Gamma scale MLE with shape four

Question

Rainfall accumulations are modeled as independent Gamma variables with known shape $k = 4$ and unknown scale $\theta$ , density $f(x \mid \theta) = \dfrac{x^{k-1} e^{-x/\theta}}{\Gamma(k)\,\theta^{k}}$ for $x > 0$ . A record of $n = 10$ storms totals $\sum_i x_i = 80$ millimeters.

Find the maximum likelihood estimate of the scale $\theta$ .

Accepted Answer

Keeping only the 	heta-dependent terms, the log-likelihood is \ell(	heta) = -nk\ln	heta - \frac{1}{	heta}\sumi xi + 	ext{const}. Setting \ell'(	heta) = 0 gives \hat	heta = \frac{\sumi xi}{nk} = \frac{\bar x}{k}. With \bar x = 80/10 = 8 and k = 4, \hat	heta = \frac{8}{4} = 2. Takeaway. The Gamma scale MLE is always the sample mean divided by the known shape.

Gamma scale MLE with shape four

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