Rayleigh variance parameter from noise magnitudes

Question

The magnitude of a two-dimensional isotropic Gaussian noise vector is Rayleigh with unknown parameter $\sigma^2$ , density $f(x \mid \sigma) = \dfrac{x}{\sigma^2} e^{-x^2/(2\sigma^2)}$ for $x > 0$ . A sample of $n = 25$ magnitudes has $\sum_i x_i^2 = 200$ .

What is the maximum likelihood estimate of the parameter $\sigma^2$ ?

Accepted Answer

Differentiating the log-likelihood with respect to v = \sigma^2 and solving gives the estimator directly in terms of \sigma^2: \widehat{\sigma^2} = \frac{1}{2n}\sumi xi^2 = \frac{200}{50} = 4. No square root is needed because the question asks for \sigma^2 itself, which is exactly the natural parameter that maximum likelihood delivers. Takeaway. For the Rayleigh, \sigma^2 is the clean MLE target: it equals half the mean squared magnitude.

Rayleigh variance parameter from noise magnitudes

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