# Huber: "Huber density" distribution In extraDistr: Additional Univariate and Multivariate Distributions

## Description

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

## Usage

 ```1 2 3 4 5 6 7``` ```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 ≤ 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) = 1/(2 * sqrt(2π) * (Φ(k) + φ(k)/k - 1/2)) * exp(-ρ(x, k))

where

ρ(x, k) = [if abs(x) <= k:] (x^2)/2 [else:] k*abs(x) - (k^2)/2

## 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

 ```1 2 3 4 5 6``` ```x <- rhuber(1e5, 5, 2, 3) hist(x, 100, freq = FALSE) curve(dhuber(x, 5, 2, 3), -20, 20, col = "red", add = TRUE, n = 5000) hist(phuber(x, 5, 2, 3)) plot(ecdf(x)) curve(phuber(x, 5, 2, 3), -20, 20, col = "red", lwd = 2, add = TRUE) ```

extraDistr documentation built on Sept. 7, 2020, 5:09 p.m.