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

# EstimatedDistribution

 EstimatedDistribution estimates the parametric distribution dist from data. EstimatedDistributionestimates the parameters p, q, ... with starting values , , ....
• 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 meta 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
 "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.
Obtain the maximum likelihood parameter estimates, assuming a gamma distribution:
Visually compare the PDFs for the original and estimated distributions:
Obtain the method of moments estimates:
Estimate parameters for a multivariate distribution:
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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 Scope   (14)
Estimate both parameters for a binomial distribution:
Estimate , assuming is known:
Estimate , assuming is known:
Get the distribution with maximum likelihood parameter estimate for a particular family:
Check goodness of fit by comparing a histogram of the data and the estimate's PDF:
Estimate parameters by maximizing the log-likelihood:
Plot the log-likelihood function to visually check that the solution is optimal:
Visualize a log-likelihood surface to find rough values for the parameters:
Supply those rough values as starting values for the estimation:
Estimate the normal approximation of Poisson data:
Obtain the estimate to 20 digits:
Estimate parameters for a continuous distribution:
Compare empirical and distribution quantiles:
Estimate parameters for a discrete distribution:
Estimate parameters for a discrete multivariate distribution:
Estimate parameters for a continuous multivariate distribution:
Compare the difference between the original and estimated PDFs:
Estimate parameters for a truncated normal:
Compare original and estimated distribution:
Estimate parameters for a constructed distribution:
Estimate parameters for a product distribution:
Estimate parameters for a copula distribution:
Compare original and estimated CDFs:
Estimate parameters for a component mixture:
Estimate the mixture probabilities assuming the component distributions are known:
 Options   (4)
Estimate parameters by matching central moments:
Other moment-based methods typically give similar results:
Estimate parameters based on default moments:
Estimate parameters from the first and fourth moments:
Obtain the maximum likelihood estimates using the default method:
Use FindMaximum to obtain the estimates:
Use EvaluationMonitor to extract the points sampled:
Visualize the sequences of sampled and values:
Use machine precision for continuous parameters by default:
Obtain a higher-precision result:
 Applications   (15)
Model lognormal distributed data with a gamma distribution:
Compare the distributions of the simulation and estimated distributions:
The number of accident claims per policy per year from an insurance company:
Model the data by a logarithmic series distribution since most policies have at most one claim:
Get word length data for several languages:
Model the word lengths for each language as binomially distributed:
Compare the actual and estimated distributions:
The word count in a text follows a Zipf distribution:
Fit a ZipfDistribution to the word frequency data:
Compare the frequency histogram with the estimated distribution:
EstimatedDistribution can be used with constructs like MixtureDistribution to create multimodal models:
The magnitudes of earthquakes in the United States in the years 1935-1989 have two modes:
Fit distribution from possible mixtures of one NormalDistribution with another:
Compare the histogram to the PDF of the estimated distribution:
Find the probability of an earthquake of magnitude 7 or higher:
Find the mean earthquake magnitude:
Simulate magnitudes of the next 30 earthquakes:
Model monthly maximum wind speeds in Boston:
Fit the data to a RayleighDistribution:
Compare the empirical quantiles and those for the fitted distributions to see where the models deviate from the data:
Model incomes at a large state university:
Assume the salaries are Dagum distributed:
Assume they follow a more general Pareto distribution:
Compare the subtle differences in the estimated distributions:
Use a beta distribution to model the proportion of Dow Jones Industrial stocks that increase in value on a given day:
Find daily change for Dow Jones Industrial stocks:
Filter out missing data and pad with zeros:
Calculate the daily ratio of companies with an increase in value:
Find parameter estimates, excluding days with zero or all companies having an increase in value:
Compare quantiles to see that the data and estimated distribution match well:
The average city and highway mileage for midsize cars follows a binormal distribution:
Assume city and highway miles per gallon are normally distributed and correlated:
Show the distribution of city and highway mileage:
Visualize the joint density with contours on a logarithmic scale:
The data contains waiting times in days between serious (magnitude at least 7.5 or over 1000 fatalities) earthquakes worldwide, recorded from 12/16/1902 to 3/4/1977:
Model waiting times by an ExponentialDistribution:
Estimate the average and median number of days between major earthquakes:
Fit the distribution to the data:
Compare the data histogram with the PDF of the estimated distribution:
Find the probability of at least 60 earthquakes in the U.S. in a year:
Mixtures can be used to model multimodal data:
A histogram of waiting times for eruptions of the Old Faithful geyser exhibits two modes:
Fit a MixtureDistribution to the data:
Compare the histogram to the PDF of the estimated distribution:
Find the probability that the waiting time is over 80 minutes:
Simulate waiting times for the next 60 eruptions:
Lognormal distribution can be used to model stock prices:
Fit the distribution to the data:
Observe that the quantiles for the data and distribution match well except for the largest values:
Consider the annual minimum daily flows given in cubic meters per second for the Mahanadi river:
Model the annual minimum mean daily flows as a MinStableDistribution:
Compare the histogram of the data to the PDF of the estimated distribution:
Simulate annual minimum mean daily flows for the next 30 years:
Use a Pareto distribution to model Australian city population sizes:
Estimate the probability that a city has a population of at least 10,000 people:
Compute the probability based on the original data:
EstimatedDistribution gives a distribution with parameter estimates inserted:
FindDistributionParameters gives parameter estimates as replacement rules:
Estimate distribution parameters by maximum likelihood:
Use DistributionFitTest to test quality of the fit:
Extract the fitted distribution:
Obtain a table of relevant test statistics and -values:
EstimatedDistribution estimates parameters in a parametric distribution:
SmoothKernelDistribution gives a nonparametric kernel density estimate:
Compare the PDFs for the nonparametric and parametric distributions:
Visualize the nonparametric density using SmoothHistogram:
EstimatedDistribution gives a maximum likelihood estimate of parameters:
Compute the likelihood using Likelihood:
Compute the log-likelihood using LogLikelihood:
Estimate parameters by matching raw moments:
Compute raw moments from the data using Moment:
Compute the same moments from the estimated distribution:
Estimate parameters for a Weibull distribution:
Use QuantilePlot to visualize empirical quantiles versus fitted distribution quantiles:
Obtain the same visualization when the estimation is done within QuantilePlot:
Good starting values may also result in quicker results:
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