cdfgpa | R Documentation |

Distribution function and quantile function of the generalized Pareto distribution.

```
cdfgpa(x, para = c(0, 1, 0))
quagpa(f, para = c(0, 1, 0))
```

`x` |
Vector of quantiles. |

`f` |
Vector of probabilities. |

`para` |
Numeric vector containing the parameters of the distribution,
in the order |

The generalized Pareto distribution with
location parameter `\xi`

,
scale parameter `\alpha`

and
shape parameter `k`

has distribution function

`F(x)=1-\exp(-y)`

where

`y=-k^{-1}\log\lbrace1-k(x-\xi)/\alpha\rbrace,`

with `x`

bounded by `\xi+\alpha/k`

from below if `k<0`

and from above if `k>0`

,
and quantile function

`x(F)=\xi+{\alpha\over k}\lbrace 1-(1-F)^k\rbrace.`

The exponential distribution is the special case `k=0`

.
The uniform distribution is the special case `k=1`

.

`cdfgpa`

gives the distribution function;
`quagpa`

gives the quantile function.

The functions expect the distribution parameters in a vector,
rather than as separate arguments as in the standard **R**
distribution functions `pnorm`

, `qnorm`

, etc.

Two parametrizations of the generalized Pareto distribution are in common use. When Jenkinson (1955) introduced the generalized extreme-value distribution he wrote the distribution function in the form

`F(x) = \exp [ - \lbrace 1 - k ( x - \xi ) / \alpha) \rbrace^{1/k}].`

Hosking and Wallis (1987) wrote the distribution function of the generalized Pareto distribution analogously as

`F(x) = 1 - \lbrace 1 - k ( x - \xi ) / \alpha) \rbrace^{1/k}`

and that is the form used in **R** package lmom. A slight inconvenience with it is that the
skewness of the distribution is a decreasing function of the shape parameter `k`

.
Perhaps for this reason, authors of some other **R** packages prefer a form in which
the sign of the shape parameter `k`

is changed and the parameters are renamed:

`F(x) = 1 - \lbrace 1 + \xi ( x - \mu ) / \sigma) \rbrace ^{-1/\xi}.`

Users should be able to mix functions from packages that use either form; just be aware that
the sign of the shape parameter will need to be changed when converting from one form to the other
(and that `\xi`

is a location parameter in one form and a shape parameter in the other).

Hosking, J. R. M., and Wallis, J. R. (1987). Parameter and quantile estimation
for the generalized Pareto distribution.
*Technometrics*, **29**, 339-349.

Jenkinson, A. F. (1955). The frequency distribution of the annual maximum
(or minimum) of meteorological elements.
*Quarterly Journal of the Royal Meteorological Society*, **81**, 158-171.

`cdfexp`

for the exponential distribution.

`cdfkap`

for the kappa distribution and
`cdfwak`

for the Wakeby distribution,
which generalize the generalized Pareto distribution.

```
# Random sample from the generalized Pareto distribution
# with parameters xi=0, alpha=1, k=-0.5.
quagpa(runif(100), c(0,1,-0.5))
```

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