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The pictures below are from Statistical Inference by casella berger, page 477-478.

I'm confused on the below example, why does the MLE estimator has a higher asymptotic variance than the sample average for small values of the gamma mean, as shown in the graph figure 10.1.1. The MLE is assumed to by asymptotically efficient, so how could any other estimator triumph in variance? Appreciate any help!

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Just above the graph, it says "Of course, we know the ARE must be greater than 1", which sounds as though they are defining the ARE the other way round from what you (and I) would expect.

However, the errata list says that the 'last display on p477' is wrong, and replaces it with the reciprocal, so in fact the graph is the ARE of the MLE vs the moment estimator, not vice versa, so it does make sense.

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  • $\begingroup$ fantastic, thank you!! $\endgroup$
    – Philip Chen
    Commented Apr 22, 2021 at 12:37

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