LaplacesDemon (version 16.1.1)

dist.Student.t: 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 \(\mu\).

sigma

This is the scale parameter \(\sigma\), which must be positive.

nu

This is the degrees of freedom parameter \(\nu\), which must be positive.

lower.tail

Logical. If lower.tail=TRUE, then probabilities are \(Pr[X \le x]\), otherwise, \(Pr[X > x]\).

log, log.p

Logical. If log=TRUE, then the logarithm of the density or probability is returned.

Value

dst gives the density, pst gives the distribution function, qst gives the quantile function, and rst generates random deviates.

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 undefined

• Variance: \(var(\theta) = \frac{\nu}{\nu - 2}\sigma^2\), 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.

See Also

dcauchy, dmvt, dmvtp, dnorm, dnormp, dnormv, dstp, and dt.

Examples

# NOT RUN {
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"))
# }