laplaceUC {VGAM} | R Documentation

## The Laplace Distribution

### Description

Density, distribution function, quantile function and random generation for the Laplace distribution with location parameter `location` and scale parameter `scale`.

### Usage

    dlaplace(x, location = 0, scale = 1, log = FALSE)
    plaplace(q, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
    qlaplace(p, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
    rlaplace(n, location = 0, scale = 1)

### Arguments

`x, q` | vector of quantiles.
`p` | vector of probabilities.
`n` | number of observations. Same as in `runif`.
`location` | the location parameter a, which is the mean.
`scale` | the scale parameter b. Must consist of positive values.
`log` | Logical. If `log = TRUE` then the logarithm of the density is returned.
`lower.tail, log.p` | Same meaning as in `pnorm` or `qnorm`.

### Details

The Laplace distribution is often known as the double-exponential distribution and, for modelling, has heavier tail than the normal distribution. The Laplace density function is

f(y) = (1/(2b)) exp( -|y-a|/b )

where -Inf<y<Inf, -Inf<a<Inf and b>0. The mean is a and the variance is 2b^2.

See `laplace`, the VGAM family function for estimating the two parameters by maximum likelihood estimation, for formulae and details. Apart from `n`, all the above arguments may be vectors and are recyled to the appropriate length if necessary.

### Value

`dlaplace` gives the density, `plaplace` gives the distribution function, `qlaplace` gives the quantile function, and `rlaplace` generates random deviates.

### Author(s)

T. W. Yee and Kai Huang

### References

Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011) Statistical Distributions, Hoboken, NJ, USA: John Wiley and Sons, Fourth edition.

### See Also

`laplace`.

### Examples

    loc <- 1; b <- 2
    y <- rlaplace(n = 100, loc = loc, scale = b)
    mean(y)  # sample mean
    loc      # population mean
    var(y)   # sample variance
    2 * b^2  # population variance

    ## Not run:  loc <- 0; b <- 1.5; x <- seq(-5, 5, by = 0.01)
    plot(x, dlaplace(x, loc, b), type = "l", col = "blue", ylim = c(0,1),
         main = "Blue is density, orange is cumulative distribution function",
         sub = "Purple are 5,10,...,95 percentiles", las = 1, ylab = "")
    abline(h = 0, col = "blue", lty = 2)
    lines(qlaplace(seq(0.05,0.95,by = 0.05), loc, b),
          dlaplace(qlaplace(seq(0.05, 0.95, by = 0.05), loc, b), loc, b),
          col = "purple", lty = 3, type = "h")
    lines(x, plaplace(x, loc, b), type = "l", col = "orange")
    abline(h = 0, lty = 2)
    ## End(Not run)

    plaplace(qlaplace(seq(0.05, 0.95, by = 0.05), loc, b), loc, b)

[Package VGAM version 1.1-1 Index]