Methods of Moment Estimation

We can approximate the Population Moment with Sample Moment.

$$\frac{1}{n}\sum _{i=1}^n{X}_i^k\sim E[X^k]$$

Summary

• Express lower order population moments in terms of parameters
• Invert expressions to express parameters in terms of population moments
• Replace the population moments with sample moments

Example

For $X_1,\dots ,X_n{\sim }^{iid}\text{Poisson}(\lambda )$, find the method of moments estimator of $\lambda$. $X_i\sim \text{Poisson}(\lambda )$ $\lambda =E[X]$ MGF estimation of $\lambda$, $\lambda =\frac{1}{n}{\sum }_{i=1}^n{X}_i=\overline{X}$