Fisk Distribution family function
Maximum likelihood estimation of the 2-parameter Fisk distribution.
fisk(lscale = "loglink", lshape1.a = "loglink", iscale = NULL,
ishape1.a = NULL, imethod = 1, lss = TRUE, gscale = exp(-5:5),
gshape1.a = seq(0.75, 4, by = 0.25), probs.y = c(0.25, 0.5, 0.75),
zero = "shape")lss |
See |
lshape1.a, lscale |
Parameter link functions applied to the
(positive) parameters a and |
iscale, ishape1.a, imethod, zero |
See |
gscale, gshape1.a |
See |
probs.y |
See |
The 2-parameter Fisk (aka log-logistic) distribution is the 4-parameter generalized beta II distribution with shape parameter q=p=1. It is also the 3-parameter Singh-Maddala distribution with shape parameter q=1, as well as the Dagum distribution with p=1. More details can be found in Kleiber and Kotz (2003).
The Fisk distribution has density
f(y) = a y^(a-1) / [b^a (1 + (y/b)^a)^2]
for a > 0, b > 0, y >= 0.
Here, b is the scale parameter scale,
and a is a shape parameter.
The cumulative distribution function is
F(y) = 1 - [1 + (y/b)^a]^(-1) = [1 + (y/b)^(-a)]^(-1).
The mean is
E(Y) = b gamma(1 + 1/a) gamma(1 - 1/a)
provided a > 1; these are returned as the fitted values. This family function handles multiple responses.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
See the notes in genbetaII.
T. W. Yee
Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.
fdata <- data.frame(y = rfisk(n = 200, shape = exp(1), scale = exp(2))) fit <- vglm(y ~ 1, fisk(lss = FALSE), data = fdata, trace = TRUE) fit <- vglm(y ~ 1, fisk(ishape1.a = exp(2)), data = fdata, trace = TRUE) coef(fit, matrix = TRUE) Coef(fit) summary(fit)
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