QF112 Week04 MOM

Overview

• Presentation Title: Method of Moments

• Instructor: Thomas Lonon

• Course: QF 112 - Quantitative Finance

• Institution: Stevens Institute of Technology

• Academic Term: Spring 2025

Introduction to Moments

• The kth moment of a random variable X is defined as:

  • E[X^k]

  • where k is a positive integer (k ∈ Z+).

Finding Moments for Distributions

• Binomial Distribution (X ∼ Bin(n, p))

  • Objective: Find the first two moments.

• Exponential Distribution (W ∼ Exp(λ))

  • Objective: Find the first two moments.

Notation and Estimation

• Introduce notation:

  • µ_k = E[X^k]

  • Estimated moments will be represented as:

  • µ̂_k = (1/n) Σ (x_j^k)

  • where n is the number of observations and x_j represents individual observations.

Parameter Relationships

• For a set of parameters θ = {θ1, θ2, ..., θp}, functions can be defined as:

  • µ_1 = g1(θ1, θ2, ..., θp)

  • ...

  • µ_p = gp(θ1, θ2, ..., θp)

Method of Moments (MOM)

• The method involves deriving alternative functions from the moments to estimate parameters, defined as:

  • θ1 = h1(µ1, µ2, ..., µp)

  • ...

  • θp = hp(µ1, µ2, ..., µp)

• Using estimates for the moments leads to:

  • θ̂1 = h1(µ̂1, µ̂2, ..., µ̂p)

  • ...

  • θ̂p = hp(µ̂1, µ̂2, ..., µ̂p)

Examples of MOM Estimation

• Binomial Distribution:

  • Use MOM to estimate parameters n and p from m observations.

• Exponential Distribution:

  • Use MOM to estimate parameter λ from n observations.

• Normal Distribution:

  • Use MOM to estimate parameters µ (mean) and σ² (variance) from n observations.

• Uniform Distribution:

  • Question: Can MOM be used to estimate the parameters a and b?

• Poisson Distribution:

  • Use MOM to estimate parameter λ from n observations.