ParameterEstimator

ParameterEstimator

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

Details

  • The following basic settings can be used:
  • "MaximumLikelihood"maximize the loglikelihood 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 distribution.
  • The method of moments solves , , where is the ^(th) sample moment and is the ^(th) moment of the distribution with parameters .
  • The different methods of moments include: "MethodOfMoments", "MethodOfCentralMoments", "MethodOfCumulants", or "MethodOfFactorialMoments".
  • 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 "MaximumLikelihood" include:
  • Automaticautomatically 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 "MethodOfMoments", "MethodOfCentralMoments", "MethodOfCumulants", and "MethodOfFactorialMoments" include:
  • Automaticautomatically 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.

Examples

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Basic Examples  (1)

Construct a distribution using maximum likelihood parameter estimates:

Use estimates based on method of moments:

Plot the difference between densities for the two estimates:

Scope  (4)

Maximum Likelihood Estimator  (2)

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:

MomentBased Estimators  (2)

Estimate parameters by matching raw moments:

Other momentbased 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:

Properties & Relations  (2)

Use FindDistributionParameters to get the maximum likelihood estimate:

Obtain the estimate by maximizing the loglikelihood 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:

Introduced in 2010
 (8.0)