Is there a way to derive Maximum Likelihood Estimators using

Is there a way to derive Maximum Likelihood Estimators using the natural parameter space for Exponential Families Justify your answer. Attach any articles and/or a page for links/references that you use to develop your answer.

Solution

For exponential families of distribution uniformly minimum variance bound estimator (UMVBUE) is the MLE.

For exponential families CRLB of any parametric function is attained and the UMVBUE attains that CRLB.

Since exponential families follows the regularity conditions UMVBUE of any parametric function which is obtained by appling Lehman Scheffe thorem is the MLE. Hence to find MLE one has to find any arbitrary unbiased estimator and then have Blackwelise it to obtain the UMVBUE or MLE.