Why are both maximum-likelihood estimators and maximum a posteriori estimators both asymp- totically efficient?

Both Maximum-Likelihood Estimators (MLE) and Maximum a Posteriori (MAP) estimators are asymptotically efficient because as the sample size increases, both estimators converge to the true parameter value.

Here's a step-by-step explanation:

1. Law of Large Numbers: The law of large numbers states that as the size of a sample increases, the mean value of the sample gets closer and closer to the mean value of the entire population. This is the fundamental principle that makes both MLE and MAP asymptotically efficient.

2. MLE and Large Samples: In the case of MLE, as the sample size increases, the likelihood function becomes sharper and its maximum tends to the true value of the parameter being estimated. This is because MLE maximizes the likelihood that the process described by the model produced the data that were actually observed.

3. MAP and Large Samples: For MAP, as the sample size increases, the posterior distribution (which is what MAP maximizes) tends to concentrate around the true parameter value. This is because the influence of the prior distribution diminishes with increasing sample size, making the MAP estimate more and more similar to the MLE estimate.

4. Asymptotic Efficiency: Asymptotic efficiency refers to the property that as the sample size goes to infinity, the estimator becomes more and more efficient (i.e., its variance goes to zero). Both MLE and MAP have this property because, as explained above, they both converge to the true parameter value as the sample size increases.

5. Cramér-Rao Lower Bound: Both MLE and MAP estimators achieve the Cramér-Rao lower bound asymptotically. This means that, in the limit of large sample sizes, they achieve the smallest possible variance among all unbiased estimators, which is another way of saying they are asymptotically efficient.

In conclusion, both MLE and MAP are asymptotically efficient because they both converge to the true parameter value as the sample size increases, and they both achieve the smallest possible variance (i.e., they are the most efficient) in the limit of large sample sizes.