# The Method Of Moments (MOM)

## Method of Moments

### Method of moments

#### Introduction

If we have $$k$$ parameters to estimate, we can solve this if we have $$k$$ equations.

We generate these

First, we link each first $$k$$ moments to functions of the parameters.

Then we replace the momenets with sample estimates.

#### Estimation

The moments of this population distribution are:

$$\mu_i =E[X^i]=g_i(\theta_1,...,\theta_k)$$

We have a sample.

$$X=[X_1,...,X_n]$$

We now define the method of moments estimator

$$\hat \mu_i=\dfrac{1}{n}\sum_{j=1}^nx_j^i$$