Half-t distribution

Density, distribution function, quantile function, and random generation for the half-t distribution.

Usage

dhalft(x, df, sigma = 1, log = FALSE)

phalft(q, df, sigma = 1, lower.tail = TRUE, log.p = FALSE)

qhalft(p, df, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rhalft(n, df, sigma = 1)

Arguments

x, q: vector of quantiles.
df: degrees of freedom, must be positive.
sigma: scale parameter, 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 $P[X > x]$.
p: vector of probabilities.
n: number of random values to return.

Value

dhalft gives the density, phalft gives the distribution function, qhalft gives the quantile function, and rhalft generates random deviates.

Details

dhalft and phalft allow for automatic differentiation with RTMB, with respect to df as well as sigma.

The half-t is the distribution of $|Y|$ for $Y = \sigma T$ and $T$ Student t on $\nu$ degrees of freedom: $$f(x;\,\nu,\sigma) = \frac{2}{\sigma} f_T(x/\sigma;\, \nu), \quad x \ge 0,$$ with distribution function $F(x) = 2 F_T(x/\sigma;\, \nu) - 1$.

Together with the half-Cauchy, which is the case df = 1, this is the standard weakly informative prior for the standard deviation of a hierarchical model, recommended by Gelman (2006) in place of the inverse-gamma. The degrees of freedom set how heavy the tail is: small df leaves large values essentially unpenalised, and as df grows the distribution approaches the half-normal, which is the folded normal with mu = 0.

The mean $$E(X) = \frac{2\sigma\sqrt{\nu}\,\Gamma\bigl(\tfrac{\nu+1}{2}\bigr)}{\sqrt{\pi}\,(\nu - 1)\,\Gamma\bigl(\tfrac{\nu}{2}\bigr)}$$ exists only for $\nu > 1$ and the variance $\sigma^2 \nu / (\nu - 2) - E(X)^2$ only for $\nu > 2$.

References

Gelman, A. (2006) Prior distributions for variance parameters in hierarchical models. Bayesian Analysis, 1, 515-534, doi:10.1214/06-BA117A.

Examples

set.seed(123)
x <- rhalft(5, df = 3, sigma = 2)
d <- dhalft(x, df = 3, sigma = 2)
p <- phalft(x, df = 3, sigma = 2)
q <- qhalft(p, df = 3, sigma = 2)