The Paralogistic Distribution

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

Usage

dparalogistic(x, scale = 1, shape1.a, log = FALSE)
pparalogistic(q, scale = 1, shape1.a, lower.tail = TRUE, log.p = FALSE)
qparalogistic(p, scale = 1, shape1.a, lower.tail = TRUE, log.p = FALSE)
rparalogistic(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

dparalogistic gives the density, pparalogistic gives the distribution function, qparalogistic gives the quantile function, and rparalogistic 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 and Kai Huang

Details

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

Note

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

Examples

pdata <- data.frame(y = rparalogistic(n = 3000, scale = exp(1), exp(2)))
fit <- vglm(y ~ 1, paralogistic(lss = FALSE, ishape1.a = 4.1),
            data = pdata, trace = TRUE)
#> Iteration 1: loglikelihood = -1269.41035
#> Iteration 2: loglikelihood = -918.540761
#> Iteration 3: loglikelihood = -905.585639
#> Iteration 4: loglikelihood = -886.243364
#> Iteration 5: loglikelihood = -886.090619
#> Iteration 6: loglikelihood = -886.008014
#> Iteration 7: loglikelihood = -886.007876
#> Iteration 8: loglikelihood = -886.00786
#> Iteration 9: loglikelihood = -886.00786
#> Taking a modified step.................
#> Iteration  9 :  loglikelihood = -886.00786
coef(fit, matrix = TRUE)
#>             loglink(shape1.a) loglink(scale)
#> (Intercept)          2.004334       0.995325
Coef(fit)
#> shape1.a    scale 
#> 7.421150 2.705604