Gamma rate MLE by invariance

Question

Some texts write the Gamma with a rate \beta = 1/	heta instead of a scale, density f(x \mid \beta) = \dfrac{\beta^{k} x^{k-1} e^{-\beta x}}{\Gamma(k)} for x 0, with known shape k = 2. A large sample of processing times has mean \bar x = 8 seconds. What is the maximum likelihood estimate of the rate \beta?

Accepted Answer

First estimate the scale. Maximizing the log-likelihood in 	heta gives \hat	heta = \bar x / k. Since the rate is a one-to-one function of the scale, \beta = 1/	heta, the invariance property of maximum likelihood says \hat\beta = \frac{1}{\hat	heta} = \frac{k}{\bar x}. With k = 2 and \bar x = 8, \hat\beta = \frac{2}{8} = 0.25 	ext{ per second}. Takeaway. Rate and scale are reciprocals, so the rate MLE is the shape divided by the sample mean, k/\bar x.

Gamma rate MLE by invariance

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