Generalized logistic distribution — cdfglo • lmom

Distribution function and quantile function of the generalized logistic distribution.

cdfglo(x, para = c(0, 1, 0))
quaglo(f, para = c(0, 1, 0))

Arguments

x: Vector of quantiles.
f: Vector of probabilities.
para: Numeric vector containing the parameters of the distribution, in the order \(\xi, \alpha, k\) (location, scale, shape).

Details

The generalized logistic distribution with location parameter \(\xi\), scale parameter \(\alpha\) and shape parameter \(k\) has distribution function $$F(x)=1/\lbrace 1+\exp(-y)\rbrace$$ where $$y=-k^{-1}\log\lbrace1-k(x-\xi)/\alpha\rbrace,$$ with \(x\) bounded by \(\xi+\alpha/k\) from below if \(k<0\) and from above if \(k>0\), and quantile function $$x(F)=\xi+{\alpha\over k}\biggl\lbrace1-\biggl({1-F \over F}\biggr)^k\biggr\rbrace.$$

The logistic distribution is the special case \(k=0\).

Value

cdfglo gives the distribution function; quaglo gives the quantile function.

Note

The functions expect the distribution parameters in a vector, rather than as separate arguments as in the standard R distribution functions pnorm, qnorm, etc.

See also

cdfkap for the kappa distribution, which generalizes the generalized logistic distribution.

Examples

# Random sample from the generalized logistic distribution
# with parameters xi=0, alpha=1, k=-0.5.
quaglo(runif(100), c(0,1,-0.5))
#>   [1]  2.83171966 -0.31878147  1.47349253  1.19930970  0.06239400 10.44824756
#>   [7] -0.49746793 -1.09359244  1.80067583 -1.64272868  1.87236215 -1.10577564
#>  [13] -1.65613173  1.83240450  2.34711680  8.53926859 -0.43874917 -1.08082591
#>  [19]  0.46873398  2.06960917 -1.60973847  1.31304431 -0.95247345 -1.74494385
#>  [25] -1.23167193  0.95658826  0.67793792 -0.60481990 -0.40927057  1.11050352
#>  [31]  1.03808133  5.05722983 -0.15773032  0.42896436 -1.11090934  0.28822005
#>  [37]  3.93678832  0.42150588  2.44585864  0.41617067  1.75461620 -0.37457735
#>  [43]  2.75876484  1.39036902 -0.63497285 -1.29085757 -1.32928378  2.00122226
#>  [49] -0.43441424 -1.52826754 15.10226056  0.47092079 -1.16384646  0.16154527
#>  [55] -1.24134619  8.22945672 -1.55616943 -1.12255670  5.67982129  9.50790977
#>  [61]  1.13147735  3.55068762 10.99905154 -1.61967554 -0.23165584  0.69273116
#>  [67] 25.74985436 -0.64480945  2.87391296  0.16841884  3.25356144  0.06334514
#>  [73]  3.16156798  2.88660967 -0.54629903 -1.97963315 -0.98383841  6.30782789
#>  [79] -0.74865508  3.44011712  7.94338006 -1.87296194  0.22408375 -0.94019490
#>  [85]  0.70604743  1.26952457  3.23393540 -0.42764551  3.77014241  2.64970137
#>  [91]  1.28659763 -1.60025187 -0.58422703  1.45739566  4.48541961 -0.98648934
#>  [97] -0.10995960 -0.41696430 -0.39155547 -0.94297391