• ↑
  • STA260 Lecture 08

MOM Estimator for Gamma Parameters

• Setup:

  • Let $X_1,\dots ,X_n$ be random samples from $\mathrm{\Gamma }(\alpha ,\beta )$.
  • Goal: Find MOM estimates for $\alpha$ (shape) and $\beta$ (scale).
  • Since $k=2$ parameters, we require 2 moments.

• Theoretical Moments:

  • First Moment (${\mu }_1^{\prime }$):
    • ${\mu }_1^{\prime }=\mathrm{E}[X]=\alpha \beta$.
  • Second Moment (${\mu }_2^{\prime }$):
    • Derived via variance relation: $\mathrm{Var}(X)=\mathrm{E}[X^2]-(\mathrm{E}[X]{)}^2$.
    • $\alpha {\beta }^2=\mathrm{E}[X^2]-(\alpha \beta {)}^2$.
    • ${\mu }_2^{\prime }=\mathrm{E}[X^2]=\alpha {\beta }^2+{\alpha }^2{\beta }^2$.

• Sample Moments:

  • $m_1^{\prime }=\overline{X}$.
  • $m_2^{\prime }=\frac{1}{n}\sum _{i=1}^n{X}_i^2=\overline{(X^2)}$.

• Derivation:

  • Step 1: Equate first moments.
    • $\alpha \beta =\overline{X}⟹\hat{\alpha }=\frac{\overline{X}}{\beta }$.
  • Step 2: Equate variance (algebraic shortcut).
    • $\mathrm{Var}(X)\approx \text{Sample Variance (biased)}$.
    • $\alpha {\beta }^2=\overline{(X^2)}-(\overline{X}{)}^2$.
  • Step 3: Substitute $\hat{\alpha }$.
    • $(\frac{\overline{X}}{\beta }){\beta }^2=\overline{(X^2)}-{\overline{X}}^2$.
    • $\overline{X}\beta =\overline{(X^2)}-{\overline{X}}^2$.

• Final Estimators:

  • ${\hat{\beta }}_{MOM}=\frac{\overline{(X^2)}-{\overline{X}}^2}{\overline{X}}$.
  • ${\hat{\alpha }}_{MOM}=\frac{{\overline{X}}^2}{\overline{(X^2)}-{\overline{X}}^2}$.

• Correction Note:

  • A common error is stating $\hat{\beta }=\frac{\overline{(X^2)}-{\overline{X}}^2}{(\overline{X}{)}^2}$; this fraction is actually the reciprocal of $\hat{\alpha }$.

• Footnotes