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")

Arguments

lss

See CommonVGAMffArguments for important information.

lshape1.a, lscale, lshape3.q

Parameter link functions applied to the (positive) parameters $a$, scale, and $q$. See Links for more choices.

iscale, ishape1.a, ishape3.q, imethod, zero

See CommonVGAMffArguments for information. For imethod = 2 a good initial value for ishape3.q is needed to obtain good estimates for the other parameters.

gscale, gshape1.a, gshape3.q

See CommonVGAMffArguments for information.

probs.y

See CommonVGAMffArguments for information.

Details

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 $a > 0$, $b > 0$, $q > 0$, $y \geq 0$. 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.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

References

Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.

Author

T. W. Yee

Note

See the notes in genbetaII.

Examples

if (FALSE) { # \dontrun{
sdata <- data.frame(y = rsinmad(n = 1000, shape1 = exp(1),
                    scale = exp(2), shape3 = exp(0)))
fit <- vglm(y ~ 1, sinmad(lss = FALSE), sdata, trace = TRUE)
fit <- vglm(y ~ 1, sinmad(lss = FALSE, ishape1.a = exp(1)),
            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), sdata, trace = TRUE)
  fit1 <- vglm(y1 ~ 1, sinmad(lss = FALSE), 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)    } # }