MLE for the scale of Gamma claim sizes

Question

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

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

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

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

Accepted Answer

The log of one density is \ln f = (k-1)\ln x - \dfrac{x}{	heta} - \ln\Gamma(k) - k\ln	heta. Summing over the sample and keeping only the 	heta-dependent pieces, \ell(	heta) = -\,nk\ln	heta - \frac{1}{	heta}\sumi xi + 	ext{const}. Differentiate and set to zero: \ell'(	heta) = -\frac{nk}{	heta} + \frac{1}{	heta^2}\sumi xi = 0 \;\Longrightarrow\; \sumi xi = nk\,	heta \;\Longrightarrow\; \hat	heta = \frac{\sumi xi}{nk} = \frac{\bar x}{k}. The mean of a Gamma is k	heta, so this just says "match the sample mean to k	heta." Here \bar x = 60/12 = 5 and k = 2: \hat	heta = \frac{5}{2} = …

MLE for the scale of Gamma claim sizes

Your answer

Discussion

Learn the concepts

Related questions