momenttimenetelmä

Momenttimenetelmä, also known as the method of moments, is a technique in statistical inference used to estimate parameters of a probability distribution. It is one of the oldest and most intuitive methods for parameter estimation. The core idea behind the method of moments is to equate population moments with corresponding sample moments and then solve for the unknown parameters.

A population moment is a theoretical value derived from the probability distribution of a random variable. For instance, the first population moment is the expected value (mean), and the second central population moment is the variance. A sample moment is calculated directly from observed data. The sample mean is an estimate of the population mean, and the sample variance is an estimate of the population variance.

The method of moments involves selecting a sufficient number of sample moments (equal to the number of unknown parameters) and setting them equal to their theoretical population counterparts. For example, if a distribution has two unknown parameters, say $\theta_1$ and $\theta_2$, one would calculate the first and second sample moments from the data. Then, these sample moments are equated to the expressions for the first and second population moments, which will be functions of $\theta_1$ and $\theta_2$. This results in a system of equations that can be solved to obtain estimates for $\theta_1$ and $\theta_2$.

While simple to understand and implement, the method of moments does not always yield the most statistically efficient estimators. In some cases, other methods, such as maximum likelihood estimation, may produce estimators with smaller variances. However, the method of moments often provides estimators that are consistent and have a closed-form solution, making them computationally convenient. It also serves as a good starting point for more complex estimation procedures.