# glogis: The Generalized Logistic Distribution (Type I: Skew-Logitic) In glogis: Fitting and Testing Generalized Logistic Distributions

## Description

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

## Usage

 ```1 2 3 4 5``` ```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 ≤ 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` = m, `scale` = s, and `shape` = g has distribution function

F(x) = 1 / (1 + exp(-(x-m)/s))^g

.

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

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```## 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) ```

### Example output

```Loading required package: zoo

Attaching package: 'zoo'

The following objects are masked from 'package:base':

as.Date, as.Date.numeric

Call:
glogisfit(x = x)

Coefficients:
Estimate Std. Error z value Pr(>|z|)
location   -1.16961    0.18840  -6.208 5.36e-10 ***
log(scale) -0.63017    0.04323 -14.578  < 2e-16 ***
log(shape)  1.29581    0.25916   5.000 5.73e-07 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Log-likelihood: -1074 on 12 Df
Goodness-of-fit statistic: 39.11 on 58 DF,  p-value: 0.9731
Number of iterations in BFGS optimization: 15
```

glogis documentation built on May 2, 2019, 4:47 p.m.