Singh-Maddala Distribution Family Function
Maximum likelihood estimation of the 3-parameter Singh-Maddala distribution.
sinmad(lscale = "loglink", lshape1.a = "loglink", lshape3.q = "loglink",
iscale = NULL, ishape1.a = NULL, ishape3.q = NULL, imethod = 1,
lss = TRUE, gscale = exp(-5:5), gshape1.a = exp(-5:5),
gshape3.q = exp(-5:5), probs.y = c(0.25, 0.5, 0.75),
zero = "shape")lss |
See |
lshape1.a, lscale, lshape3.q |
Parameter link functions applied to the
(positive) parameters a, |
iscale, ishape1.a, ishape3.q, imethod, zero |
See |
gscale, gshape1.a, gshape3.q |
See |
probs.y |
See |
The 3-parameter Singh-Maddala distribution is the 4-parameter generalized beta II distribution with shape parameter p=1. It is known under various other names, such as the Burr XII (or just the Burr distribution), Pareto IV, beta-P, and generalized log-logistic distribution. More details can be found in Kleiber and Kotz (2003).
Some distributions which are special cases of the 3-parameter Singh-Maddala are the Lomax (a=1), Fisk (q=1), and paralogistic (a=q).
The Singh-Maddala distribution has density
f(y) = aq y^(a-1) / [b^a (1 + (y/b)^a)^(1+q)]
for a > 0, b > 0, q > 0, y >= 0.
Here, b is the scale parameter scale,
and the others are shape parameters.
The cumulative distribution function is
F(y) = 1 - [1 + (y/b)^a]^(-q).
The mean is
E(Y) = b gamma(1 + 1/a) gamma(q - 1/a) / gamma(q)
provided -a < 1 < aq; 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.
sdata <- data.frame(y = rsinmad(n = 1000, shape1 = exp(1),
scale = exp(2), shape3 = exp(0)))
fit <- vglm(y ~ 1, sinmad(lss = FALSE), data = sdata, trace = TRUE)
fit <- vglm(y ~ 1, sinmad(lss = FALSE, ishape1.a = exp(1)),
data = sdata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
summary(fit)
# Harder problem (has the shape3.q parameter going to infinity)
set.seed(3)
sdata <- data.frame(y1 = rbeta(1000, 6, 6))
# hist(with(sdata, y1))
if (FALSE) {
# These struggle
fit1 <- vglm(y1 ~ 1, sinmad(lss = FALSE), data = sdata, trace = TRUE)
fit1 <- vglm(y1 ~ 1, sinmad(lss = FALSE), data = sdata, trace = TRUE,
crit = "coef")
Coef(fit1)
}
# Try this remedy:
fit2 <- vglm(y1 ~ 1, data = sdata, trace = TRUE, stepsize = 0.05, maxit = 99,
sinmad(lss = FALSE, ishape3.q = 3, lshape3.q = "logloglink"))
coef(fit2, matrix = TRUE)
Coef(fit2)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.