Estimating a Rayleigh scale from the second moment

Question

The magnitude of a two-dimensional random displacement follows a Rayleigh distribution with unknown scale $\sigma$ , whose second moment is $\mathbb{E}[X^2] = 2\sigma^2$ . From your data the average of the squares is $\frac{1}{n}\sum X_i^2 = 18$ .

Use the method of moments to estimate $\sigma$ by matching the second moment.

Accepted Answer

Why the second moment is convenient. The Rayleigh mean does depend on \sigma, but it carries an awkward factor of \sqrt{\pi/2}. The second moment is far cleaner: it is exactly 2\sigma^2, with no messy constant. Matching it gives a tidy estimate. Match the second moment. \mathbb{E}[X^2] = 2\sigma^2. Set that equal to the sample average of the squares m2 = 18: 2\sigma^2 = 18 \implies \sigma^2 = 9 \implies \hat\sigma = 3. Check. A Rayleigh with scale 3 has second moment 2 \cdot 3^2 = 18, matching the data. A note on moment choice. The method of moments does not force you to use the lowest moment…

Estimating a Rayleigh scale from the second moment

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