Method of moments (statistics)

Suppose that the problem is to estimate ${\displaystyle k}$ unknown parameters ${\displaystyle \theta _{1},\theta _{2},\dots ,\theta _{k}}$ characterizing the distribution ${\displaystyle f_{W}(w;\theta )}$ of the random variable ${\displaystyle W}$.[1] Suppose the first ${\displaystyle k}$ moments of the true distribution (the "population moments") can be expressed as functions of the ${\displaystyle \theta }$s:

${\displaystyle {\begin{aligned}\mu _{1}&\equiv \operatorname {E} [W]=g_{1}(\theta _{1},\theta _{2},\ldots ,\theta _{k}),\\[4pt]\mu _{2}&\equiv \operatorname {E} [W^{2}]=g_{2}(\theta _{1},\theta _{2},\ldots ,\theta _{k}),\\&\,\,\,\vdots \\\mu _{k}&\equiv \operatorname {E} [W^{k}]=g_{k}(\theta _{1},\theta _{2},\ldots ,\theta _{k}).\end{aligned}}}$

Suppose a sample of size ${\displaystyle n}$ is drawn, resulting in the values ${\displaystyle w_{1},\dots ,w_{n}}$. For ${\displaystyle j=1,\dots ,k}$, let

${\displaystyle {\widehat {\mu }}_{j}={\frac {1}{n}}\sum _{i=1}^{n}w_{i}^{j}}$

be the j-th sample moment, an estimate of ${\displaystyle \mu _{j}}$. The method of moments estimator for ${\displaystyle \theta _{1},\theta _{2},\ldots ,\theta _{k}}$ denoted by ${\displaystyle {\widehat {\theta }}_{1},{\widehat {\theta }}_{2},\dots ,{\widehat {\theta }}_{k}}$ is defined as the solution (if there is one) to the equations:

${\displaystyle {\begin{aligned}{\widehat {\mu }}_{1}&=g_{1}({\widehat {\theta }}_{1},{\widehat {\theta }}_{2},\ldots ,{\widehat {\theta }}_{k}),\\[4pt]{\widehat {\mu }}_{2}&=g_{2}({\widehat {\theta }}_{1},{\widehat {\theta }}_{2},\ldots ,{\widehat {\theta }}_{k}),\\&\,\,\,\vdots \\{\widehat {\mu }}_{k}&=g_{k}({\widehat {\theta }}_{1},{\widehat {\theta }}_{2},\ldots ,{\widehat {\theta }}_{k}).\end{aligned}}}$

The method of moments is fairly simple and yields consistent estimators (under very weak assumptions), though these estimators are often biased.

It is an alternative to the method of maximum likelihood.

However, in some cases the likelihood equations may be intractable without computers, whereas the method-of-moments estimators can be computed much more quickly and easily. Due to easy computability, method-of-moments estimates may be used as the first approximation to the solutions of the likelihood equations, and successive improved approximations may then be found by the Newton–Raphson method. In this way the method of moments can assist in finding maximum likelihood estimates.

In some cases, infrequent with large samples but not so infrequent with small samples, the estimates given by the method of moments are outside of the parameter space (as shown in the example below); it does not make sense to rely on them then. That problem never arises in the method of maximum likelihood. Also, estimates by the method of moments are not necessarily sufficient statistics, i.e., they sometimes fail to take into account all relevant information in the sample.

When estimating other structural parameters (e.g., parameters of a utility function, instead of parameters of a known probability distribution), appropriate probability distributions may not be known, and moment-based estimates may be preferred to maximum likelihood estimation.

An example application of the method of moments is to estimate polynomial probability density distributions. In this case, an approximate polynomial of order ${\displaystyle N}$ is defined on an interval ${\displaystyle [a,b]}$. The method of moments then yields a system of equations, whose solution involves the inversion of a Hankel matrix.[2]

• Generalized method of moments
• Decoding methods