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STA260 Lecture 08

STA260 Lecture 08 Raw
STA260 Post-Lecture 08
STA260 Lecture 08 Flashcards

• Topics Covered

  • Method of Moments Estimation Procedure
  • MOM Estimator for Gamma Parameters
  • MOM Estimator for Normal Parameters
  • Maximum Likelihood Estimation (MLE) Principle
  • Invariance Property of MLE
  • MLE for Poisson Parameter

• Practice Questions

  • Remember/Understand:
    • Define the $k$-th sample moment $m_k^{\prime }$. How does it differ from ${\mu }_k^{\prime }$?
    • Why do we typically maximize $\ell (\theta )$ rather than $L(\theta )$?
    • What is the Invariance Property of MLE?
  • Apply/Analyze:
    • Uniform Distribution: Find MOM estimator for $\theta$ where $X\sim U[0,\theta ]$.
    • Exponential Distribution: Find MLE for $\lambda$ given $f(x)=\lambda e^{-\lambda x}$.
    • Comparison: Compare ${\hat{\sigma }}_{MOM}^2$ vs $S^2$. Which is unbiased and why?
  • Evaluate/Create:
    • How does the MOM estimator change if the mean $\mu$ is known but variance is unknown?
    • Design a scenario where MOM might produce an impossible estimate (e.g., negative variance).

• Study Plan

  • Immediate:
    • Review Gamma MOM derivation and corrected $\hat{\alpha },\hat{\beta }$.
    • Distinguish Likelihood function (of $\theta$) vs PDF (of $x$).
  • Practice:
    • Complete Uniform Distribution MOM question.
    • Perform full MLE derivation for Exponential Distribution.
    • Verify bias of MOM variance estimator.
  • Deep Dive:
    • Focus on Invariance Property of MLE.
    • Create flashcard comparing ${\hat{\sigma }}_{MOM}^2$ vs $S^2$.

• Footnotes