• ↑
  • STA260

STA260 Lecture 07

STA260 Lecture 07 Raw
STA260 Lecture 07 Flashcards

• Review:

  • Concept of the Bias of an Estimator.
  • Comparison of $S_{n-1}^2$ (unbiased) vs $S_n^2$ (biased).

• Consistency:

  • Defined the Consistent Estimator.
  • Used Chebyshev's Inequality to prove consistency.
  • Proof Example: Showing $\hat{p}=\frac{X}{n}\stackrel{P}{\to }p$ for a Bernoulli Distribution.
    • $lim\mathrm{Var}(\hat{p})=lim\frac{p(1-p)}{n}=0$.

• Theorems:

  • Theorem 5: Unbiasedness + Vanishing Variance = Consistency.
  • Theorem 6: Consistency is preserved under linear operations and continuous mappings (Continuous Mapping Theorem).
  • Discussed biased yet consistent estimators (e.g., $\tilde{X}=\frac{1}{n}\sum X_i+\frac{1}{n}$).

• Estimation Methods:

  • Introduced Method of Moments Estimation.
  • Applications:
    • Exponential Distribution.
    • Continuous Uniform Distribution.
    • Gamma Distribution.

• Footnotes