PL {PLreg}

Power Logit Distributions

Description

Density, distribution function, quantile function and random generation for power logit distributions.

Usage

dPL(x, mu, sigma, lambda, zeta = 2, family, log = FALSE)

pPL(q, mu, sigma, lambda, zeta = 2, family, lower.tail = TRUE, log.p = FALSE)

qPL(p, mu, sigma, lambda, zeta = 2, family, lower.tail = TRUE, log.p = FALSE)

rPL(n, mu, sigma, lambda, zeta = 2, family)

Arguments

x, q

vector of quantiles.

mu

vector of medians.

sigma

vector of dispersion parameters.

lambda

vector of skewness parameters.

zeta

vector of extra parameters.

family

string that specifies the family used to define the power logit distribution. The family is NO, TF, LO, PE, SHN, Hyp and SLASH for normal, Student-t, type II logistic, power exponential, sinh-normal, hyperbolic and slash distribution, respectively.

log, log.p

logical; if TRUE, probabilities p are given as log(p). Default is FALSE.

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.

Details

If zeta is not specified, it assumes the default value 2.
The power logit distribution has density

f(y; \mu, \sigma, \lambda) = \lambda r(z^2)/(\sigma y (1-y^\lambda)),

for y \in (0,1), in which z = [logit(y^\lambda) - logit(\mu^\lambda)]/\sigma, r(\cdot) is the density generator and \mu \in (0,1), \sigma>0 and \lambda>0 are the median, dispersion and skewness of the distribution.
It is possible to consider \lambda=0 to obtain the limiting case, the log-log distribution. This distribution has density

f(y; \mu, \sigma, \lambda) = r(z^2)/(\sigma y (-log(y))),

for y \in (0,1), in which z = [-log(-log(y)) - (-log(-log(y)))]/\sigma.
The family argument defines the density generator r(\cdot), which may depend on an extra parameter (zeta).

Value

dPL gives the density, pPL gives the distribution function, qPL gives the quantile function, and rPL generates random variables.

References

Queiroz, F. F. and Ferrari, S. L. P. (2022). Power logit regression for modeling bounded data. arXiv preprint arXiv:2202.01697.

Examples

dPL(0.2, mu = 0.3, sigma = 1, lambda=1, zeta = 2, family = "PE")
mu = 0.3; sigma = 1; lambda = 2
set.seed(1)
PLsample = rPL(1000, mu, sigma, lambda, family = "SN",  zeta = 2.5)
hist(PLsample, prob = TRUE, breaks = 15, main = "", las = 1)
curve(dPL(x, mu, sigma, lambda, family = "SN",  zeta = 2.5),
      from = 0.01, to = 0.8, add = TRUE, col = "red")
rug(PLsample)

x = seq(0.01, 0.9,0.01)
y = dPL(x, mu, sigma, lambda, family = "Hyp",  zeta = 2)
plot(x, y, type = "l", lwd = 2, las = 1)

x1 = seq(0.01, 0.4, 0.01)
y1 = dPL(x1, mu, sigma, lambda, family = "Hyp",  zeta = 2)
polygon(c(x1, 0.4, 0), c(y1, 0, 0), col = "lightblue")
text(mu-0.025, 1, paste("P(Y<0.4) = ",
                        round(pPL(0.4, mu, sigma, lambda,
                        family = "Hyp",
                        zeta = 2),2)),
     font = 1, cex = 0.8)

plot(x, pPL(x, mu, sigma, lambda, family = "PE",  zeta = 1.3),
     type = "l", las = 1, lwd = 2,
     ylab = expression(P(Y<y)),
     xlab = "y")
p = pPL(0.5, mu, sigma, lambda, family = "PE",  zeta = 1.3)
q = qPL(p, mu, sigma, lambda, family = "PE",  zeta = 1.3)
points(q, p, pch = 16, col = 2, cex = 1.5)
text(0.55, 0.83, paste("(", 0.5, ",", round(p, 2), ")"), font = 2,
     cex = 0.8, col = "red")