Methodofmoments

The method of moments is a parameter estimation technique in statistics that uses the moments of a distribution to infer its parameters. Moments are expectations of powers of the random variable, with the first moment being the mean and the second moment related to the variance. In this method, the sample moments are computed from data and then equated to the theoretical moments of a chosen distribution, expressed as functions of the distribution’s parameters. Solving these equations yields estimates for the parameters. In cases with more moments than parameters, a subset of moment equations is used, or a least-squares approach is applied.

The method traces its lineage to early developments in mathematical statistics, notably Karl Pearson in the early 20th century, as a simple alternative to likelihood-based methods. It provides a straightforward, algebraic route to parameter values without invoking likelihoods or optimization.

Examples illustrate the idea. For a normal distribution with unknown mean mu and variance sigma^2, the first moment gives mu = x̄, and the second moment around mu yields sigma^2 = (1/n) ∑(x_i − x̄)^2, giving MoM estimates mû = x̄ and sigmâ^2 = (1/n) ∑(x_i − x̄)^2. For a gamma distribution with shape k and scale θ, moments E[X] = kθ and E[X^2] = k(k+1)θ^2 lead to equations in k and θ solved from the sample mean and second moment.

Advantages include simplicity and closed-form solutions in many cases. Limitations involve potential bias and inefficiency relative to maximum likelihood estimation, sensitivity to outliers, and the possibility that moment equations do not yield feasible or unique parameter values.