→ An alternative way to get an estimator quick and dirty.

Method of Moments

alg. Method of Moments (MOM):

let . Then to get estimators for :

  1. let
  2. gather data on to get an emperical estimate

Observe that because , this means that

  1. let function which maps can be inverted
  2. Get system of equations for as many ’s as necessary
  3. Solve the system of equations as

Limitations on MoM:

  • Estimators may not make sense (negative numbers, complex number, etc.)
  • Illegal

def. Moment Generating Functions

(alternative specification of pdf. Makes it easy to calculate things.)

Moment generating functions are defined as such:

…where is differentiable -times around zero to be able to generate the -th moment.

You can build an MGF from a pmf of pdf:

  • for pmfs:
  • for pdfs:

It can generate the -th moment like such:

MGFs are unique to each distribution. If an RV has the same MGF, they are distributed identically.

thm. Linear Combination of MGFs. Given r.v. , and are iid,: