dagum: Dagum Distribution Family Function In VGAM: Vector Generalized Linear and Additive Models

 dagum R Documentation

Dagum Distribution Family Function

Description

Maximum likelihood estimation of the 3-parameter Dagum distribution.

Usage

``````dagum(lscale = "loglink", lshape1.a = "loglink", lshape2.p =
"loglink", iscale = NULL, ishape1.a = NULL, ishape2.p =
NULL, imethod = 1, lss = TRUE, gscale = exp(-5:5), gshape1.a
= seq(0.75, 4, by = 0.25), gshape2.p = exp(-5:5), probs.y =
c(0.25, 0.5, 0.75), zero = "shape")
``````

Arguments

 `lss` See `CommonVGAMffArguments` for important information. `lshape1.a, lscale, lshape2.p` Parameter link functions applied to the (positive) parameters `a`, `scale`, and `p`. See `Links` for more choices. `iscale, ishape1.a, ishape2.p, imethod, zero` See `CommonVGAMffArguments` for information. For `imethod = 2` a good initial value for `ishape2.p` is needed to obtain a good estimate for the other parameter. `gscale, gshape1.a, gshape2.p` See `CommonVGAMffArguments` for information. `probs.y` See `CommonVGAMffArguments` for information.

Details

The 3-parameter Dagum distribution is the 4-parameter generalized beta II distribution with shape parameter `q=1`. It is known under various other names, such as the Burr III, inverse Burr, beta-K, and 3-parameter kappa distribution. It can be considered a generalized log-logistic distribution. Some distributions which are special cases of the 3-parameter Dagum are the inverse Lomax (`a=1`), Fisk (`p=1`), and the inverse paralogistic (`a=p`). More details can be found in Kleiber and Kotz (2003).

The Dagum distribution has a cumulative distribution function

`F(y) = [1 + (y/b)^{-a}]^{-p}`

which leads to a probability density function

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

for `a > 0`, `b > 0`, `p > 0`, `y \geq 0`. Here, `b` is the scale parameter `scale`, and the others are shape parameters. The mean is

`E(Y) = b \, \Gamma(p + 1/a) \, \Gamma(1 - 1/a) / \Gamma(p)`

provided `-ap < 1 < a`; 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`.

From Kleiber and Kotz (2003), the MLE is rather sensitive to isolated observations located sufficiently far from the majority of the data. Reliable estimation of the scale parameter require `n>7000`, while estimates for `a` and `p` can be considered unbiased for `n>2000` or 3000.

T. W. Yee

References

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

`Dagum`, `genbetaII`, `betaII`, `sinmad`, `fisk`, `inv.lomax`, `lomax`, `paralogistic`, `inv.paralogistic`, `simulate.vlm`.

Examples

``````## Not run:
ddata <- data.frame(y = rdagum(n = 3000, scale = exp(2),
shape1 = exp(1), shape2 = exp(1)))
fit <- vglm(y ~ 1, dagum(lss = FALSE), data = ddata, trace = TRUE)
fit <- vglm(y ~ 1, dagum(lss = FALSE, ishape1.a = exp(1)),
data = ddata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
summary(fit)

## End(Not run)``````

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