# EstimatedDistribution

EstimatedDistribution[data,dist]

estimates the parametric distribution dist from data.

EstimatedDistribution[data,dist,{{p,p0},{q,q0},}]

estimates the parameters p, q, with starting values p0, q0, .

EstimatedDistribution[data,dist,idist]

estimates distribution dist with starting values taken from the instantiated distribution idist.

# Details and Options

• EstimatedDistribution returns the symbolic distribution dist with parameter estimates inserted for any non-numeric values.
• The data must be a list of possible outcomes from the given distribution dist.
• The distribution dist can be any parametric univariate, multivariate, or derived distribution with unknown parameters.
• The following options can be given:
•  AccuracyGoal Automatic the accuracy sought ParameterEstimator "MaximumLikelihood" what parameter estimator to use PrecisionGoal Automatic the precision sought WorkingPrecision Automatic the precision used in internal computations
• The following basic settings can be used for ParameterEstimator:
•  "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 attempts to 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 .
• Method-of-moment-based estimators may not satisfy all restrictions on parameters.

# Examples

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## Basic Examples(3)

Obtain the maximum likelihood parameter estimates, assuming a gamma distribution:

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Visually compare the PDFs for the original and estimated distributions:

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Obtain the method of moments estimates:

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Estimate parameters for a multivariate distribution:

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Estimated parameters from data with quantities:

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# See Also

Introduced in 2010
(8.0)