I am trying to understand the mathematical properties of supervised learning and semi-supervised learning. Let us consider the case for the mean $\mu$. Then the supervised learning estimator can just be given as the the sample mean $$ \hat{\mu}_{s}=1/n\sum_{i=1}^n Y_i$$ (Here we assume $Y$ is just a standard regression model, say $E[Y|X]=\beta_0+\beta X$.) Now the semi-supervised estimator becomes $$ \hat{\mu}_{ss}=1/N \sum_{j=1}^N (\hat{\beta}_0+\hat{\beta}X_j).$$ Here $N$ is the amount of unlabelled data we have with $N>n$.
After a bit of work, I see that the semisupervised is asymptotically linear (and so of course asymptotically normal). However now I would like to compare the two estimators to see which is more efficient. How do I do this? What are the asymptotic standard errors of both the supervised and semisupervised estimators. Thanks!