Student t distribution with location and scale

Density, distribution function, quantile function, and random generation for the t distribution with location and scale parameters.

Usage

dt2(x, mu, sigma, df, log = FALSE)

pt2(q, mu, sigma, df, lower.tail = TRUE, log.p = FALSE)

rt2(n, mu, sigma, df)

qt2(p, mu, sigma, df, lower.tail = TRUE, log.p = FALSE)

pt.ad(q, df)

Arguments

x, q

vector of quantiles

mu

location parameter

sigma

scale parameter, must be positive.

df

degrees of freedom, must be positive.

log, log.p

logical; if TRUE, probabilities/ densities \(p\) are returned as \(\log(p)\).

lower.tail

logical; if TRUE (default), probabilities are \(P[X \le x]\), otherwise \(P[X > x]\).

n

number of random values to return.

p

vector of probabilities

Value

dt2 gives the density, pt2 gives the distribution function, qt2 gives the quantile function, and rt2 generates random deviates.

pt.ad is an AD-compatible replacement for stats::pt. It is used internally and is reached automatically whenever an argument is an AD variable, so stats::pt is left untouched for ordinary use.

Details

This implementation of dt2 allows for automatic differentiation with RTMB.

$$f(x;\,\mu,\sigma,\nu) = \frac{1}{\sigma}\,f_t\!\left(\frac{x-\mu}{\sigma};\,\nu\right),$$ where \(f_t(\cdot;\nu)\) is the Student-\(t\) PDF with \(\nu\) degrees of freedom.

Examples

x <- rt2(1, 1, 2, 5)
d <- dt2(x, 1, 2, 5)
p <- pt2(x, 1, 2, 5)
q <- qt2(p, 1, 2, 5)