In what situation can one obtain an estimator that reaches the Cramer-Rao lower bound, i.e. an
efficient estimator?
I know the rules for finding UMVUEs, and I know they are efficient if they reach CRLB. But how can I use the Cramer-Rao result directly to find UMVUEs?
Thanks.
Best Answer
CRLB is proved using the Cauchy-Schwarz inequality. Equality holds in C.S. inequality whenever the two vectors are linearly dependent. The vectors in this case are: $\frac{\partial\log f(x;\theta)}{\partial^T\theta}$ and $\hat{\theta}-\theta$, So $\frac{\partial\log f(x;\theta)}{\partial^T\theta}=C(\theta)(\hat{\theta}-\theta)$. Where $C(\theta)$ is a matrix (if $\theta$ is a scalar then $C(\theta)$ is also scalar. On this case when the equality holds, the bound is reached by $\hat{\theta}$. By differentiating the above equation with respect to $\theta$ and applying the Expectation operator to both sides you will see that $C(\theta)=I(\theta)$, where $I(\theta)$ stands for the Fisher Information Matrix. So, the condition for efficiency is $\frac{\partial\log f(x;\theta)}{\partial^T\theta}=I(\theta)(\phi(x)-\theta)$. The derivative of the log-likelihood function takes the above form iff the efficient estimator is given by $\phi(x)$.