Exponential family in testing and estimation In the Annals of Statistics paper "Defining the curvature of a statistical problem(with applications to second order efficiency)" by Bradley Efron, he claims the following two statements in the first paragraph.


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*The locally most powerful test of $\theta=\theta_0$ versus $\theta>\theta_0$ is uniformly most powerful in an exponential family.

*The MLE for $\theta$ is a sufficient statistic in an exponential family and achieves the Cramer-Rao lower bound if we have chosen the right function of $\theta$ to estimate.
Can someone elaborate on these two statements as I am a Mathematician who has only a little knowledge of Statistics?
 A: This is a rather broad question, but I will try to give in informal reply.
Generally, these statements establish the fact that the exponential family is a "well behaved" parametric family of distributions. 
The emphasis in the first is in that the exponential family satisfies the regularity conditions needed so that a uniformly most powerful test exist. This means there is a test statistic and a rejection region, that have a type I error rate no greater than $\alpha$, and no other statistic nor rejection region have more power than these.
Now to the second. The Cramer-Rao bound, or the "information bound" states that any unbiased estimator, cannot be arbitrarily precise, i.e., it's variance is bounded from below. So the novelty in the second statement is that, in those cases where the MLE is actually unbiased, it is also efficient in the sense it has the minimal possible variance. 
A: Here is a partial answer, maybe someone could complement it.
"Statistical sufficency" means that no other statistic uses more information from the sample. Definition of sufficiency. Maximum likelihood estimate $\hat{\theta}$ is a sufficient statistic.
Cramer-Rao lower bound states the lowest value (lower bound) for the variance. It is computed from the second differentiation (if it exists) of Fisher's information . When you use an estimate, you expect it to have the lowest variance, be unbiased, and sufficient. UMVUE estimate are Uniformly, Minimum Variance, Unbiased, estimator. This way, Cramer-Rao lower bound provides the minimum value for the variance of an estimator.
By the way, MLE for $\sigma^2$ is $\hat{\sigma}^2=\dfrac{1}{n}\sum(X-E\{X\})^2$, which is a biased estimate.
