Paralogistic Distribution Family Function

Maximum likelihood estimation of the 2-parameter paralogistic distribution.

paralogistic(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")

Arguments

lss

See CommonVGAMffArguments for important information.

lshape1.a, lscale

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

iscale, ishape1.a, imethod, zero

See CommonVGAMffArguments for information. For imethod = 2 a good initial value for ishape1.a is needed to obtain good estimates for the other parameter.

gscale, gshape1.a

See CommonVGAMffArguments for information.

probs.y

See CommonVGAMffArguments for information.

Details

The 2-parameter paralogistic distribution is the 4-parameter generalized beta II distribution with shape parameter $p=1$ and $a=q$. It is the 3-parameter Singh-Maddala distribution with $a=q$. More details can be found in Kleiber and Kotz (2003).

The 2-parameter paralogistic has density $$f(y) = a^2 y^{a-1} / [b^a \{1 + (y/b)^a\}^{1+a}]$$ for $a > 0$, $b > 0$, $y \geq 0$. Here, $b$ is the scale parameter scale, and $a$ is the shape parameter. The mean is $$E(Y) = b \, \Gamma(1 + 1/a) \, \Gamma(a - 1/a) / \Gamma(a)$$ provided $a > 1$; 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{
pdata <- data.frame(y = rparalogistic(n = 3000, exp(1), scale = exp(1)))
fit <- vglm(y ~ 1, paralogistic(lss = FALSE), data = pdata, trace = TRUE)
fit <- vglm(y ~ 1, paralogistic(ishape1.a = 2.3, iscale = 5),
            data = pdata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
summary(fit) } # }