R: Huber's Least Favourable Distribution

dhuber {VGAM} | R Documentation

## Huber's Least Favourable Distribution

### Description

Density, distribution function, quantile function and random generation for Huber's least favourable distribution, see Huber and Ronchetti (2009).

### Usage

     dhuber(x, k = 0.862, mu = 0, sigma = 1, log = FALSE)
    edhuber(x, k = 0.862, mu = 0, sigma = 1, log = FALSE)
     rhuber(n, k = 0.862, mu = 0, sigma = 1)
     qhuber(p, k = 0.862, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
     phuber(q, k = 0.862, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)

### Arguments

`x, q` | numeric vector, vector of quantiles.
`p` | vector of probabilities.
`n` | number of random values to be generated. If `length(n) > 1` then the length is taken to be the number required.
`k` | numeric. Borderline value of central Gaussian part of the distribution. This is known as the tuning constant, and should be positive. For example, `k = 0.862` refers to a 20% contamination neighborhood of the Gaussian distribution. If `k = 1.40` then this is 5% contamination.
`mu` | numeric. distribution mean.
`sigma` | numeric. Distribution scale (`sigma = 1` defines the distribution in standard form, with standard Gaussian centre).
`log` | Logical. If `log = TRUE` then the logarithm of the result is returned.
`lower.tail, log.p` | Same meaning as in `pnorm` or `qnorm`.

### Details

Details are given in `huber2`, the VGAM family function for estimating the parameters `mu` and `sigma`.

### Value

`dhuber` gives out a vector of density values.

`edhuber` gives out a list with components `val` (density values) and `eps` (contamination proportion).

`rhuber` gives out a vector of random numbers generated by Huber's least favourable distribution.

`phuber` gives the distribution function, `qhuber` gives the quantile function.

### Author(s)

Christian Hennig wrote `[d,ed,r]huber()` (from smoothmest) and slight modifications were made by T. W. Yee to replace looping by vectorization and addition of the `log` argument. Arash Ardalan wrote `[pq]huber()`, and two arguments for these were implemented by Kai Huang. This helpfile was adapted from smoothmest.

### See Also

`huber2`.

### Examples

    set.seed(123456)
    edhuber(1:5, k = 1.5)
    rhuber(5)

    ## Not run:  mu <- 3; xx <- seq(-2, 7, len = 100)  # Plot CDF and PDF
    plot(xx, dhuber(xx, mu = mu), type = "l", col = "blue", las = 1, ylab = "",
         main = "blue is density, orange is cumulative distribution function",
         sub = "Purple lines are the 10,20,...,90 percentiles",
         ylim = 0:1)
    abline(h = 0, col = "blue", lty = 2)
    lines(xx, phuber(xx, mu = mu), type = "l", col = "orange")
    probs <- seq(0.1, 0.9, by = 0.1)
    Q <- qhuber(probs, mu = mu)
    lines(Q, dhuber(Q, mu = mu), col = "purple", lty = 3, type = "h")
    lines(Q, phuber(Q, mu = mu), col = "purple", lty = 3, type = "h")
    abline(h = probs, col = "purple", lty = 3)
    phuber(Q, mu = mu) - probs  # Should be all 0s

    ## End(Not run)

[Package VGAM version 1.1-3 Index]