This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
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# ParameterEstimator

 ParameterEstimator is an option to EstimatedDistribution and FindDistributionParameters that specifies what parameter estimator to use.
• 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, 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.
• 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.
Construct a distribution using maximum likelihood parameter estimates:
Use estimates based on method of moments:
Plot the difference between densities for the two estimates:
Construct a distribution using maximum likelihood parameter estimates:
 Out[2]=
Use estimates based on method of moments:
 Out[3]=
Plot the difference between densities for the two estimates:
 Out[4]=
 Scope   (4)
Obtain the maximum likelihood estimates using the default method:
Use NMaximize to obtain the estimates:
Obtain the maximum likelihood estimates using the default method:
Use FindMaximum with EvaluationMonitor to extract sampled points:
Visualize the sequences of sampled and values:
Estimate parameters by matching raw moments:
Other moment-based methods typically give similar results:
Estimate parameters using method of moments with default moments:
Use the first and fourth moments:
Use the second and third factorial moments:
Use FindDistributionParameters to get the maximum likelihood estimate:
Obtain the estimate by maximizing the log-likelihood directly:
Compute the maximized value from the FindDistributionParameters estimate:
Obtain the method-of-moments estimate for data:
Solve for parameters by matching moments when a closed form exists:
Obtain the symbolic result:
Compute the moments for data:
Substitute data moments to get the method-of-moments estimate:
Solve the moment equations numerically:
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