dist.Student.t {LaplacesDemon} | R Documentation |

## Student t Distribution: Univariate

### Description

These functions provide the density, distribution function, quantile
function, and random generation for the univariate Student t
distribution with location parameter `\mu`

, scale parameter
`\sigma`

, and degrees of freedom parameter `\nu`

.

### Usage

```
dst(x, mu=0, sigma=1, nu=10, log=FALSE)
pst(q, mu=0, sigma=1, nu=10, lower.tail=TRUE, log.p=FALSE)
qst(p, mu=0, sigma=1, nu=10, lower.tail=TRUE, log.p=FALSE)
rst(n, mu=0, sigma=1, nu=10)
```

### Arguments

`x` , `q` |
These are each a vector of quantiles. |

`p` |
This is a vector of probabilities. |

`n` |
This is the number of observations, which must be a positive integer that has length 1. |

`mu` |
This is the location parameter |

`sigma` |
This is the scale parameter |

`nu` |
This is the degrees of freedom parameter |

`lower.tail` |
Logical. If |

`log` , `log.p` |
Logical. If |

### Details

Application: Continuous Univariate

Density:

`p(\theta) = \frac{\Gamma[(\nu+1)/2]}{\Gamma(\nu/2)} \sqrt{\nu \pi} \sigma[1 + \frac{1}{\nu}[\frac{\theta - \mu}{\sigma}]^2]^{(-\nu + 1)/2}`

Inventor: William Sealy Gosset (1908)

Notation 1:

`\theta \sim \mathrm{t}(\mu, \sigma, \nu)`

Notation 2:

`p(\theta) = \mathrm{t}(\theta | \mu, \sigma, \nu)`

Parameter 1: location parameter

`\mu`

Parameter 2: scale parameter

`\sigma > 0`

Parameter 3: degrees of freedom

`\nu > 0`

Mean:

`E(\theta) = \mu`

, for`\nu > 1`

, otherwise undefinedVariance:

`var(\theta) = \frac{\nu}{\nu - 2}\sigma^2`

, for`\nu > 2`

Mode:

`mode(\theta) = \mu`

The Student t-distribution is often used as an alternative to the normal distribution as a model for data. It is frequently the case that real data have heavier tails than the normal distribution allows for. The classical approach was to identify outliers and exclude or downweight them in some way. However, it is not always easy to identify outliers (especially in high dimensions), and the Student t-distribution is a natural choice of model-form for such data. It provides a parametric approach to robust statistics.

The degrees of freedom parameter, `\nu`

, controls the kurtosis
of the distribution, and is correlated with the scale parameter
`\sigma`

. The likelihood can have multiple local maxima
and, as such, it is often necessary to fix `\nu`

at a fairly
low value and estimate the other parameters taking this as given.
Some authors report that values between 3 and 9 are often good
choices, and some authors suggest 5 is often a good choice.

In the limit `\nu \rightarrow \infty`

, the
Student t-distribution approaches
`\mathcal{N}(\mu, \sigma^2)`

. The
case of `\nu = 1`

is the Cauchy distribution.

The `pst`

and `qst`

functions are similar to those in the
`gamlss.dist`

package.

### Value

`dst`

gives the density,
`pst`

gives the distribution function,
`qst`

gives the quantile function, and
`rst`

generates random deviates.

### See Also

`dcauchy`

,
`dmvt`

,
`dmvtp`

,
`dnorm`

,
`dnormp`

,
`dnormv`

,
`dstp`

, and
`dt`

.

### Examples

```
library(LaplacesDemon)
x <- dst(1,0,1,10)
x <- pst(1,0,1,10)
x <- qst(0.5,0,1,10)
x <- rst(100,0,1,10)
#Plot Probability Functions
x <- seq(from=-5, to=5, by=0.1)
plot(x, dst(x,0,1,0.1), ylim=c(0,1), type="l", main="Probability Function",
ylab="density", col="red")
lines(x, dst(x,0,1,1), type="l", col="green")
lines(x, dst(x,0,1,10), type="l", col="blue")
legend(1, 0.9, expression(paste(mu==0, ", ", sigma==1, ", ", nu==0.5),
paste(mu==0, ", ", sigma==1, ", ", nu==1),
paste(mu==0, ", ", sigma==1, ", ", nu==10)),
lty=c(1,1,1), col=c("red","green","blue"))
```

*LaplacesDemon*version 16.1.6 Index]