Parametric Inference and Method of moments

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

There are two main methods of generating parametric estimators:

  1. Method of Moments
  2. Maximum Likelihood Method

The Method of Moments

Suppose the parameter θ = (θ1, . . . ,θk) has k components.

jth moment

αj ≡ αj(θ) ≡ Eθ(Xj) = ∫ xjdFθ(x)

jth sample moment

jth sample moment

The method of moments estimators is θ̂n is defined to be the value of θ such that
α1(θ̂n) = α̂1
α2(θ̂n) = α̂2

αk(θ̂n) = α̂k

Example:
Let X1, X2, . . . , Xn ~ Normal(µ,  σ2).
Then,
α1 = Eθ(X1) = µ and
α2 = Eθ(X12) = Vθ(X1) + (Eθ(X1))2 = σ2 + µ2 . . . (using V(X) = (E(X2) – (E(X))2)
We need to solve –
method of moments ex
Solving the two equations we get –
mm ex2
mm ex3

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