Let X = $(X_1, \dots, X_n)$ be a sample from Weibull distribution $W(\alpha, \beta)$ with fixed and known $\alpha$. Find MLE of parametric function $g(\beta) = \beta^{\alpha}$. Check if bias is equal to $0$. Show it is consistent and asymptotically normal.
Weibull's density is: $$ f(x, \alpha, \beta) = \alpha \beta^{-\alpha}x^{\alpha-1}e^{-(\frac{x}{\beta})^\alpha} = $$ so $$ l(p) = \alpha^n\beta^{-n\alpha}(x_1 \cdot \dots \cdot x_n)^{\alpha -1}e^{-\frac{1}{\beta^{\alpha}}(x_1 \cdot \dots \cdot x_n)^\alpha} \\ L(p) = n\ln \alpha - n \alpha \ln \beta + (\alpha -1)(\ln x_1 + \dots + \ln x_n)-(\frac{x_1}{\beta})^\alpha - \dots - (\frac{x_n}{\beta})^\alpha \\ (L(p))' = \frac{-n \alpha}{\beta} + \alpha(\frac{x_1}{\beta})^{\alpha-1}\frac{x_1}{\beta^2}+\dots + \alpha(\frac{x_n}{\beta})^{\alpha-1}\frac{x_n}{\beta^2}$$ so $$ \frac{n \alpha}{\beta} = \alpha \frac{\frac{x_1^\alpha}{\beta^{\alpha-1}}+\dots+\frac{x_n^\alpha}{\beta^{\alpha-1}}}{\beta^2} \\ \beta n = \frac{x_1^\alpha + \dots + x_n^\alpha}{\beta^{\alpha-1}} $$ and finally $$ \beta^\alpha = \frac{x_1^\alpha + \dots + x_n^\alpha}{n} $$
so the MLE of g is $\frac{x_1^\alpha + \dots + x_n^\alpha}{n}$.
How can I proceed with calculating bias, consistency or the asymptotical normality?
self-study
tag if this is homework. Search for invariance property of MLE. What is $\theta$ here? $\endgroup$