Student t Distribution: Univariate

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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

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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 <= x], otherwise, Pr[X > x].

log, log.p

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

Details

  • Application: Continuous Univariate

  • Density: p(theta) = [{Gamma[(nu + 1) / 2]} / Gamma(nu/2)]*sqrt(nu*pi)*sigma*[1 + (1/nu)*[(theta - mu)/sigma]^2]^[(-nu+ 1)/2]

  • Inventor: William Sealy Gosset (1908)

  • Notation 1: theta ~ t(mu, sigma,nu)

  • Notation 2: p(theta) = 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) = [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 -> infinity, the Student t-distribution approaches 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

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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"))

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