Method of moments (MM)

Also called the substitution method

The rth sample moment is

m_{r}^{'} = \frac{\sum x_{i}^{r}}{m}

If we have $k$ parameters in the population distribution, we need $k$ equations to solve for the $k$ unknown parameters

1 parameter, Exponential( $θ$ )

E (X) = θ = \bar{X} ⟶ \hat{θ} = \bar{X}

2 parameters, $E (X) = g_{1} (θ_{1}, θ_{2}), E (X^{2}) = g_{2} (θ_{1}, θ_{2})$

\begin{aligned} E (X) & = \bar{X} = g_{1} (θ_{1}, θ_{2}) & ⟶ & {\hat{θ}}_{1} = h_{1} (\bar{X}, \bar{X^{2}}) \\ E (X^{2}) & = \bar{X^{2}} = g_{2} (θ_{1}, θ_{2}) & ⟶ & θ_{2} = h_{2} (\bar{X}, \bar{X^{2}}) \end{aligned}

We can also use the variance rather than the second moment

S_{0}^{2} = \frac{1}{n} \sum_{i = 1}^{n} (X_{i} - \bar{X})^{2}

For a $Γ (α, β)$

\begin{aligned} E (X) & = α β, & v a r (X) & = α β^{2} \\ \hat{β} & = \frac{S_{0}^{2}}{\bar{X}} & \hat{α} & = \frac{\bar{X}}{\hat{β}} = \frac{(\bar{X})^{2}}{S_{0}^{2}} \end{aligned}

Using moments does not guarantee unbiasedness, seen as we used the second central moment $S_{0}^{2}$ instead of the sample variance $S_{1}^{2}$ to find the estimators