# Why efficiency matters?

Suppose we are trying to estimate the quantity $\theta$ and we have that the estimator $\hat\theta_n$. Suppose it is efficient, i.e. is variance is the smallest among certain class of other possible estimators of $\theta$, say that this class is a class of unbiased estimators.

Efficient estimators are naturally desired, since they are the "best" in some sense. But what do we lose when we use estimator which is not efficient? Suppose we have two estimators which are asymptotically normal, then we can say that the confidence interval of the efficient estimator is narrower than the one of non-efficient one. But surely there is a better explanation than such hand-waving? Is there some quantification of what is lost?

My question was motivated by this quote by Ch. Sims found here:

Frequentist inference could be approached in the same way: Deﬁne your model, derive fully efﬁcient estimators, pay no attention to anything else

Note that I would not like to ignite another frequentist vs Bayesian war, I just feel that there might be some deep result I am not familiar with.

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As a practical matter, as you surely know, one can equate loss of efficiency with cost of data collection: to achieve a given power in a study where the variance of the estimator scales like $1/n$, a reduction in efficiency by a factor $t \lt 1$ typically requires collecting $1/t$ times as much data. For example, the statistician who can find an estimator that is twice as efficient as one proposed by a client has (caeteris paribus) just halved the cost of the client's data collection.
There are subtleties involving asymptotic efficient estimators. For example, an AEE can be inefficient for all finite values of $n$. But I hope your question isn't bearing on this issue.