L1pack (version 0.41)

Laplace: 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)

Value

dlaplace, plaplace, and qlaplace are respectively the density, distribution function and quantile function of the Laplace distribution. rlaplace

generates random deviates from the Laplace.

The length of the result is determined by n for rlaplace, and is the maximum of the lengths of the numerical parameters for the other functions.

Arguments

x, q

vector of quantiles.

location, scale

location and scale parameters. Scale 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.

Author

Felipe Osorio and Tymoteusz Wolodzko

Details

If location or scale are not specified, they assume the default values of 0 and 1 respectively.

The Laplace distribution with location \(\mu\) and scale \(\phi\) has density $$ f(x) = \frac{1}{\sqrt{2}\phi} \exp(-\sqrt{2}|x-\mu|/\phi)$$

References

Kotz, S., Kozubowski, T.J., and Podgorski, K. (2001). The Laplace Distributions and Generalizations. Birkhauser, Boston.

Phillips, R.F. (2002). Least absolute deviations estimation via the EM algorithm. Statistics and Computing 12, 281-285.

See Also

Distributions for other standard distributions and rmLaplace for the random generation from the multivariate Laplace distribution.

Examples

x <- rlaplace(1000)
## Q-Q plot for Laplace data against true theoretical distribution:
qqplot(qlaplace(ppoints(1000)), x, main = "Laplace Q-Q plot",
  xlab = "Theoretical quantiles", ylab = "Sample quantiles")
abline(c(0,1), col = "red", lwd = 2)