Index: The Book of Statistical Proofs ▷ General Theorems ▷ Frequentist statistics ▷ Likelihood theory ▷ Method of moments

Definition: Let measured data $y$ follow a probability distribution with probability mass or probability density $p(y \vert \theta)$ governed by unknown parameters $\theta_1, \ldots, \theta_k$. Then, method-of-moments estimation, also referred to as “method of moments” or “matching the moments”, consists in


1) expressing the first $k$ moments of $y$ in terms of $\theta$

\[\label{eq:mom} \begin{split} \mu_1 &= f_1(\theta_1, \ldots, \theta_k) \\ &\vdots \\ \mu_k &= f_k(\theta_1, \ldots, \theta_k) \; , \end{split}\]


2) calculating the first $k$ sample moments from $y$

\[\label{eq:mom-samp} \hat{\mu}_1(y), \ldots, \hat{\mu}_k(y)\]


3) and solving the system of $k$ equations

\[\label{eq:mome} \begin{split} \hat{\mu}_1(y) &= f_1(\hat{\theta}_1, \ldots, \hat{\theta}_k) \\ &\vdots \\ \hat{\mu}_k(y) &= f_k(\hat{\theta}_1, \ldots, \hat{\theta}_k) \end{split}\]

for $\hat{\theta}_1, \ldots, \hat{\theta}_k$, which are subsequently refered to as “method-of-moments estimates”.

 
Sources:

Metadata: ID: D151 | shortcut: mome | author: JoramSoch | date: 2021-04-29, 07:51.