Gamma scale MLE straight from the sample mean

Question

Service durations are modeled as independent Gamma variables with known shape $k = 2$ and unknown scale $\theta$ , density $f(x \mid \theta) = \dfrac{x^{k-1} e^{-x/\theta}}{\Gamma(k)\,\theta^{k}}$ for $x > 0$ . A large sample has mean $\bar x = 9$ minutes.

What is the maximum likelihood estimate of the scale $\theta$ ?

Accepted Answer

For fixed shape the scale MLE follows from \ell'(	heta) = -\dfrac{nk}{	heta} + \dfrac{1}{	heta^2}\sumi xi = 0, which rearranges to \hat	heta = \frac{\bar x}{k}. With \bar x = 9 and k = 2, \hat	heta = \frac{9}{2} = 4.5 	ext{ minutes}. Takeaway. Once you know the Gamma scale MLE equals \bar x / k, only the sample mean and the shape are needed.

Gamma scale MLE straight from the sample mean

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