extraDistr (version 1.8.6)

Huber: "Huber density" distribution

Description

Density, distribution function, quantile function and random generation for the "Huber density" distribution.

Usage

dhuber(x, mu = 0, sigma = 1, epsilon = 1.345, log = FALSE)
phuber(q, mu = 0, sigma = 1, epsilon = 1.345, lower.tail = TRUE,
  log.p = FALSE)
qhuber(p, mu = 0, sigma = 1, epsilon = 1.345, lower.tail = TRUE,
  log.p = FALSE)
rhuber(n, mu = 0, sigma = 1, epsilon = 1.345)

Arguments

x, q

vector of quantiles.

mu, sigma, epsilon

location, and scale, and shape parameters. Scale and shape must be positive.

log, log.p

logical; if TRUE, probabilities p are given 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 observations. If length(n) > 1, the length is taken to be the number required.

Details

Huber density is connected to Huber loss and can be defined as:

$$ f(x) = \frac{1-\epsilon}{\sqrt{2\pi}} e^{-\rho_k(x)} $$

where

$$ \rho_k(x) = \left\{\begin{array}{ll} \frac{1}{2} x^2 & |x|\le k \\ k|x|- \frac{1}{2} k^2 & |x|>k \end{array}\right. $$

and \(\epsilon\) satisfies

$$ \frac{2\phi(k)}{k} - 2\Phi(-k) = \frac{\epsilon}{1-\epsilon} $$

References

Huber, P.J. (1964). Robust Estimation of a Location Parameter. Annals of Statistics, 53(1), 73-101.

Huber, P.J. (1981). Robust Statistics. Wiley.

Schumann, D. (2009). Robust Variable Selection. ProQuest.

Examples

x <- rhuber(1e5, 5, 2, 3)
hist(x, 100, freq = FALSE)
curve(dhuber(x, 5, 2, 3), -20, 20, col = "red", add = TRUE)
hist(phuber(x, 5, 2, 3))
plot(ecdf(x))
curve(phuber(x, 5, 2, 3), -20, 20, col = "red", lwd = 2, add = TRUE)