Efficient estimator in OLS and statistical power in t-test Could please anybody  help me to understand the difference between those two concepts? I know that statistical power refers to $1-\beta$, the type II error and that efficiency of estimator is related to the amount of variance of the estimator. 
Do they imply the same concept? Are those interchangeable with each other? 
 A: For a test, the power is the probability to say "no" when it is "no". It depends on:


*

*the "distance" between the real parameter $\theta$ and the parameter $\theta_0$ of the null hypothesis : how much the real distribution is different from the tested one. "When it's no" is not a binary concept but a continuous one.

*the number of samples

*the $\alpha$ being chosen


Typically, testing if a coin is unbiased has very small power when the real coin has head probability equal to 0.49999 (and say usual $\alpha$). Expect if you have billions of trials.
The efficiency of an estimator is (can be) measured by the average standard error: $\sqrt{E((\hat\theta-\theta)^2)}$. You could define the efficiency as the inverse of the error. The smaller error, the better estimator. The efficiency also grows with the sample size but unlike a test it is does not depend on the distance to a null hypothesis (there is no null hypothesis here) nor a $\alpha$ convention.
So both functions grow with the sample size, but they are different functions. Crucially one of them depends on a d($\theta_0$,$\theta)$ which does not exist for the other one. In a case such as t-test you will find a $\sqrt{n}$ term somewhere in both functions where $n$ is the sample size but apart from this the functions are just different.
