The MLE of a function is the function of the MLE

Question

You have already computed the maximum likelihood estimate $\hat\sigma^2$ of a return variance. One desk now needs the volatility $\sigma$ , and a risk analyst needs the coefficient of variation and the odds-style quantity $\sigma^2/(1+\sigma^2)$ .

Do you have to re-run a full maximum likelihood derivation for each of these, or is there a shortcut?

Accepted Answer

The invariance property. No re-derivation is needed. If \hat	heta is the MLE of 	heta, then for any function g, the MLE of g(	heta) is simply g(\hat	heta). So \widehat{\sigma} = \sqrt{\hat\sigma^2}, \qquad \widehat{\sigma^2/(1+\sigma^2)} = \frac{\hat\sigma^2}{1+\hat\sigma^2}, and so on. Plug the MLE into whatever transformation you want. Why it holds. The likelihood is maximized at a particular parameter point. Reparameterizing the problem (say from \sigma^2 to \sigma) is just relabeling the horizontal axis; it slides the location of the peak's label but not the peak itself. Formally, for…

The MLE of a function is the function of the MLE

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