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 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 symbolic 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: , , , 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:
  • 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 , , , and 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.

ExamplesExamplesopen allclose all

Basic Examples  (1)Basic Examples  (1)

Construct a distribution using maximum likelihood parameter estimates:

In[1]:=
Click for copyable input
In[2]:=
Click for copyable input
Out[2]=

Use estimates based on method of moments:

In[3]:=
Click for copyable input
Out[3]=

Plot the difference between densities for the two estimates:

In[4]:=
Click for copyable input
Out[4]=
Introduced in 2010
(8.0)