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  • STA260 Lecture 08

MOM Estimator for Normal Parameters

• Setup:

  • $X_1,\dots ,X_n\stackrel{iid}{\sim }N(\mu ,{\sigma }^2)$.
  • Goal: Find MOM estimates for $\mu ,{\sigma }^2$.

• Step 1: First Moment:

  • Theoretical: ${\mu }_1^{\prime }=\mu$.
  • Sample: $m_1^{\prime }=\overline{X}$.
  • Estimator: ${\hat{\mu }}_{MOM}=\overline{X}$.

• Step 2: Second Moment:

  • Theoretical: ${\mu }_2^{\prime }={\sigma }^2+{\mu }^2$.
  • Sample: $m_2^{\prime }=\overline{(X^2)}$.
  • Equate: ${\hat{\sigma }}^2+{\hat{\mu }}^2=\overline{(X^2)}$.

• Derivation:

  • Substitute $\hat{\mu }=\overline{X}$:
    • ${\hat{\sigma }}^2=\overline{(X^2)}-{\overline{X}}^2$.
  • Rewrite as summation:
    • ${\hat{\sigma }}_{MOM}^2=\frac{1}{n}\sum (X_i-\overline{X}{)}^2$.

• Bias Property:

  • The MOM estimator for variance is biased.
  • $\mathrm{E}[{\hat{\sigma }}^2]=\frac{n-1}{n}{\sigma }^2\ne {\sigma }^2$.
  • It divides by $n$, whereas the unbiased sample variance ($S^2$) divides by $n-1$.

• Footnotes