BUILT-IN MATHEMATICA SYMBOL

FindDistributionParameters

finds the parameter estimates for the distribution dist from data.

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

,

, ....

MORE INFORMATION

FindDistributionParameters returns a list of replacement rules for the parameters in dist.

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 Laplace distribution:

Obtain the method of moments estimates:

Estimate parameters for a multivariate distribution:

Compare the difference between the original and estimated PDFs:

Obtain the maximum likelihood parameter estimates assuming a Laplace distribution:

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Obtain the method of moments estimates:

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Estimate parameters for a multivariate distribution:

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Compare the difference between the original and estimated PDFs:

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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:

Mark the optimal point on the contour plot:

Estimate the normal approximation of Poisson data:

Obtain estimate to 20 digits:

Estimate parameters for a continuous distribution:

Estimate parameters for a discrete distribution:

Compare the fitted and empirical CDFs:

Estimate parameters for a discrete multivariate distribution:

Estimate parameters for a continuous multivariate distribution:

Visualize the density functions for the marginal distributions:

Obtain the covariance matrix from the formula:

Estimate parameters for a truncated normal:

Estimate parameters for a constructed distribution:

Visualize the optimal point:

Estimate parameters for a product distribution:

Estimate parameters for a copula distribution:

Estimate parameters for a component mixture:

Estimate the mixture probabilities assuming the component distributions are known:

Visualize the two estimates against the data:

Options (4)

Estimate parameters by matching cumulants:

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 (16)

Estimate Laplace parameters for data from an ExponentialPowerDistribution:

Use the Laplace estimate as a starting point for estimating exponential power parameters:

Compare the data with the Laplace and exponential power estimates:

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:

Estimate the parameter

for a logarithmic series distribution for policy claims shifted by 1:

See that the estimate gives a maximal result:

Get word length data for several languages:

Model the word lengths for each language as binomially distributed with

:

Compare the actual and estimated distributions:

Bootstrap the distribution of p values based on these 9 results:

Estimate the expected value of p and a standard deviation for the estimate:

The word count in a text follows a Zipf distribution:

Fit a ZipfDistribution to the word frequency data:

Fit a truncated ZipfDistribution to counts at most 50 using rhohat as a starting value:

Visualize the CDFs up to the truncation value:

Estimate the proportion of the original data not included in the truncated model:

Find estimates for a multimodal MixtureDistribution model:

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:

Extract the means of the components:

The components' means are far enough apart that they are still the modes:

Model monthly maximum wind speeds in Boston:

Fit the data to a RayleighDistribution:

An ExtremeValueDistribution:

Compare the empirical and fitted quantiles 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:

Visualize the likelihood contours and mark the optimal point:

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:

Extract the estimated average city and highway mileages:

Extract the estimated correlation between city and highway mileages:

Visualize the joint density on a logarithmic scale with the mean mileage marked with a blue point:

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:

Compute the maximized log-likelihood:

Visualize the log-likelihood profiles near the optimal parameter values:

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 mixture of gamma and normal distributions to the data:

Compare the histogram to the PDF of the estimated distribution:

Lognormal distribution can be used to model stock prices:

Fit the distribution to the data:

Visualize the profile likelihoods, fixing one parameter at the fitted value:

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:

Simulate annual minimum mean daily flows for the next 30 years:

Use a Pareto distribution to model Australian city population sizes:

Get the probability that a city has a population at least 10000 under a Pareto distribution:

Compute the probability given the parameter estimates:

Compute the probability based on the original data:

Properties & Relations (6)

FindDistributionParameters gives estimates as replacement rules:

EstimatedDistribution gives a distribution with parameter estimates inserted:

Estimate distribution parameters by maximum likelihood:

Use DistributionFitTest to test quality of the fit:

Extract the fitted distribution parameter:

Obtain a table of relevant test statistics and p-values:

Estimate parameters in a parametric distribution:

Get a nonparametric kernel density estimate using SmoothKernelDistribution:

Compare the PDFs for the nonparametric and parametric distributions:

Visualize the nonparametric density using SmoothHistogram:

Get 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 beta distribution for the estimated parameters:

Estimate parameters for a Weibull distribution:

Use QuantilePlot to visualize the empirical quantiles vs. the theoretical quantiles:

Obtain the same visualization when the estimation is done within QuantilePlot:

Possible Issues (3)

Solutions of method-of-moment equations can give parameters that are not valid:

Good starting values may be needed to obtain a good solution:

Good starting values may result in quicker results: