Method of moments (statistics)

The method of moments is in the inferential statistics an estimation method, with which one can estimate the parameters of the theoretical distribution of a population based on a sample. This pressing these parameters as a function of the moments of the distribution. In a second step, one uses the empirical moments instead of moments in the equations and thus obtain the moment estimator. By substituting the values ​​of a sample of the moment estimator provides an estimate of the corresponding parameter of the theoretical distribution.

The method of moments estimation technique is the oldest and dates back to Karl Pearson. It is very easy to use, but has the disadvantage that the resulting estimators are not always unbiased.

Method

A single parameter can be calculated directly. Wherein a plurality of parameters to be estimated often results in a system of equations which can be solved with respect to the unknown parameters.

By inserting the values ​​of a sample is then values ​​for the parameters of the theoretical distribution is replaced.

Example

It is believed that the population is normal. Starting from a random sample to the expected value and variance of the corresponding random variable to be estimated.

When the normal distribution applies and. The unknown parameters depend then as follows from and to:

If, now, the estimation functions, we obtain the following estimates:

One can see that is not unbiased, since the sum of squares is divided by instead. ( See proof here.) The estimator is asymptotically unbiased, however.