Contents

prob.tLocationScaleDistribution class

Package: prob
Superclasses: prob.ToolboxFittableParametricDistribution

t Location-Scale probability distribution object

Description

prob.tLocationScaleDistribution is an object consisting of parameters, a model description, and sample data for a t location-scale probability distribution.

Create a probability distribution object with specified parameter values using makedist. Alternatively, fit a distribution to data using fitdist or the Distribution Fitting app.

Construction

pd = makedist('tLocationScale') creates a t location-scale probability distribution object using the default parameter values.

pd = makedist('tLocationScale','mu',mu,'sigma',sigma,'nu',nu) creates a t location-scale probability distribution object using the specified parameter values.

Input Arguments

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mu — Location parameter0 (default) | scalar value

Location parameter for the t location-scale distribution, specified as a scalar value.

Data Types: single | double

sigma — Scale parameter1 (default) | positive scalar value

Scale parameter for the t location-scale distribution, specified as a positive scalar value.

Data Types: single | double

nu — Degrees of freedom5 (default) | positive scalar value

Degrees of freedom for the t location-scale distribution, specified as a positive scalar value.

Data Types: single | double

Properties

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muLocation parameterscalar value

Location parameter of the t location-scale distribution, stored as a scalar value.

Data Types: single | double

sigmaScale parameterpositive scalar value

Scale parameter of the t location-scale distribution, stored as a positive scalar value.

Data Types: single | double

nuDegrees of freedompositive scalar value

Degrees of freedom of the t location-scale distribution, stored as a positive scalar value.

Data Types: single | double

DistributionNameProbability distribution nameprobability distribution name string

Probability distribution name, stored as a valid probability distribution name string. This property is read-only.

Data Types: char

InputDataData used for distribution fittingstructure

Data used for distribution fitting, stored as a structure containing the following:

  • data: Data vector used for distribution fitting.

  • cens: Censoring vector, or empty if none.

  • freq: Frequency vector, or empty if none.

This property is read-only.

Data Types: struct

IsTruncatedLogical flag for truncated distribution0 | 1

Logical flag for truncated distribution, stored as a logical value. If IsTruncated equals 0, the distribution is not truncated. If IsTruncated equals 1, the distribution is truncated. This property is read-only.

Data Types: logical

NumParametersNumber of parameterspositive integer value

Number of parameters for the probability distribution, stored as a positive integer value. This property is read-only.

Data Types: single | double

ParameterCovarianceCovariance matrix of the parameter estimatesmatrix of scalar values

Covariance matrix of the parameter estimates, stored as a p-by-p matrix, where p is the number of parameters in the distribution. The (i,j) element is the covariance between the estimates of the ith parameter and the jth parameter. The (i,i) element is the estimated variance of the ith parameter. If parameter i is fixed rather than estimated by fitting the distribution to data, then the (i,i) elements of the covariance matrix are 0. This property is read-only.

Data Types: single | double

ParameterDescriptionDistribution parameter descriptionscell array of strings

Distribution parameter descriptions, stored as a cell array of strings. Each cell contains a short description of one distribution parameter. This property is read-only.

Data Types: char

ParameterIsFixedLogical flag for fixed parametersarray of logical values

Logical flag for fixed parameters, stored as an array of logical values. If 0, the corresponding parameter in the ParameterNames array is not fixed. If 1, the corresponding parameter in the ParameterNames array is fixed. This property is read-only.

Data Types: logical

ParameterNamesDistribution parameter namescell array of strings

Distribution parameter names, stored as a cell array of strings. This property is read-only.

Data Types: char

ParameterValuesDistribution parameter valuesvector of scalar values

Distribution parameter values, stored as a vector. This property is read-only.

Data Types: single | double

TruncationTruncation intervalvector of scalar values

Truncation interval for the probability distribution, stored as a vector containing the lower and upper truncation boundaries. This property is read-only.

Data Types: single | double

Methods

Inherited Methods

cdf Cumulative distribution function of probability distribution object
icdfInverse cumulative distribution function of probability distribution object
iqrInterquartile range of probability distribution object
median Median of probability distribution object
pdfProbability density function of probability distribution object
randomGenerate random numbers from probability distribution object
truncateTruncate probability distribution object
meanMean of probability distribution object
negloglikNegative loglikelihood of probability distribution object
paramciConfidence intervals for probability distribution parameters
proflikProfile likelihood function for probability distribution object
std Standard deviation of probability distribution object
varVariance of probability distribution object

Definitions

t Location-Scale Distribution

The t location-scale distribution is useful for modeling data distributions with heavier tails (more prone to outliers) than the normal distribution. It approaches the normal distribution as ν approaches infinity, and smaller values of ν yield heavier tails.

The t location-scale distribution uses the following parameters.

ParameterDescriptionSupport
muLocation parameter<μ<
sigmaScale parameterσ>0
nuShape parameterν>0

The probability density function (pdf) is

f(x|μ,σ,ν)=Γ(ν+12)σνπΓ(ν2)[ν+(xμσ)2ν](ν+12);<x<,

where Γ() is the Gamma function.

Examples

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Create a t Location-Scale Distribution Object Using Default Parameters

Create a t location scale distribution object using the default parameter values.

pd = makedist('tLocationScale')
pd = 

  tLocationScaleDistribution

  t Location-Scale distribution
       mu = 0
    sigma = 1
       nu = 5

Create a t Location-Scale Distribution Object Using Specified Parameters

Create a t location-scale distribution object by specifying the parameter values.

pd = makedist('tLocationScale','mu',-2,'sigma',1,'nu',20)
pd = 

  tLocationScaleDistribution

  t Location-Scale distribution
       mu = -2
    sigma =  1
       nu = 20

Compute the interquartile range of the distribution.

r = iqr(pd)
r =

    1.3739
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