MomenteSchätzung

MomenteSchätzung, which translates to "moment estimation," is a method used in statistics to estimate the parameters of a probability distribution. It is one of the simplest and most intuitive estimation techniques. The core idea is to equate sample moments with the corresponding theoretical moments of the distribution.

For a given probability distribution with unknown parameters, the theoretical moments are functions of these parameters. For example, the first theoretical moment is the expected value (mean), and the second theoretical moment is the expected value of the square. The sample moments are calculated directly from the observed data. The k-th sample moment is the average of the k-th powers of the observed values.

The method of moments involves setting up a system of equations where each sample moment is set equal to its corresponding theoretical moment. The number of equations is typically equal to the number of unknown parameters that need to be estimated. Solving this system of equations for the unknown parameters yields the moment estimates.

While conceptually straightforward, moment estimation has certain properties. The estimators are often consistent, meaning they converge to the true parameter values as the sample size increases. However, they are not always the most efficient estimators, particularly when compared to maximum likelihood estimators, which often have lower variance. Despite this, moment estimation remains a valuable tool, especially for its ease of calculation and its ability to provide initial estimates that can be refined by other methods. It is particularly useful when the theoretical moments are easy to compute and the resulting system of equations is simple to solve.