Paralogistic | R Documentation |
Density function, distribution function, quantile function, random generation,
raw moments and limited moments for the Paralogistic distribution with
parameters shape
and scale
.
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)
x, q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If |
shape, scale |
parameters. Must be strictly positive. |
rate |
an alternative way to specify the scale. |
log, log.p |
logical; if |
lower.tail |
logical; if |
order |
order of the moment. |
limit |
limit of the loss variable. |
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.
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.
levparalogis
computes the limited expected value using
betaint
.
See Kleiber and Kotz (2003) for alternative names and parametrizations.
The "distributions"
package vignette provides the
interrelations between the continuous size distributions in
actuar and the complete formulas underlying the above functions.
Vincent Goulet vincent.goulet@act.ulaval.ca and Mathieu Pigeon
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.
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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