TDist {wiqid}

The Generalized Student's t Distribution

Description

Density, distribution function, quantile function and random generation for the generalised t distribution with df degrees of freedom, using location and scale, or mean and sd. These are wrappers for stats::dt, etc.

Note: In earlier versions of wiqid the scale argument to *t2 functions was incorrectly named sd; they are not the same. These function now give a warning with the correct value of the sd. New *t3 functions do use sd which is only defined for df > 2.

Usage

dt2(x, location, scale, df)
pt2(x, location, scale, df, lower.tail=TRUE, log.p=FALSE)
qt2(p, location, scale, df, lower.tail=TRUE, log.p=FALSE)
rt2(n, location, scale, df)

dt3(x, mean, sd, df)
pt3(x, mean, sd, df, lower.tail=TRUE, log.p=FALSE)
qt3(p, mean, sd, df, lower.tail=TRUE, log.p=FALSE)
rt3(n, mean, sd, df)

Arguments

x

vector of parameter values

location

location of the t-distribution

mean

mean of the t-distribution

scale

scale parameter of the t-distribution

sd

standard deviation of the t-distribution, only defined for df > 2.

df

degrees of freedom

lower.tail

logical; if TRUE (default), cumulative probabilities up to x, otherwise, above x.

log.p

logical; if TRUE, probabilities p are given as log(p).

p

probability.

n

number of random draws required.

Value

dtx gives the density, ptx gives the cumulative probability, qtx gives the quantile function, and rtx generates random deviates.

Author(s)

Mike Meredith

See Also

See the stats functions dt, pt, qt, rt.

Examples

## Plot the t-distribution with varying sd and scale
require(graphics)
xx <- seq(-5, 15, length=201)
density <- dt3(xx, mean=5, sd=1, 4)
plot(xx, density, type='l', lwd=2, main="t-distribution with df = 4")
lines(xx, dt3(xx, 5, 2, 4), lwd=2, col=2)
lines(xx, dt3(xx, 5, 3, 4), lwd=2, col=3)
lines(xx, dt3(xx, 5, 4, 4), lwd=2, col=4)
legend('topleft', paste0("sd = ", 1:4), lwd=2, lty=1, col=1:4,
  bty='n')
lines(xx, dt2(xx, 5, 1, 4), lwd=2, lty=2, col=1)
lines(xx, dt2(xx, 5, 2, 4), lwd=2, lty=2, col=2)
lines(xx, dt2(xx, 5, 3, 4), lwd=2, lty=2, col=3)
lines(xx, dt2(xx, 5, 4, 4), lwd=2, lty=2, col=4)
legend('topright', paste0("scale = ", 1:4), lwd=2, lty=2, col=1:4,
  bty='n')

# Generate random numbers
rand2 <- rt2(1e6, location=5, scale=2, df=4)
mean(rand2)
sd(rand2)  # about 2.83
rand3 <- rt3(1e6, mean=5, sd=2, df=4)
mean(rand3)
sd(rand3)  # close to 2

# check pt* and qt*
prob <- pt2(x=7, location=5, scale=3, df=4)
qt2(p=prob, location=5, scale=3, df=4)