Consistency and Asymptotic Normality¶

Overview¶

An estimator \(\hat{\theta}_n\) is consistent for \(\theta\) if \(\hat{\theta}_n \xrightarrow{P} \theta\) as \(n \to \infty\).

An estimator is consistent if:

\[ \text{Bias}(\hat{\theta}_n) \to 0 \quad \text{and} \quad \text{Var}(\hat{\theta}_n) \to 0 \quad \text{as } n \to \infty \]

An estimator is asymptotically normal if:

\[ \sqrt{n}(\hat{\theta}_n - \theta) \xrightarrow{d} N(0, \sigma^2) \]

for some \(\sigma^2\). This allows construction of approximate confidence intervals and tests for large samples.