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MLE for the scale of Gamma claim sizes

Insurance claim sizes (in thousands of dollars) are modeled as independent Gamma variables with a known shape k=2k = 2 and unknown scale θ\theta, density

f(xθ)=xk1ex/θΓ(k)θk,x>0.f(x \mid \theta) = \frac{x^{k-1} e^{-x/\theta}}{\Gamma(k)\,\theta^{k}}, \qquad x > 0.

A batch of n=12n = 12 claims has total size ixi=60\sum_i x_i = 60 (thousand dollars).

Derive the maximum likelihood estimator of the scale θ\theta and compute it.

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