Shifted (3-Parameter) Loglogistic Distribution — shifted

Density, distribution function, quantile function and random generation for the shifted log-logistic (generalised log-logistic) distribution with location mu (real), scale sigma > 0 and shape xi (real). The two-parameter logistic distribution is recovered as xi -> 0.

Usage

dshifted_loglogistic(x, mu = 0, sigma = 1, xi = 0, log = FALSE)

pshifted_loglogistic(
  q,
  mu = 0,
  sigma = 1,
  xi = 0,
  lower.tail = TRUE,
  log.p = FALSE
)

qshifted_loglogistic(
  p,
  mu = 0,
  sigma = 1,
  xi = 0,
  lower.tail = TRUE,
  log.p = FALSE
)

rshifted_loglogistic(n, mu = 0, sigma = 1, xi = 0)

Arguments

x, q: Numeric vector of quantiles.
mu: Location parameter (real; equals the median).
sigma: Scale parameter (sigma > 0).
xi: Shape parameter (real; xi = 0 gives the logistic distribution).
log, log.p: Logical. See Distributions.
lower.tail: Logical. See Distributions.
p: Vector of probabilities.
n: Number of random draws.

Value

Numeric vector.

Parameterisation

This implementation uses the GEV-style parameterisation of Hosking & Wallis (1997) and the Flood Estimation Handbook (Robson & Reed 1999), in which $\mu$ is a pure location parameter (the median), $\sigma$ a pure scale parameter and $\xi$ a pure shape parameter: $$F(x \mid \mu, \sigma, \xi) = \{1 + (1 + \xi z)^{-1/\xi}\}^{-1}, \qquad z = (x - \mu) / \sigma,$$ with corresponding density $$f(x \mid \mu, \sigma, \xi) = \frac{(1 + \xi z)^{-(1/\xi + 1)}} {\sigma \{1 + (1 + \xi z)^{-1/\xi}\}^{2}}.$$

The support depends on $\xi$:

$\xi > 0$: $x \ge \mu - \sigma/\xi$ (bounded below).
$\xi < 0$: $x \le \mu - \sigma/\xi$ (bounded above).
$\xi = 0$: $x \in \mathbb{R}$ (logistic limit).

The median is always $\mu$; the mean exists when $|\xi| < 1$ and is $\mu + \sigma (\alpha\csc\alpha - 1)/\xi$, $\alpha = \pi\xi$. Reducing further, the family contains:

the standard log-logistic when $\xi = 1$ (reparameterised);
the logistic distribution as $\xi \to 0$;
the generalised Pareto family at $\xi = -1$.

Why this parameterisation? An alternative "simple-shift" form, $Y - \delta \sim \mathrm{LogLogistic}$, exists in the literature (Geskus 2001) and is closer in spirit to brms::shifted_lognormal()'s positive shift ndt. We deliberately follow the GEV-style parameterisation because

it provides a smooth limit to the logistic distribution at $\xi = 0$;
the parameters $(\mu, \sigma, \xi)$ are orthogonally interpretable (location / scale / shape);
it is the canonical form in hydrology and extreme-value applications (Hosking & Wallis 1997).

References

Geskus, R. B. (2001). Methods for estimating the AIDS incubation time distribution when date of seroconversion is censored. Statistics in Medicine, 20(5), 795-812.

Hosking, J. R. M., & Wallis, J. R. (1997). Regional Frequency Analysis: An Approach Based on L-Moments. Cambridge University Press. ISBN 0-521-43045-3.

Robson, A., & Reed, D. (1999). Flood Estimation Handbook, Volume 3: Statistical Procedures for Flood Frequency Estimation. Institute of Hydrology, Wallingford, UK.

Examples

dshifted_loglogistic(c(1, 2, 5),    mu = 0, sigma = 1, xi = 0.5)
#> [1] 0.14201183 0.08000000 0.01993592
pshifted_loglogistic(c(1, 2, 5),    mu = 0, sigma = 1, xi = 0.5)
#> [1] 0.6923077 0.8000000 0.9245283
qshifted_loglogistic(c(0.25, 0.75), mu = 0, sigma = 1, xi = 0.5)
#> [1] -0.8452995  1.4641016
set.seed(1); rshifted_loglogistic(5, mu = 0, sigma = 1, xi = 0.5)
#> [1] -0.7975251 -0.4602975  0.3161316  4.2910001 -0.9947467