# alaplaceUC: The Laplace Distribution In VGAM: Vector Generalized Linear and Additive Models

 alaplaceUC R Documentation

## The Laplace Distribution

### Description

Density, distribution function, quantile function and random generation for the 3-parameter asymmetric Laplace distribution with location parameter `location`, scale parameter `scale`, and asymmetry parameter `kappa`.

### Usage

```dalap(x, location = 0, scale = 1, tau = 0.5, kappa = sqrt(tau/(1-tau)),
log = FALSE)
palap(q, location = 0, scale = 1, tau = 0.5, kappa = sqrt(tau/(1-tau)),
lower.tail = TRUE, log.p = FALSE)
qalap(p, location = 0, scale = 1, tau = 0.5, kappa = sqrt(tau/(1-tau)),
lower.tail = TRUE, log.p = FALSE)
ralap(n, location = 0, scale = 1, tau = 0.5, kappa = sqrt(tau/(1-tau)))
```

### Arguments

 `x, q` vector of quantiles. `p` vector of probabilities. `n` number of observations. If `length(n) > 1` then the length is taken to be the number required. `location` the location parameter xi. `scale` the scale parameter sigma. Must consist of positive values. `tau` the quantile parameter tau. Must consist of values in (0,1). This argument is used to specify `kappa` and is ignored if `kappa` is assigned. `kappa` the asymmetry parameter kappa. Must consist of positive values. `log` if `TRUE`, probabilities `p` are given as `log(p)`. `lower.tail, log.p` Same meaning as in `pnorm` or `qnorm`.

### Details

There are many variants of asymmetric Laplace distributions (ALDs) and this one is known as the ALD by Kotz et al. (2001). See `alaplace3`, the VGAM family function for estimating the three parameters by maximum likelihood estimation, for formulae and details. The ALD density may be approximated by `dextlogF`.

### Value

`dalap` gives the density, `palap` gives the distribution function, `qalap` gives the quantile function, and `ralap` generates random deviates.

### Author(s)

T. W. Yee and Kai Huang

### References

Kotz, S., Kozubowski, T. J. and Podgorski, K. (2001). The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance, Boston: Birkhauser.

`alaplace3`, `dextlogF`, `extlogF1`.

### Examples

```x <- seq(-5, 5, by = 0.01)
loc <- 0; sigma <- 1.5; kappa <- 2
## Not run:  plot(x, dalap(x, loc, sigma, kappa = kappa), type = "l",
main = "Blue is density, orange is the CDF",
ylim = c(0, 1), sub = "Purple are 5, 10, ..., 95 percentiles",
las = 1, ylab = "", cex.main = 0.5, col = "blue")
abline(h = 0, col = "blue", lty = 2)
lines(qalap(seq(0.05, 0.95, by = 0.05), loc, sigma, kappa = kappa),
dalap(qalap(seq(0.05, 0.95, by = 0.05), loc, sigma, kappa = kappa),
loc, sigma, kappa = kappa), col="purple", lty=3, type = "h")
lines(x, palap(x, loc, sigma, kappa = kappa), type = "l", col = "orange")
abline(h = 0, lty = 2)
## End(Not run)

pp <- seq(0.05, 0.95, by = 0.05)  # Test two functions
max(abs(palap(qalap(pp, loc, sigma, kappa = kappa),
loc, sigma, kappa = kappa) - pp))  # Should be 0
```

VGAM documentation built on July 6, 2022, 5:05 p.m.