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# method of moments

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An alternative to the method of maximum likelihood as a method of estimating the parameters of a distribution. Each moment of a distribution can be expressed as a function of the parameters of the distribution, and often this implies that the parameters can be expressed as simple functions of the moments. In such cases, replacing the moments with their sample estimates provides estimates of the population parameters.

For example, the two-parameter gamma distribution with probability density function proportional to xα−1ex/β has its first moment, μ1 (equal to the mean, μ), given by μ=αβ and its second central moment, μ2, given by μ2=αβ2. Solving these equations, we get α=μ2/μ2 and β=μ2/μ. The method of moments replaces the unknown quantities μ and μ2 with the corresponding sample quantities and so that, for example, the estimator of α is α̃ given by .

Subjects: Probability and Statistics.

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