In statistics, the method of moments is a method of estimation of population parameters such as mean, variance, median, etc., by equating sample moments with unobservable population moments and then solving those equations for the population moments. Statistics is a type of data analysis whose practice includes the planning, summarizing, and interpreting of observations of a system possibly followed by predicting or forecasting of future events based on a mathematical model of the system being observed. ... Estimation is approximate or uncertain calculation of a result, often based on approximate, uncertain, incomplete, or noisy inputs. ... See also moment (physics). ...

Example

Suppose X₁, ..., X_n are independent identically distributed random variables with a gamma distribution with probability density function In probability theory, a sequence or other collection of random variables is independent and identically distributed (i. ... A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ... In probability theory and statistics, the gamma distribution is a continuous probability distribution. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

for x > 0, and 0 for x < 0.

The first moment, i.e., the expected value, of a random variable with this probability distribution is In probability (and especially gambling), the expected value (or (mathematical) expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds...

and the second moment, i.e., the expected value of its square, is

These are the "population moments".

The first and second "sample moments" are respectively

Equating the population moments with the sample moments, we get

and

Solving these two equations for α and β, we get

and

We then use these two quantities as estimates, based on the sample, of the two unobservable population parameters α and β.

Advantages and disadvantages of this method

In some respects, this method was superseded by Fisher's method of maximum likelihood, because maximum likelihood estimators have higher probability of being close to the quantities to be estimated. Sir Ronald Fisher Sir Ronald Aylmer Fisher, FRS (February 17, 1890 – July 29, 1962) was an evolutionary biologist, geneticist and statistician. ... Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution of a given data set. ... In statistics, an estimator is a function of the known data that is used to estimate an unknown parameter; an estimate is the result from the actual application of the function to a particular set of data. ...

However, in some cases, as in the above example of the gamma distribution, the likelihood equations may be intractable without computers, whereas the method-of-moments estimators can be quickly and easily calculated by hand as shown above.

Estimates by the method of moments 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 and the method of maximum likelihood are symbiotic. In numerical analysis, Newtons method (or the Newton-Raphson method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ...

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; 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 sometimes fail to take into account all relevant information in the sample; i.e., they are sometimes not sufficient statistics. In statistics, one often considers a family of probability distributions for a random variable X (and X is often a vector whose components are scalar-valued random variables, frequently independent) parameterized by a scalar- or vector-valued parameter, which let us call θ. ...

See also

• Generalized method of moments