BUILT-IN MATHEMATICA SYMBOL

# ParameterEstimator

ParameterEstimator
is an option to EstimatedDistribution and FindDistributionParameters that specifies what parameter estimator to use.

## DetailsDetails

• The following basic settings can be used:
•  "MaximumLikelihood" maximize the log-likelihood function "MethodOfMoments" match raw moments "MethodOfCentralMoments" match central moments "MethodOfCumulants" match cumulants "MethodOfFactorialMoments" match factorial moments
• The maximum likelihood method will maximize the log-likelihood function where are the distribution parameters and is the PDF of the symbolic distribution.
• The method of moments solves , , where is the sample moment and is the moment of the distribution with parameters .
• The different methods of moments include: , , , or .
• With ParameterEstimator->{mm, "MomentOrders"->list}, the moment orders specified by list are used for the method of moments estimator mm.
• For univariate distributions, the moment orders should be a list of positive integers.
• For -dimensional distributions, the moment orders should be length lists of non-negative integers with each list summing to a positive number.
• ParameterEstimator->{"estimator", Method->"solver"} specifies what underlying equation or optimization solver to use.
• Possible solver settings for include:
•  Automatic automatically chosen solver "FindMaximum" use FindMaximum to maximize log-likelihood "FindRoot" use FindRoot to solve likelihood equations "NMaximize" use NMaximize to maximize log-likelihood
• Possible solver settings for , , , and include:
•  Automatic automatically chosen solver "FindRoot" use FindRoot to solve moment equations "NSolve" use NSolve to solve moment equations "Solve" use Solve to solve moment equations
• The Automatic setting uses a solver or combination of solvers based on the distribution and the parameters to be estimated.
• With the setting ParameterEstimator->{"estimator", Method->{"solver", opts}}, additional options can be given for the solver.
• Solver methods such as Solve, NSolve, and NMaximize that do not rely on starting values will not make use of starting values given to EstimatedDistribution or FindDistributionParameters.

## ExamplesExamplesopen allclose all

### Basic Examples (1)Basic Examples (1)

Construct a distribution using maximum likelihood parameter estimates:

 Out[2]=

Use estimates based on method of moments:

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Plot the difference between densities for the two estimates:

 Out[4]=