Maximum Likelihood Estimates (MLEs)

See also: Fitting distributions to data, Fitting in ModelRisk, Method of Moments (MoM)

The maximum likelihood estimates of a distribution type are the values of its parameters that produce the maximum joint probability density or mass for the observed data X given the chosen probability model.

Maximum likelihood estimation starts with the mathematical expression known as a likelihood function of the sample data. This expression contains the unknown parameters to be estimated. Those values of the parameter that maximize the sample likelihood are known as the maximum likelihood estimates which are determined by setting the partial derivative of the likelihood function to zero (i.e. finding the location of the functions peak with respect to the estimated parameters).

With the ModelRisk probability calculation functions you can calculate the likelihood (i.e. joint probability) of given data for all of the distributions included.

To directly calculate the MLE estimates use the ModelRisk fitting functions.

Advantages of MLE over other fitting methods

Maximum likelihood provides a consistent approach to parameter estimation problems. This means that maximum likelihood estimates can be developed for a large variety of estimation situations (essentially whenever you can produce a probability equation relating the parameter to a sufficient statistic of the observations), including missing or censored data.

Maximum likelihood methods have desirable mathematical and optimality properties: they become minimum variance unbiased estimators as the sample size increases. They often have approximate normal distributions and with a Taylor series expansion calculation they can be used to generate Normal distributions of uncertainty.


Maximum likelihood estimates can be heavily biased for small samples. The optimality properties may not apply for small samples. It is also sensitive to the choice of starting values when using numerical estimation. 



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