This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
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finds the parameter estimates for the distribution dist from data.
finds the parameters p, q, ... with starting 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:
AccuracyGoalAutomaticthe accuracy sought
ParameterEstimator"MaximumLikelihood"what parameter estimator to use
PrecisionGoalAutomaticthe precision sought
WorkingPrecisionAutomaticthe 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 ^(th) sample moment and is the ^(th) 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 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:
Click for copyable input
Obtain the method of moments estimates:
Click for copyable input
Estimate parameters for a multivariate distribution:
Click for copyable input
Click for copyable input
Compare the difference between the original and estimated PDFs:
Click for copyable input
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:
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:
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:
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:
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:
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:
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