Question: Let $X_1, ..., X_n \sim \text{Exp}(\lambda)$ where $E(X) = \lambda$. Find MLE for $\log(\lambda)$, and show if it is biased or not.
From the description, $f_X(x) = \lambda^{-1}\exp(-\lambda^{-1}x)$
I know that $\bar{X}$ is the MLE for $\lambda$, and since $\log$ is a bijection, and overall a "good" function, invariance of MLE applies. So MLE of $\log(\lambda) = \log$ of MLE of $\lambda = \log(\bar{X})$.
I know $\bar{X}$ is an unbiased estimator of $\lambda$, and I doubt $E(\log(\bar{X})) = \log(\lambda)$ because of Jensen's inequality.
Is there a better way of proving this (preferably not calculating the integral), especially when Jensen's inequality doesn't apply?
In addition, the follow-up question asks me to show that MLE of $\log(\lambda)$ is "CAN" for $\log(\lambda)$, and then identify its asymptotic normal variance.
I suppose "AN" in "CAN" means asymptotic normal. Does any one know what the "C" might stand for? Also, how do I calculate its asymptotic normal variance?
Thanks!
In your particular case the sample mean of an iid sample of RVs with exponencial distribution has a Gamma distribution with parameters $n$ and $\lambda/n$ so all you need to do is to compute the expected value of the log of a Gamma. Wich is a well know result. See this post: https://stats.stackexchange.com/questions/370880/what-is-the-expected-value-of-the-logarithm-of-gamma-distribution/371031
Now for the other question: I am not sure what you mean for "whenñ jensen's inequality doesn't apply". You mean functions that are not concave or convex over all the support?
Finally: CAN means Consistent Asymptotically Normal. You could use the usual asymptotic results for MLE estimators or in my opinion a (simpler) combination of the CLT and the Delta Method. You know that
$$\sqrt{n}\left(\bar{X}-\mathbb{E}X\right)\to \mathcal{N}\left(0,V(X)\right)$$
Your MLE estimator is $\log \bar{X}$ so use the Delta Method and conclude.