STA260 Lecture 15

• Review
  • Efficiency of estimators
    • If we have multiple unbiased estimators, we want the one with the smallest variance
      • An estimator $\hat{\theta }$ is more efficient than another estimator $\tilde{\theta }$ if $\mathrm{Var}(\hat{\theta })<\mathrm{Var}(\tilde{\theta })$
    • If we have multiple biased estimators, we want the one with the smallest MSE
      • An estimator $\hat{\theta }$ is more efficient than another estimator $\tilde{\theta }$ if $\mathrm{MSE}(\hat{\theta })<\mathrm{MSE}(\tilde{\theta })$
    • If we have multiple estimators: ${\hat{\theta }}_1,\dots ,{\hat{\theta }}_k$ for $\theta$, and several of those estimators are unbiased: $\mathrm{\exists }i:\mathrm{E}[{\hat{\theta }}_i]=\theta$. Then out of the unbiased estimators, we choose the one(s) with the least variance.
  • Cramér-Rao Lower Bound
• Relative Efficiency
• Cramér-Rao Lower Bound
• Sufficiency