BUILT-IN MATHEMATICA SYMBOL

EstimatedDistribution

estimates the parametric distribution dist from data.

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

,

, ....

MORE INFORMATION

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

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 (2)

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:

An ExtremeValueDistribution:

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:

The number of earthquakes per year can be modeled by SinghMaddalaDistribution:

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:

Properties & Relations (6)

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:

Possible Issues (1)

Good starting values may also result in quicker results: