Log-normal variance from summary sums

Question

Trade sizes are log-normal, so yi = \ln Xi is normal with mean \mu and variance \sigma^2. From n = 25 trades you have the summary sums \sumi yi = 100 and \sumi yi^2 = 420. What is the maximum likelihood estimate of \sigma^2?

Accepted Answer

The log-normal variance MLE is the normal one on the logs, \hat\sigma^2 = \frac{1}{n}\sumi (yi - \bar y)^2. Use the computational identity \sumi (yi - \bar y)^2 = \sumi yi^2 - n\bar y^2. Here \bar y = 100/25 = 4, so \sumi (yi - \bar y)^2 = 420 - 25(4)^2 = 420 - 400 = 20, \qquad \hat\sigma^2 = \frac{20}{25} = 0.8. Takeaway. With the log-sum and log-sum-of-squares you can form \hat\sigma^2 directly via \big(\sum y^2 - n\bar y^2\big)/n.

Log-normal variance from summary sums

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