Weibull scale MLE with shape two

Question

Fatigue lifetimes are Weibull with known shape $k = 2$ and unknown scale $\lambda$ , density $f(x \mid \lambda) = \dfrac{k}{\lambda}\left(\dfrac{x}{\lambda}\right)^{k-1} e^{-(x/\lambda)^k}$ for $x > 0$ . A sample of $n = 50$ parts has $\sum_i x_i^{\,2} = 800$ .

Estimate the scale $\lambda$ by maximum likelihood.

Accepted Answer

The Weibull scale MLE with known shape is \hat\lambda = \left(\frac{1}{n}\sumi xi^{\,k}\right)^{1/k}. With k = 2, \hat\lambda = \sqrt{\frac{800}{50}} = \sqrt{16} = 4. Takeaway. For shape two, the scale MLE is the root-mean-square of the observations.

Weibull scale MLE with shape two

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