FellerPareto {actuar} | R Documentation |

Density function, distribution function, quantile function, random generation,
raw moments and limited moments for the Feller Pareto distribution
with parameters `min`

, `shape1`

, `shape2`

, `shape3`

and
`scale`

.

dfpareto(x, min, shape1, shape2, shape3, rate = 1, scale = 1/rate, log = FALSE) pfpareto(q, min, shape1, shape2, shape3, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE) qfpareto(p, min, shape1, shape2, shape3, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE) rfpareto(n, min, shape1, shape2, shape3, rate = 1, scale = 1/rate) mfpareto(order, min, shape1, shape2, shape3, rate = 1, scale = 1/rate) levfpareto(limit, min, shape1, shape2, shape3, rate = 1, scale = 1/rate, order = 1)

`x, q` |
vector of quantiles. |

`p` |
vector of probabilities. |

`n` |
number of observations. If |

`min` |
lower bound of the support of the distribution. |

`shape1, shape2, shape3, 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 Feller-Pareto distribution with parameters `min`

*= m*,
`shape1`

*= a*, `shape2`

*= b*,
`shape3`

*= c* and `scale`

*= s*, has
density:

*
f(x) = Gamma(a + c)/(Gamma(a) * Gamma(c)) (b ((x - m)/s)^(bc - 1))/
(s [1 + ((x - m)/s)^b]^(a + c))*

for *x > m*, *-Inf < m < Inf*,
*a > 0*, *b > 0*,
*c > 0* and *s > 0*.
(Here *Gamma(a)* is the function implemented
by **R**'s `gamma()`

and defined in its help.)

The Feller-Pareto is the distribution of the random variable

*
m + s ((1 - X)/X)^(1/b),*

where *X* has a beta distribution with parameters *a*
and *c*.

The Feller-Pareto defines a large family of distributions encompassing
the transformed beta family and many variants of the Pareto
distribution. Setting *min = 0* yields the
transformed beta distribution.

The Feller-Pareto distribution also has the following direct special cases:

A Pareto IV distribution when

`shape3 == 1`

;A Pareto III distribution when

`shape1 shape3 == 1`

;A Pareto II distribution when

`shape1 shape2 == 1`

;A Pareto I distribution when

`shape1 shape2 == 1`

and`min = scale`

.

The *k*th raw moment of the random variable *X* is
*E[X^k]* for nonnegative integer values of *k < shape1 * shape2*.

The *k*th limited moment at some limit *d* is *E[min(X, d)^k]* for nonnegative integer values of *k*
and *shape1 - j/shape2*, *j = 1, …, k*
not a negative integer.

Note that the range of admissible values for *k* in raw and
limited moments is larger when *min == 0*.

`dfpareto`

gives the density,
`pfpareto`

gives the distribution function,
`qfpareto`

gives the quantile function,
`rfpareto`

generates random deviates,
`mfpareto`

gives the *k*th raw moment, and
`levfpareto`

gives the *k*th moment of the limited loss
variable.

Invalid arguments will result in return value `NaN`

, with a warning.

`levfpareto`

computes the limited expected value using
`betaint`

.

For the Feller-Pareto and other Pareto distributions, we use the classification of Arnold (2015) with the parametrization of Klugman et al. (2012).

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 Nicholas Langevin

Arnold, B.C. (2015), *Pareto Distributions*, Second Edition, CRC
Press.

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.

Abramowitz, M. and Stegun, I. A. (1972), *Handbook of
Mathematical Functions*, Dover.

`dtrbeta`

for the transformed beta distribution.

exp(dfpareto(2, 1, 2, 3, 4, 5, log = TRUE)) p <- (1:10)/10 pfpareto(qfpareto(p, 1, 2, 3, 4, 5), 1, 2, 3, 4, 5) ## variance mfpareto(2, 1, 2, 3, 4, 5) - mfpareto(1, 1, 2, 3, 4, 5)^2 ## case with shape1 - order/shape2 > 0 levfpareto(10, 1, 2, 3, 4, scale = 1, order = 2) ## case with shape1 - order/shape2 < 0 levfpareto(20, 10, 0.1, 14, 2, scale = 1.5, order = 2)

[Package *actuar* version 3.1-4 Index]