# FindDistributionParameters

FindDistributionParameters[data,dist]

finds the parameter estimates for the distribution dist from data.

FindDistributionParameters[data,dist,{{p,p0},{q,q0},}]

finds the parameters p, q, with starting values p0, q0, .

# Details and Options • 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 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 Laplace distribution:

Obtain the method of moments estimates:

Estimate parameters for a multivariate distribution:

Compare the difference between the original and estimated PDFs:

Estimate parameters from quantity data:

## 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 from res:

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:

Mark the optimal point on the contour plot:

Estimate the normal approximation of Poisson data:

Obtain estimate to 20 digits:

### Univariate Parametric Distributions(2)

Estimate parameters for a continuous distribution:

Estimate parameters for a discrete distribution:

Compare the fitted and empirical CDFs:

### Multivariate Parametric Distributions(2)

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:

### Derived Distributions(6)

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 for a distribution in specified units:

## Options(4)

### ParameterEstimator(3)

Estimate parameters by matching cumulants:

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(17)

### Use One Parameter Estimator to Get Starting Values for Another(1)

Get the method of moments estimate:

Use the method of moments estimate as the starting value for ml estimation:

Obtain ml estimates for a gamma distribution:

Use those as starting values for the method of moments:

### Obtain Starting Values for Another Estimation(1)

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:

### Parameter 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:

Estimate the parameter for a logarithmic series distribution for policy claims shifted by 1:

See that the estimate gives a maximal result:

### Word Lengths in Different Languages(1)

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:

### Text Frequency(1)

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:

### Earthquake Magnitudes(1)

Find estimates for a multimodal MixtureDistribution model:

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:

Extract the means of the components:

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

### Wind Speed Analysis(1)

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:

### 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:

### Market Change in Stock Values(1)

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:

Number of days for each financial entity:

Extract values from time series for each entity and normalize numeric quantities:

Check if each entity has the same length of data:

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:

### 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:

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:

### 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:

Compute the maximized loglikelihood:

Visualize the loglikelihood profiles near the optimal parameter values:

### 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 mixture of gamma and normal distributions to the data:

Compare the histogram to the PDF of the estimated distribution:

### Stock Price Distribution(1)

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:

### 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:

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

### Population Sizes(1)

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(8)

FindDistributionParameters gives estimates as replacement rules:

EstimatedDistribution gives a distribution with parameter estimates inserted:

FindProcessParameters returns a list of parameter estimates for a random process:

FindDistributionParameters returns a list of parameter estimates for a distribution:

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 pvalues:

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 loglikelihood 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 versus the theoretical quantiles:

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

FindDistributionParameters 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 result in quicker results:
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