Long time studied these concepts. But is not CRLB about the variance of an estimator? I mean a biased estimator can have better precision than a less biased alternate in theory?

Dear Professor Amir, CRLB is about variance of unbiased estimators. For exponential population mean, sample mean is not only unbiases but also attains CRLB. This appears to be contradiction as we teach students that mean for highly skewed distributions (e.g. exponential) is not be used as a central tendency measure. Not sure what I point I am missing in this regards.

Javed you are probably not missing something here you are sharing an issue that you face in everyday "teaching practice". And I am also trying to benefit from this conversation, by picking some analogies from social science "research practice". So how about a large enough sample? what happens to the distribution of the mean (mean which is a function of independently and identically distributed random variables with an exponential distribution)?

unbiasedness and efficiency are small sample properties, no asymptotic involved.

Yes. To summarize, the mean has nice distributional properties for almost all standard distributions, which is why it is preferred as an estimator in comparison to the median. The median is usually preferred when the parent population distribution is unknown. It is better to use the mean as an estimator of the location parameter whenever the probability distribution of the population is known. Also, it is not easy to find the exact distribution of the median in a closed form for most of the distributions. Hope this will make no contradiction with the CRLB.