Lecture 6 slides: Efficient estimators and Rao-Cramer bound

Let X = (X1, ..., X ˆ n) be a random sample from distribution fθ. Let θ = δ(X) be an estimator of θ. Let T (X) be a su cient statistic for θ. As we have seen already, MSE provides one way to compare the quality of di erent estimators. In particular, estimators with smaller MSE are said to be more e cient. On the other hand, once we know T (X), we can discard X. How do these concepts relate to each other? The theorem below shows that for any estimator θ̂ = δ(X), there is another estimator which depends on data X only through T (X) and is at least as e cient as θ̂: