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# Distribution Fitting - Maximum Likelihood Estimation of Parameters

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Maximum likelihood estimates of the parameters are calculated by maximizing the likelihood function with respect to the parameters. The likelihood function is simply the sum of the log of the probability density function (PDF) for each observation.

The maximum likelihood parameter estimates are then calculated using the Newton-Raphson method. This is a fast iterative process that uses both the first and second derivatives to move to a point at which no further improvement in the likelihood is possible, resulting in an optimal estimate of the parameters. Initial parameter values are derived from method of moments estimates.

For some data sets, the likelihood function for threshold models is unbounded, and the maximum likelihood methodology fails. In this context, a threshold is estimated using a bias correction method. This is an iterative methodology that evaluates the threshold based on the difference between the minimum value of the variate and the prediction for the minimum value, conditional on the current values of the parameters.

The Anderson Darling statistic measures how well the data fits a particular continuous distribution - the smaller the AD value, the better the fit. In general, when comparing several distributions, the distribution with the smallest AD statistic has the best fit to the data. For some distributions, tabulated data exists on the distribution of the AD statistic, which permits the reporting of its associated probability based on interpolation. However, for the majority of distributions, tabulated data doesn't exist. In such cases, the only feasible method of reporting p-values is by simulation, given the specified distribution and the associated parameter values. During batch estimation, DiscoverSim provides approximate p-values for each of the 1, 2, 3 and 4 parameter distributions that have the lowest AD statistic. The p-values are derived based on simulation, using a method of moments estimator rather than maximum likelihood which allows the simulation to occur in a reasonable time.

The Chi-Square goodness of fit test and associated p-value are used when distribution fitting with discrete distributions.

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