# inv.paralogistic: Inverse Paralogistic Distribution Family Function In VGAM: Vector Generalized Linear and Additive Models

 inv.paralogistic R Documentation

## Inverse Paralogistic Distribution Family Function

### Description

Maximum likelihood estimation of the 2-parameter inverse paralogistic distribution.

### Usage

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 a good estimate for the other parameter. gscale, gshape1.a See CommonVGAMffArguments for information. probs.y See CommonVGAMffArguments for information.

### Details

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

The inverse paralogistic distribution has density

f(y) = a^2 y^{a^2-1} / [b^{a^2} \{1 + (y/b)^a\}^{a+1}]

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(a + 1/a) \, \Gamma(1 - 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.

### Note

See the notes in genbetaII.

T. W. Yee

### References

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

Inv.paralogistic, genbetaII, betaII, dagum, sinmad, fisk, inv.lomax, lomax, paralogistic, simulate.vlm.

### Examples

idata <- data.frame(y = rinv.paralogistic(3000, exp(1), sc = exp(2)))
fit <- vglm(y ~ 1, inv.paralogistic(lss = FALSE), idata, trace = TRUE)
fit <- vglm(y ~ 1, inv.paralogistic(imethod = 2, ishape1.a = 4),
data = idata, trace = TRUE, crit = "coef")
coef(fit, matrix = TRUE)
Coef(fit)
summary(fit)

VGAM documentation built on Sept. 19, 2023, 9:06 a.m.