Method of moments for a gamma in shape-rate form

Question

The size of an insurance claim (in thousands) is modeled as a Gamma distribution written in shape-rate form, with density proportional to $x^{\alpha - 1} e^{-\beta x}$ . Here $\alpha$ is the shape and $\beta$ is the rate. From your data the sample mean is $4$ and the sample variance is $2$ .

Use the method of moments to estimate $\alpha$ and $\beta$ .

Accepted Answer

The two model moments. In shape-rate form the Gamma has \mathbb{E}[X] = \frac{\alpha}{\beta}, \qquad \operatorname{Var}(X) = \frac{\alpha}{\beta^2}. Call the sample mean m1 and the sample variance v. A clean way to solve. Divide the variance by the mean and the \alpha cancels: \frac{\operatorname{Var}(X)}{\mathbb{E}[X]} = \frac{\alpha/\beta^2}{\alpha/\beta} = \frac{1}{\beta} \implies \hat\beta = \frac{m1}{v}. Then get \alpha from the mean, \hat\alpha = \hat\beta \, m1 = m1^2 / v. Plug in. With m1 = 4 and v = 2: \hat\beta = \frac{4}{2} = 2, \qquad \hat\alpha = \frac{4^2}{2} = \frac{16}{2} = 8.…

Method of moments for a gamma in shape-rate form

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