Method of Moments

Powerful ideas in Statistical Inference.

By Vamshi Jandhyala in mathematics

November 6, 2020

The method of moments estimator $\theta_n$ is defined to be the value of $\theta$ such that

$$ \begin{align*} \alpha_1(\hat{\theta_n}) &= \hat{\alpha_1} \\
\alpha_2(\hat{\theta_n}) &= \hat{\alpha_2} \\
&\vdots \\
\alpha_k(\hat{\theta_n}) &= \hat{\alpha_k} \\
\end{align*} $$

where $\hat{\alpha_j} = \frac{1}{n}\sum_{i=1}^n X_i^j$ and $\alpha_j(\theta) = \mathbb{E}_{\theta}(X^j)$.

The above defines a system of $k$ equations with $k$ unknowns.

Problem

Let $f(x) = \left(\frac{\alpha m^{\alpha}}{x^{\alpha+1}} \right)\mathbb{I}{x ≥ m}$ where α and m are two unknown parameters. Estimate the values of these parameters using method of moments.

The mean of the pareto distribution is given by

$$ \begin{cases} \infty, \alpha \leq 1 \\
\frac{\alpha m}{\alpha -1}, \alpha > 1 \end{cases} $$

The variance of the pareto distribution is given by

$$ \begin{cases} \infty, \alpha \leq 2 \\
\frac{m^2 \alpha}{(\alpha-1)^2(\alpha-2)}, \alpha > 2 \end{cases} $$

Using the method of moments, we get the following equations

$$ \begin{align} \frac{\alpha m}{\alpha-1} &= A \\
\frac{m^2 \alpha}{(\alpha-1)^2(\alpha-2)} + A^2 &= B \end{align} $$

where

$$ A = \frac{1}{n}\sum_{i=1}^n x_i \\
B = \frac{1}{n}\sum_{i=1}^n x_i^2 $$

Solving the above equations for $\alpha$ and $m$, we get

$$ \alpha = 1 + \sqrt{\frac{B}{B-A^2}} \\
m = \frac{A \sqrt{B}}{\sqrt{B} + \sqrt{B-A^2}} $$