---
tag: 09J1H07VAH
kind: concept
name: Method of moments
aliases: [method-of-moments-statistics]
msc: [62F10]
uses: [0PH0VHTQS0, 0TTNRP5PFQ, 0QNRBHWV35, 0XJVF81QF6]
sources:
  - url: https://en.wikipedia.org/wiki/Method_of_moments_(statistics)
    locator: lead section
same_as:
  wikidata: Q1925212
  wikipedia: Method of moments (statistics)
---

## Summary

The method of moments is a way to build [[estimators|0TTNRP5PFQ]] of unknown parameters.
Suppose that the law of an i.i.d. sample $X_1, \dots, X_n$ depends on $k$ real parameters $\theta = (\theta_1, \dots, \theta_k)$, and that the [[moments|0PH0VHTQS0]] $\mu_j(\theta) = E_\theta[X_1^j]$ exist for $j \le k$.
The method replaces each $\mu_j(\theta)$ with the sample moment $\hat{m}_j = \frac{1}{n} \sum_{i=1}^{n} X_i^j$, and solves the $k$ equations $\mu_j(\hat{\theta}) = \hat{m}_j$ for $\hat{\theta}$.

## Remarks

Example: for $X_i \sim \operatorname{Exp}(\lambda)$, $E[X_1] = 1/\lambda$, so the method gives $\hat{\lambda} = 1/\bar{X}_n$.
Example: for $X_i \sim \mathcal{N}(\mu, \sigma^2)$, it gives $\hat{\mu} = \bar{X}_n$ and $\hat{\sigma}^2 = \frac{1}{n}\sum_i (X_i - \bar{X}_n)^2$.
If the map $\theta \mapsto (\mu_1(\theta), \dots, \mu_k(\theta))$ has a continuous inverse, the law of large numbers makes the estimators [[consistent|0XJVF81QF6]].
The estimators are often simple, but they are usually less efficient than maximum likelihood estimators.



## Explanations

### L1 human

The method of moments chooses the parameters so that the model has the same averages as the data.
For example, if the data have mean $2$, an exponential model with mean $1/\lambda$ gets the rate $\lambda = 1/2$.