Log-normal log-scale variance MLE

Question

Insurance claim sizes are log-normal, so \ln Xi is normal with mean \mu and variance \sigma^2. The natural logs of four claims are yi = 1, 3, 3, 5. What is the maximum likelihood estimate of \sigma^2?

Accepted Answer

Because the log-normal density factors into a parameter-free 1/x term and a normal density in y = \ln x, the MLEs are the normal ones computed on the logs: \hat\mu = \bar y, \qquad \hat\sigma^2 = \frac{1}{n}\sumi (yi - \bar y)^2. Here \bar y = (1 + 3 + 3 + 5)/4 = 3. The squared deviations are 4, 0, 0, 4 summing to 8, so \hat\sigma^2 = \frac{8}{4} = 2. Takeaway. The log-normal variance parameter is the average squared deviation of the logs from their mean.

Log-normal log-scale variance MLE

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