t Location-Scale Distribution

Overview

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.

Parameters

The t location-scale distribution uses the following parameters.

Parameter	Description	Support
μ	Location parameter	–∞ < μ < ∞
σ	Scale parameter	σ > 0
ν	Shape parameter	ν > 0

To estimate distribution parameters, use mle. Alternatively, fit a tLocationScaleDistribution object to data using fitdist or the Distribution Fitter app.

Probability Density Function

The probability density function (pdf) of the t location-scale distribution is

$$\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\sigma \sqrt{\nu \pi }\Gamma \left(\frac{\nu }{2}\right)}{\left[\frac{\nu +{\left(\frac{x-\mu }{\sigma }\right)}^2}{\nu }\right]}^{-\left(\frac{\nu +1}{2}\right)}$$

where Γ( • ) is the gamma function, µ is the location parameter, σ is the scale parameter, and ν is the shape parameter.

To compute the probability density function, use pdf and specify 'tLocationScale'. Alternatively, you can create a tLocationScaleDistribution object using fitdist or makedist, then use the pdf to work with the object.

Cumulative Distribution Function

To compute the probability density function, use cdf and specify 'tLocationScale'. Alternatively, you can create a tLocationScaleDistribution object using fitdist or makedist, then use the cdf to work with the object.

Descriptive Statistics

The mean of the t location-scale distribution is

$$\text{mean}=\mu ,$$

where μ is the location parameter. The mean is only defined for shape parameter values ν > 1. For other values of ν, the mean is undefined.

The variance of the t location-scale distribution is

$$\mathrm{var}={\sigma }^2\frac{\nu }{\nu -2},$$

where μ is the location parameter and ν is the shape parameter. The variance is only defined for values of ν > 2. For other values of ν, the variance is undefined.

To compute the mean and variance, create a tLocationScaleDistribution object using fitdist or makedist. You can also use the Distribution Fitter app.

Relationship to Other Distributions

If x has a t location-scale distribution, with parameters µ, σ, and ν, then

$$\frac{x-\mu }{\sigma }$$

has a Student's t distribution with ν degrees of freedom.