Weibull scale MLE with shape four

Question

Component wear times are Weibull with known shape $k = 4$ 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 $n = 10$ items has $\sum_i x_i^{\,4} = 160$ .

Estimate $\lambda$ by maximum likelihood.

Accepted Answer

Using \hat\lambda = \left(\frac{1}{n}\sumi xi^{\,k}\right)^{1/k} with k = 4: the average fourth power is 160/10 = 16, so \hat\lambda = 16^{1/4} = 2. Takeaway. Raise each observation to the shape, average, and take the matching root to get the Weibull scale.

Weibull scale MLE with shape four

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