# inv.paralogisticUC: The Inverse Paralogistic Distribution In VGAM: Vector Generalized Linear and Additive Models

## Description

Density, distribution function, quantile function and random generation for the inverse paralogistic distribution with shape parameters `a` and `p`, and scale parameter `scale`.

## Usage

 ```1 2 3 4``` ```dinv.paralogistic(x, scale = 1, shape1.a, log = FALSE) pinv.paralogistic(q, scale = 1, shape1.a, lower.tail = TRUE, log.p = FALSE) qinv.paralogistic(p, scale = 1, shape1.a, lower.tail = TRUE, log.p = FALSE) rinv.paralogistic(n, scale = 1, shape1.a) ```

## Arguments

 `x, q` vector of quantiles. `p` vector of probabilities. `n` number of observations. If `length(n) > 1`, the length is taken to be the number required. `shape1.a` shape parameter. `scale` scale parameter. `log` Logical. If `log = TRUE` then the logarithm of the density is returned. `lower.tail, log.p` Same meaning as in `pnorm` or `qnorm`.

## Details

See `inv.paralogistic`, which is the VGAM family function for estimating the parameters by maximum likelihood estimation.

## Value

`dinv.paralogistic` gives the density, `pinv.paralogistic` gives the distribution function, `qinv.paralogistic` gives the quantile function, and `rinv.paralogistic` generates random deviates.

## Note

The inverse paralogistic distribution is a special case of the 4-parameter generalized beta II distribution.

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`.
 ```1 2 3 4 5``` ```idata <- data.frame(y = rinv.paralogistic(n = 3000, exp(1), scale = exp(2))) fit <- vglm(y ~ 1, inv.paralogistic(lss = FALSE, ishape1.a = 2.1), data = idata, trace = TRUE, crit = "coef") coef(fit, matrix = TRUE) Coef(fit) ```