Generalized normal distribution

Distribution function and quantile function of the generalized normal distribution.

cdfgno(x, para = c(0, 1, 0))
quagno(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 normal distribution with location parameter $\xi$, scale parameter $\alpha$ and shape parameter $k$ has distribution function $$F(x)=\Phi(y)$$ where $$y=-k^{-1}\log\lbrace1-k(x-\xi)/\alpha\rbrace$$ and $\Phi(y)$ is the distribution function of the standard normal distribution, with $x$ bounded by $\xi+\alpha/k$ from below if $k<0$ and from above if $k>0$.

The generalized normal distribution contains as special cases the usual three-parameter lognormal distribution, corresponding to $k<0$, with a finite lower bound and positive skewness; the normal distribution, corresponding to $k=0$; and the reverse lognormal distribution, corresponding to $k>0$, with a finite upper bound and negative skewness. The two-parameter lognormal distribution, with a lower bound of zero and positive skewness, is obtained when $k<0$ and $\xi+\alpha/k=0$.

Value

cdfgno gives the distribution function; quagno 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.

Examples

# Random sample from the generalized normal distribution
# with parameters xi=0, alpha=1, k=-0.5.
quagno(runif(100), c(0,1,-0.5))
#>   [1]  0.066570567  1.316817902 -0.845239758 -0.291251711 -0.542739806
#>   [6] -0.005374598 -1.287289513 -0.985905824  5.520351667  2.391237019
#>  [11] -0.876500929 -0.902038695 -0.313208440 -0.071351861  0.963373936
#>  [16] -0.023939546 -0.940400913  2.416713766  0.622735762  0.005683691
#>  [21]  0.076533952 -0.553647407  0.644294729  3.065631221  0.461695602
#>  [26]  1.026172965 -1.032464070  1.642480174  1.201918737  0.699786795
#>  [31] -0.311772357  0.059631832 -0.074164197 -0.950550846  0.040611207
#>  [36] -0.122426502  0.128245035  0.423326038 -0.785987607 -0.475117850
#>  [41]  1.705588991  0.235466222  0.371112914 -0.551225566 -0.420993363
#>  [46] -0.374367387  0.850583521 -1.365075927 -1.068847469 -0.541550066
#>  [51]  0.252092665 -0.466685674 -0.962576691  0.852855549  0.905662659
#>  [56] -0.233415707 -1.084341717  0.803257651  0.072330583 -0.514913974
#>  [61] -0.547364062  0.113176769 -0.689315776  0.408848394  0.803222178
#>  [66] -0.525648807  1.560265307  1.370687193 -1.139382560 -0.280911781
#>  [71] -0.726537334  2.062456343  1.865009056 -0.703592929  1.466834972
#>  [76] -1.340269222  1.651109535 -0.257194499 -1.022237234  0.414445647
#>  [81]  1.993238678  1.251821550  0.003128391  0.276407259 -0.076389746
#>  [86]  1.657872702  0.493933770 -0.252119342  0.133168922 -0.589430133
#>  [91] -0.679460911 -1.049703079 -0.445194085  4.853995511 -0.898881385
#>  [96]  0.658590878 -0.626605352 -0.852257800  2.141540685  1.734166733

# The generalized normal distribution with parameters xi=1, alpha=1, k=-1,
# is the standard lognormal distribution.  An illustration:
fval<-seq(0.1,0.9,by=0.1)
cbind(fval, lognormal=qlnorm(fval), g.normal=quagno(fval, c(1,1,-1)))
#>       fval lognormal  g.normal
#>  [1,]  0.1 0.2776062 0.2776062
#>  [2,]  0.2 0.4310112 0.4310112
#>  [3,]  0.3 0.5919101 0.5919101
#>  [4,]  0.4 0.7761984 0.7761984
#>  [5,]  0.5 1.0000000 1.0000000
#>  [6,]  0.6 1.2883304 1.2883304
#>  [7,]  0.7 1.6894457 1.6894457
#>  [8,]  0.8 2.3201254 2.3201254
#>  [9,]  0.9 3.6022245 3.6022245