# Weibull Scale Parameter Meaning and Estimation

Wikipedia: http://en.wikipedia.org/wiki/Weibull_distribution gives a nice description on what the shape parameter (they call it k) means in the Weibull distribution, but I can't find anywhere what the scale parameter (denoted as λ on Wikipedia) means or how to estimate it. Obviously for estimation one could employ a MLE method, but I'm thinking more in the lines of real world examples. For instance if a bug has an average lifespan of a year and we know λ (scale parameter), can I just find k by solving for it in the equation for the mean or would that produce bias? Thanks for the help.

For instance the moment estimator, based on the sample of size $n$: $$\hat{\lambda}_{MM}=\exp{\left(\frac{1}{n}\sum_{i=1}^n\log(X_i)+\gamma\frac{\sqrt{6}}{\pi}\sqrt{\frac{1}{n-1}\sum_{i=1}^n(\log(X_i)-\overline{\log(X)})^2}\right)},$$ where $\gamma$ - is Euler's constant.
The estimator is asymptotically unbiased, with variance: $$\mathrm{Var}(\hat{\lambda}_{MM})=1.2\frac{k^{-2}}{n}+k^{-2}O(n^{-\frac{3}{2}})$$ $\hat{\lambda}_{MM}$ has the asymptotic efficiency of $95$%.
Maximum likelihood estimator of $\lambda$ (when $k$ is unknown) is given by the statistics: $$\hat{\lambda}_{ML}=\left(\frac{1}{n}\sum_{i=1}^n X_i^{\hat{k}_{ML}}\right)^{\frac{1}{\hat{k}_{ML}}}$$ And $\hat{k}_{ML}$ is the solution of the following equation: $$\hat{k}_{ML}=\left(\left(\sum_{i=1}^n X_i^{\hat{k}_{ML}}\log(X_i)\right)\left(\sum_{i=1}^n X_i^{\hat{k}_{ML}}\right)^{-1}\!\!\!\!-\frac{1}{n}\sum_{i=1}^n\log(X_i)\right)^{-1}$$ Other estimators are even more peculiar but much harder in understanding ;).