# asymptotics of maximum likelihood

(3.4 hours to learn)

## Summary

Under certain regularity conditions, the maximum likelihood estimator is consistent, i.e. it asymptotically approaches the true value. Its sampling distribution (when rescaled appropriately) approaches a normal distribution whose variance is determined by the Fisher information. Because of the Cramer-Rao bound, this is the best we can do. The asymptotic analysis is useful for constructing confidence intervals for parameter estimates.

## Context

This concept has the prerequisites:

## Goals

• Understand basic properties of maximum likelihood estimators:
• they are consistent (they approach the correct value in the limit)
• asymptotically, their sampling distribution (rescaled appropriately) approaches a normal distribution whose variance is the inverse Fisher information
• they are efficient (no unbiased estimator has smaller variance asymptotically)
• Note: for the multivariate version of the asymptotic normality result, you'll want to know about the Fisher information matrix .