sdlrm (version 0.1.2)

sdl: The Modified Skew Discrete Laplace Distribution

Description

Probability mass function, distribution function, quantile function, and a random generation for the modified skew discrete Laplace (SDL) distribution with mean mu, dispersion parameter phi, and mode xi.

Usage

dsdl(x, mu, phi, xi = 0, log = FALSE)
psdl(q, mu, phi, xi = 0, lower.tail = TRUE)
qsdl(p, mu, phi, xi = 0, lower.tail = TRUE)
rsdl(n, mu, phi, xi = 0)

Value

dsdl returns the probability mass function, psdl

gives the distribution function, qsdl gives the quantile function, and rsdl generates random observations.

Arguments

x, q

vector of integer quantiles.

mu

vector of means.

phi

vector of dispersion parameters (greater than abs(mu - xi)).

xi

the mode of the distribution, an integer value.

log

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 random values to return.

Author

Rodrigo M. R. de Medeiros <>

Details

The SDL distribution was introduced by Kozubowski and Inusah (2006) as the discrete part of the continuous skew Laplace distribution centered at zero (Kotz et al., 2001, Ch. 3). Although the SDL distribution has attractive properties, the discrete version of the zero-centered skew Laplace distribution induces that the mode of the resulting model is always equal to zero.

To overcome this limitation, Medeiros and Bourguignon (2025) proposed to obtain the discrete version of the Laplace skew distribution without setting its location parameter to zero, defining a new probability model that generalizes the SDL distribution.

This set of functions represents the probability mass function, the cumulative distribution function, the quantile function, and a random number generator for the modified SDL distribution parameterized in terms of mu (mean), phi (a dispersion parameter), and xi (the mode of the distribution).

Let \(X\) be a discrete random variable following a SDL distribution with mean \(\mu\), dispersion parameter \(\phi\), and mode \(\xi\). The probability mass function of X is

\( \textrm{P}(X = x) = \left\{\begin{array}{ll} \dfrac{1}{1 + \phi}\left(\dfrac{\phi - \mu + \xi}{2+ \phi - \mu + \xi}\right)^{-(x - \xi)}, & x \in \{\xi - 1, \xi - 2, \ldots\}, \\ \\ \dfrac{1}{1 + \phi}\left(\dfrac{\phi + \mu - \xi}{2+ \phi + \mu - \xi}\right)^{x - \xi}, & x \in \{\xi, \xi + 1, \xi + 2, \ldots\}. \end{array}\right. \)

The parametric space of this parameterization satisfies the constraint \(\mu \in \mathbb{R}\), \(\phi > |\mu - \xi| \), and \(\xi \in \mathbb{Z}\). Additionally, the expected value and the variance of \(X\) are given, respectively, by

\( \textrm{E}(Y) = \mu \quad \mbox{ and } \quad \textrm{Var}(Y) = \dfrac{\phi(\phi + 2) + (\mu - \xi)^2}{2}. \)

References

Kotz, S., Kozubowski, T. J., and Podgórski, K. (2001). The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance. Birkhauser, Boston.

Kozubowski, T. J., and Inusah, S. (2006). A skew Laplace distribution on integers. Annals of the Institute of Statistical Mathematics, 58, 555---571.

Medeiros, R. M. R., and Bourguignon, M. (2025). Modified skew discrete Laplace regression models for integer valued data with applications to paired samples. Manuscript submitted for publication.

Examples

### Probability function ###

# Parameters
mu <- c(-4, 2, 4)
phi <- 6.5
xi <- 2

xvals <- -30:30

# Skewed-left distribution (mu < xi)
plot(xvals, dsdl(xvals, mu[1], phi, xi),
     type = "h", xlab = "x", ylab = "Pmf")

# Symmetric distribution (mu = xi)
plot(xvals, dsdl(xvals, mu[2], phi, xi),
     type = "h", xlab = "x", ylab = "Pmf")

# Skewed-right distribution (mu > 0)
plot(xvals, dsdl(xvals, mu[3], phi, xi),
     type = "h", xlab = "x", ylab = "Pmf")

### Difference between paired samples of non-negative observations  ###

# Parameters
mu <- 3
phi <- 4
xi <- 0

# Paired samples of a pre-post treatment experimental study
before <- rgeom(1000, 2 / (2 + phi - mu))
after <- rgeom(1000, 2 / (2 + phi + mu))

# Response variable
y <- after - before

# Barplot
obj <- barplot(prop.table(table(y)),
               xlab = "Response",
               ylab = "Proportion",
               col = "white",
               ylim = c(0, mean(y == 0) + 0.01))

# Sdl model for the differences
points(obj, dsdl(sort(unique(y)), mu, phi, xi), col = "red", pch = 16)