# Parametric Inference and Method of moments

Till now, we have covered the estimation of statistical functionals i.e. functions of CDF – F_{x}. They are Non-parametric Inference. Now we shift our attention to parametric models.

There are two main methods of generating parametric estimators:

- Method of Moments
- Maximum Likelihood Method

### The Method of Moments

Suppose the parameter θ = (θ_{1}, . . . ,θ_{k}) has k components.

#### j^{th} moment

α_{j} ≡ α_{j}(θ) ≡ E_{θ}(X^{j}) = ∫ x^{j}dF_{θ}(x)

#### j^{th} sample moment

The method of moments estimators is θ̂

_{n}is defined to be the value of θ such that

α_{1}(θ̂_{n}) = α̂_{1}

α_{2}(θ̂_{n}) = α̂_{2}

…

α_{k}(θ̂_{n}) = α̂_{k}

**Example**:

Let X_{1}, X_{2}, . . . , X_{n} ~ Normal(µ, σ^{2}).

Then,

α_{1} = E_{θ}(X_{1}) = µ and

α_{2} = E_{θ}(X_{1}^{2}) = V_{θ}(X_{1}) + (E_{θ}(X_{1}))^{2} = σ^{2} + µ^{2} . . . (using V(X) = (E(X^{2}) – (E(X))^{2})

We need to solve –

Solving the two equations we get –