# tLocationScaleDistribution

t Location-Scale probability distribution object

## Description

A `tLocationScaleDistribution` object consists of parameters, a model description, and sample data for a t location-scale probability 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
`mu`Location parameter$-\infty <\mu <\infty$
`sigma`Scale parameter$\sigma >0$
`nu`Shape parameter$\nu >0$

## Creation

There are several ways to create a `tLocationScaleDistribution` probability distribution object.

## Properties

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### Distribution Parameters

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

Data Types: `single` | `double`

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

Data Types: `single` | `double`

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

Data Types: `single` | `double`

### Distribution Characteristics

Logical flag for truncated distribution, specified as a logical value. If `IsTruncated` equals `0`, the distribution is not truncated. If `IsTruncated` equals `1`, the distribution is truncated.

Data Types: `logical`

Number of parameters for the probability distribution, specified as a positive integer value.

Data Types: `double`

Covariance matrix of the parameter estimates, specified 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 `i`th parameter and the `j`th parameter. The (`i`,`i`) element is the estimated variance of the `i`th 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.

Data Types: `double`

Logical flag for fixed parameters, specified 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.

Data Types: `logical`

Distribution parameter values, specified as a vector of scalar values.

Data Types: `single` | `double`

Truncation interval for the probability distribution, specified as a vector of scalar values containing the lower and upper truncation boundaries.

Data Types: `single` | `double`

### Other Object Properties

Probability distribution name, specified as a character vector.

Data Types: `char`

Data used for distribution fitting, specified 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.

Data Types: `struct`

Distribution parameter descriptions, specified as a cell array of character vectors. Each cell contains a short description of one distribution parameter.

Data Types: `char`

Distribution parameter names, specified as a cell array of character vectors.

Data Types: `char`

## Object Functions

 `cdf` Cumulative distribution function `gather` Gather properties of Statistics and Machine Learning Toolbox object from GPU `icdf` Inverse cumulative distribution function `iqr` Interquartile range of probability distribution `mean` Mean of probability distribution `median` Median of probability distribution `negloglik` Negative loglikelihood of probability distribution `paramci` Confidence intervals for probability distribution parameters `pdf` Probability density function `plot` Plot probability distribution object `proflik` Profile likelihood function for probability distribution `random` Random numbers `std` Standard deviation of probability distribution `truncate` Truncate probability distribution object `var` Variance of probability distribution

## Examples

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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 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 ```

## Version History

Introduced in R2013a
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