 = \frac{18}{8} = 2.25, \qquad \hat\sigma = 1.5 	ext{ cm}. Takeaway. Square the observations, average, halve, and take the square root: that is the Rayleigh scale estimate.

Rayleigh scale MLE for dart distances

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