Paralogistic: The Paralogistic Distribution

Description

Density function, distribution function, quantile function, random generation, raw moments and limited moments for the Paralogistic distribution with parameters shape and scale.

Usage

dparalogis(x, shape, rate = 1, scale = 1/rate, log = FALSE)
pparalogis(q, shape, rate = 1, scale = 1/rate,
           lower.tail = TRUE, log.p = FALSE)
qparalogis(p, shape, rate = 1, scale = 1/rate,
           lower.tail = TRUE, log.p = FALSE)
rparalogis(n, shape, rate = 1, scale = 1/rate)
mparalogis(order, shape, rate = 1, scale = 1/rate)
levparalogis(limit, shape, rate = 1, scale = 1/rate,
             order = 1)

Arguments

x, q

vector of quantiles.

vector of probabilities.

number of observations. If length(n) > 1, the length is taken to be the number required.

shape, scale

parameters. Must be strictly positive.

rate

an alternative way to specify the scale.

log, log.p

logical; if TRUE, probabilities/densities $p$ are returned as $\log(p)$.

lower.tail

logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.

order

order of the moment.

limit

limit of the loss variable.

Value

dparalogis gives the density, pparalogis gives the distribution function, qparalogis gives the quantile function, rparalogis generates random deviates, mparalogis gives the $k$th raw moment, and levparalogis gives the $k$th moment of the limited loss variable.

Invalid arguments will result in return value NaN, with a warning.

Details

The paralogistic distribution with parameters shape $= \alpha$ and scale $= \theta$ has density: $$f(x) = \frac{\alpha^2 (x/\theta)^\alpha}{% x [1 + (x/\theta)^\alpha)^{\alpha + 1}}$$ for $x > 0$, $\alpha > 0$ and $\theta > 0$.

The $k$th raw moment of the random variable $X$ is $E[X^k]$, $-\alpha < k < \alpha^2$.

The $k$th limited moment at some limit $d$ is $E[\min(X, d)^k]$, $k > -\alpha$ and $\alpha - k/\alpha$ not a negative integer.

References

Kleiber, C. and Kotz, S. (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley.

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

Examples

Run this code

# NOT RUN {
exp(dparalogis(2, 3, 4, log = TRUE))
p <- (1:10)/10
pparalogis(qparalogis(p, 2, 3), 2, 3)

## variance
mparalogis(2, 2, 3) - mparalogis(1, 2, 3)^2

## case with shape - order/shape > 0
levparalogis(10, 2, 3, order = 2)

## case with shape - order/shape < 0
levparalogis(10, 1.25, 3, order = 2)
# }

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