The Inverse Paralogistic Distribution

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

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.

Value

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

References

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

Author

T. W. Yee

Details

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

Note

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

See also

inv.paralogistic, genbetaII.

Examples

idata <- data.frame(y = rinv.paralogistic(3000, exp(1), sc = exp(2)))
fit <- vglm(y ~ 1, inv.paralogistic(lss = FALSE, ishape1.a = 2.1),
            data = idata, trace = TRUE, crit = "coef")
#> Iteration 1: coefficients = 0.97480889, 2.13537682
#> Iteration 2: coefficients = 0.95783366, 2.00234203
#> Iteration 3: coefficients = 0.99310216, 2.02658553
#> Iteration 4: coefficients = 0.98766397, 2.00803677
#> Iteration 5: coefficients = 0.98935571, 2.01119010
#> Iteration 6: coefficients = 0.98906705, 2.01030457
#> Iteration 7: coefficients = 0.98912272, 2.01045821
#> Iteration 8: coefficients = 0.98911241, 2.01042876
#> Iteration 9: coefficients = 0.98911434, 2.01043422
#> Iteration 10: coefficients = 0.98911398, 2.01043320
#> Iteration 11: coefficients = 0.98911405, 2.01043339
coef(fit, matrix = TRUE)
#>             loglink(shape1.a) loglink(scale)
#> (Intercept)          0.989114       2.010433
Coef(fit)
#> shape1.a    scale 
#> 2.688851 7.466553