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Consider $X$ a Gumbel distributed random variable with pdf $$f(x)=\exp(-(x-\mu)-\exp( \mu-x) ).$$ Find the MLE for the parameter $\mu$ based on $n$ i.i.d. observations from a Gumbel population. Show that this estimator is consistent.

My attempt:

I already found that $\hat{\mu}=\ln(n)-\ln\left(\sum_{i=1}^n \exp(-X_i) \right)$. To check for consistency, I'd have to find the distribution of $\hat{\mu}$ and prove $P(|\hat{\mu}-\mu|>\epsilon)\to 0$ for $n\to +\infty$. The expression for $\hat{\mu}$ makes me think that this is probably not the way to tackle this problem.

How can I find the distribution? Should I approach it differently? A question in general: what are some useful methods to explicitly show consistency of MLE?

Thanks.

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  • $\begingroup$ For checking consistency, no need to find exact distribution (unless it is obvious). In some cases, asymptotic distribution or the first two moments for large $n$ may come in handy. $\endgroup$ Jan 18, 2020 at 15:41

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Usually (in the so-called regular settings) the consistency of MLEs is proved by using the strong or the weak law of large numbers, i.e., \begin{align} \hat{\mu}_n &= \ln n - \ln\left( \sum e^{-X_i} ) \right)\\ &=\ln n - \ln\left(n \frac{1}{n}\sum e^{-X_i} ) \right)\\ &=\ln n - \ln n - \ln \left(\frac{1}{n}\sum e^{-X_i} \right)\\ &=- \ln \left(\frac{1}{n}\sum e^{-X_i} ) \right), \end{align} by the WLLN $$ \hat{\mu}_n \xrightarrow{p} - \ln ( \mathbb{E}[e^{-X}]) = -\ln (\Gamma[2]e^{-\mu})=\mu. $$ For the last part see here 2-sided Laplace transform of $\exp(-(t + e^{-t}))$

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