# glogis

##### The Generalized Logistic Distribution (Type I: Skew-Logitic)

Density, distribution function, quantile function and random
generation for the logistic distribution with parameters
`location`

and `scale`

.

- Keywords
- distribution

##### Usage

```
dglogis(x, location = 0, scale = 1, shape = 1, log = FALSE)
pglogis(q, location = 0, scale = 1, shape = 1, lower.tail = TRUE, log.p = FALSE)
qglogis(p, location = 0, scale = 1, shape = 1, lower.tail = TRUE, log.p = FALSE)
rglogis(n, location = 0, scale = 1, shape = 1)
sglogis(x, location = 0, scale = 1, shape = 1)
```

##### Arguments

- x, q
vector of quantiles.

- p
vector of probabilities.

- n
number of observations. If

`length(n) > 1`

, the length is taken to be the number required.- location, scale, shape
location, scale, and shape parameters (see below).

- 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, \(P[X > x]\).

##### Details

If `location`

, `scale`

, or `shape`

are omitted, they assume the
default values of `0`

, `1`

, and `1`

, respectively.

The generalized logistic distribution with `location`

\(= \mu\),
`scale`

\(= \sigma\), and `shape`

\(= \gamma\) has distribution function
$$
F(x) = \frac{1}{(1 + e^{-(x-\mu)/\sigma})^\gamma}%
$$.

The mean is given by `location + (digamma(shape) - digamma(1)) * scale`

, the variance by
`(psigamma(shape, deriv = 1) + psigamma(1, deriv = 1)) * scale^2)`

and the skewness by
`(psigamma(shape, deriv = 2) - psigamma(1, deriv = 2)) / (psigamma(shape, deriv = 1) + psigamma(1, deriv = 1))^(3/2))`

.

`[dpq]glogis`

are calculated by leveraging the `[dpq]logis`

and adding the shape parameter. `rglogis`

uses inversion.

##### Value

`dglogis`

gives the probability density function,
`pglogis`

gives the cumulative distribution function,
`qglogis`

gives the quantile function, and
`rglogis`

generates random deviates.
`sglogis`

gives the score function (gradient of the log-density with
respect to the parameter vector).

##### References

Johnson NL, Kotz S, Balakrishnan N (1995)
*Continuous Univariate Distributions*, volume 2.
John Wiley \& Sons, New York.

Shao Q (2002). Maximum Likelihood Estimation for Generalised Logistic Distributions.
*Communications in Statistics -- Theory and Methods*, **31**(10), 1687--1700.

Windberger T, Zeileis A (2014). Structural Breaks in Inflation Dynamics within the
European Monetary Union. *Eastern European Economics*, **52**(3), 66--88.

##### Examples

```
# NOT RUN {
## PDF and CDF
par(mfrow = c(1, 2))
x <- -100:100/10
plot(x, dglogis(x, shape = 2), type = "l", col = 4, main = "PDF", ylab = "f(x)")
lines(x, dglogis(x, shape = 1))
lines(x, dglogis(x, shape = 0.5), col = 2)
legend("topleft", c("generalized (0, 1, 2)", "standard (0, 1, 1)",
"generalized (0, 1, 0.5)"), lty = 1, col = c(4, 1, 2), bty = "n")
plot(x, pglogis(x, shape = 2), type = "l", col = 4, main = "CDF", ylab = "F(x)")
lines(x, pglogis(x, shape = 1))
lines(x, pglogis(x, shape = 0.5), col = 2)
## artifical empirical example
set.seed(2)
x <- rglogis(1000, -1, scale = 0.5, shape = 3)
gf <- glogisfit(x)
plot(gf)
summary(gf)
# }
```

*Documentation reproduced from package glogis, version 1.0-1, License: GPL-2 | GPL-3*