Estimating the variance of a function of MLE

Question

Say I have the following likelihood :

$$ l(\alpha, \lambda) = n(\log \alpha + \log \lambda) + (\alpha -1 )\sum x_i - \lambda \sum x_i^\alpha $$

which is that of Weibull distribution.

The question is,

Estimate the variance of MLE of $(\mu, \sigma) = (-(\log\lambda) / \alpha, 1/\alpha)$

---

Try

I can estimate the MLE of $(\alpha, \lambda)$, $(\hat{\alpha}, \hat{\lambda})$, by solving

$$ \left(\frac{\partial l}{\partial \alpha}, \frac{\partial l}{\partial \lambda} \right) \overset{set}{=} (0,0) $$

and I can estimate the variance, $\widehat{Var}(\hat{\alpha}, \hat{\lambda}) = I^{-1}(\hat{\alpha}, \hat{\lambda})$, i.e. plug-in estimator.

And by the invariance property of MLE, we have

$$ (\hat{\mu}, \hat{\sigma}) = \left(-(\log\hat{\lambda}) / \hat{\alpha}, 1/\hat{\alpha} \right) $$

But I'm stuck at finding $\widehat{Var}(\hat{\mu}, \hat{\sigma})$.

Answer 1

Since your parameters of interest are non-linear functions of your estimates, I would suggest you apply the delta method. See Hayashi, pp. 93-94.