Here is my understanding of these topics:
MLE is better when you know about the type of distribution that generated data (ex: formula for mean of normal distribution is different from mean of binomial distribution which is different of mean of negative binomial distribution ..... and all variance formulas are also different)
Moments/OLS is better when you do not know which distribution generated data (ex: if you collect data irregardless of which distributions it came from, you can take the basic mean .... and this formula will never change: $\mu = \frac{1}{n} \sum_{i=1}^{n} x_i$ . Moments are less sensitive to wrong choice and outliers.
I am confused how to objectively compare advantages of MLE vs Moments/OLS. MLE estimator has minimum variance when distribution class is correct, consistent, normal for large sample size and sometimes unbiased ... but I think same also is for Moment/OLS (ex: BLUE best linear unbiased estimator)
This is very confusing because everything seems similar and contradicts. I am looking for a rule to use to help myself in analysis, ex: When you know distribution use MLE because of x,y,z reasons ....and when you dont know distribution, use OLS/Moments because of a,b,c reasons.
What exactly are the advantages? How do I calculate the risk-payoff? Ex: in some situation where I am not confident about the distribution class.... the possible gain in some performance indicator from using MLE/OLS = a and the possible loss from using MLE/OLS = b ...the possible gain in same performance indicator from using OLS/MLE = a^-1 and the possible loss from using OLS/MLE = b^-1.
For example, performance indicators could be: smaller bias, smaller variance, smaller sample size needed for same performance, etc
Can this be done?
