Distribution Fitting Methods

During distribution fitting, Crystal Ball computes Maximum Likelihood Estimators (MLEs) to fit most of the probability distributions to a data set. In effect, this method chooses values for the parameters of the distributions that maximize the probability of producing the actual data set. Sometimes, however, the MLEs do not exist for some distributions (for example, gamma, beta). In these cases, Crystal Ball resorts to other natural parameter estimation techniques.

When the MLEs do exist, they exhibit desirable properties:

They are minimum-variance estimators of the parameters.
As the data set grows, the biases in the MLEs tend to zero.

For several of the distributions (for example, uniform, exponential), it is possible to remove the biases after computing the MLEs to yield minimum-variance unbiased estimators (MVUEs) of the distribution parameters. These MVUEs are the best possible estimators.