Method of moments

Builds on Moment Estimator Sample mean Consistent estimator

Summary

The method of moments is a way to build estimators 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 $\mu_j(\theta) = E_\theta[X_1^j]$ 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. The estimators are often simple, but they are usually less efficient than maximum likelihood estimators.

Reading path

To understand this page from the ground up, read these 74 pages in order. The foundations come first.

Construction of the reals from Cauchy sequences concept
Element definition
Integers object
Ordered field definition
Proposition concept
Random experiment concept
Rational numbers object
Real numbers object
Set definition
Ring of integers statement
Disjunction definition
Implication definition
Predicate definition
Ordered pair definition
Union definition
Sample space definition
The rationals form a field statement
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Subset definition
The rationals form an ordered field statement
Uniqueness quantifier definition
Binary relation definition
Universal set definition
Order on the reals definition
Power set definition
Complement definition
Function definition
Interval definition
Injective function definition
Preimage of a set definition
Sigma-algebra definition
Countable set definition
Event definition
Borel sigma-algebra definition
Probability axioms concept
Probability measure definition
Probability distribution definition
Probability space definition
Conditional probability definition
Random variable definition
Cumulative distribution function definition
Discrete random variable definition
Law of a random variable definition
Independence of two events definition
Convergence in probability definition
Random variable with a density definition
Joint probability distribution definition
Probability mass function definition
Mutual independence of events definition
Expected value definition
Identically distributed random variables definition
Independence of two random variables definition
Probability density function definition
Marginal distribution definition
Independent sigma-algebras definition
Variance definition
Law of the unconscious statistician statement
Linearity of expectation statement
Independence of random variables definition
Independence means the joint law is the product of the marginals statement
Covariance definition
Moment definition
Independent and identically distributed random variables definition
Variance of a sum statement
Expectation of a product of independent random variables statement
Statistic definition
Independent random variables are uncorrelated statement
Estimator definition
Variance of a sum of independent random variables statement
Sample mean definition
Consistent estimator definition
Method of moments concept

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2 pages build on this one.

Tag	`09J1H07VAH`
URI	`m2://09J1H07VAH`
Aliases	`method-of-moments-statistics`
Kind	concept
Writing	W0 stub
MSC	62F10
Uses	Moment, Estimator, Sample mean, Consistent estimator
Same as	Wikidata Q1925212 Wikipedia Method of moments (statistics)

Method of moments

Summary

Remarks

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