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 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 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:

Visually compare the PDFs for the original and estimated distributions:

Obtain the method of moments estimates:

Estimate parameters for a multivariate distribution:

Estimated parameters from data with quantities:

Scope(15)

Basic Uses(5)

Estimate both parameters for a binomial distribution:

Estimate p, assuming n is known:

Estimate n, assuming p 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:

Perform goodness-of-fit tests with null distribution dist:

Perform tests correcting for estimation of the parameter:

Estimate parameters by maximizing the loglikelihood:

Plot the loglikelihood function to visually check that the solution is optimal:

Visualize a loglikelihood 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:

Univariate Parametric Distributions(2)

Estimate parameters for a continuous distribution:

Compare empirical and distribution quantiles:

Estimate parameters for a discrete distribution:

Multivariate Parametric Distributions(2)

Estimate parameters for a discrete multivariate distribution:

Estimate parameters for a continuous multivariate distribution:

Compare the difference between the original and estimated PDFs:

Derived Distributions(6)

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:

Estimate parameters for quantity distribution in specified units:

Options(4)

ParameterEstimator(3)

Estimate parameters by matching central moments:

Other momentbased 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:

WorkingPrecision(1)

Use machine precision for continuous parameters by default:

Obtain a higher-precision result:

Applications(14)

Estimation of Similarly Shaped Distributions(1)

Model lognormal distributed data with a gamma distribution:

Compare the distributions of the simulation and estimated distributions:

Accident Claims(1)

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:

Word Lengths in Different Languages(1)

Get word length data for several languages:

Model the word lengths for each language as binomially distributed:

Compare the actual and estimated distributions:

Text Frequency(1)

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:

Earthquake Magnitudes(1)

EstimatedDistribution can be used with constructs like MixtureDistribution to create multimodal models:

The magnitudes of earthquakes in the United States in the selected years 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:

Wind Speed Analysis(1)

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:

Distribution of Incomes(1)

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:

Automobile Fuel Efficiency(1)

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:

Earthquake Waiting Times(1)

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:

Earthquake Frequency(1)

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 US in a year:

Time between Geyser Eruptions(1)

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:

Stock Price Distribution(1)

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:

Water Flow Rates(1)

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:

Population Sizes(1)

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(8)

EstimatedDistribution gives a distribution with parameter estimates inserted:

FindDistributionParameters gives parameter estimates as replacement rules:

EstimatedProcess estimates a parametric process:

EstimatedDistribution estimates a parametric distribution:

Estimate distribution parameters by maximum likelihood:

Use DistributionFitTest to test the 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 loglikelihood 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:

EstimatedDistribution ignores time stamps in TimeSeries and EventSeries:

The same as:

For TemporalData, all the path structure is ignored:

The same as:

Possible Issues(3)

Solutions of method of moment equations can give parameters that are not valid: For a continuous distribution: Good starting values may be needed to obtain a good solution: Good starting values may also result in quicker results: