t Location-Scale Distribution
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- Fit1 (data,'t',start=list(df=3)) fitdistis using the R functions rt,dt,ptand qt. But those functions do not support a location and scale parameter (and hence the above code will only optimise the df parameter and provide a very bad fit). So the solution is to use a version of the t-distribution that does provide the parameters you want.
A scale family of distributions has densities of the form g(xj˙) = 1 ˙ x ˙ where ˙0. ˙is the scale parameter. A location family of distributions has densities of the form g(x j ) = (x ) where 1 location family and N(0;˙2) distributions form a scale family.
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
Multivariate T Location Scale
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
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
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
has a Student's t distribution with ν degrees of freedom.
See Also
Related Topics
Scaled and shifted t distribution.
Student's t distribution for 'df' degrees of freedom, shifted by 'mean' andscaled by 'sd'.
- Keywords
- distribution
Usage
Arguments
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
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
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
has a Student's t distribution with ν degrees of freedom.
See Also
Related Topics
Scaled and shifted t distribution.
Student's t distribution for 'df' degrees of freedom, shifted by 'mean' andscaled by 'sd'.
- Keywords
- distribution
Usage
Arguments
vector of quantiles.
vector of probabilities.
number of observations. If length(n) > 1, the length is taken to be the number required.
degrees of freedom (> 0, maybe non-integer). df = Inf is allowed.
mean value for the shifted, scaled distribution.
Scale factor for the shifted, scaled distribution.
non-centrality parameter delta; currently except for rt(), only for abs(ncp) <= 37.62. If omitted, use the central t distribution.
logical; if TRUE (default), probabilities are P[X <= x]; otherwise, P[X > x].
Student T Location Scale
logical; if TRUE, probabilities p are given as log(p).
Details
These are wrappers for the corresponding t distribution functions in package stats.
The scaled, shifted t distribution has mean mean and variance sd^2 * df/(df-2)
T Location Scale Distribution
The scaled, shifted t distribution is used for Monte Carlo evaluation when a value x has been assigned a standard uncertainty u associated with with df degrees of freedom; the corresponding distribution function for that is then t.scaled withmean=x, sd=u and df=df.
Value
dt.scaled gives the density, pt.scaled gives the distribution function, qt.scaled gives the quantile function, and rt.scaled generates random deviates.
Invalid arguments will result in return value NaN, with a warning.
See Also
Aliases
- dt.scaled
- pt.scaled
- qt.scaled
- rt.scaled
Examples
Documentation reproduced from package metRology, version 0.9-28-1, License: GPL (>= 2)Community examples
