Description Usage Arguments Details Value Author(s) References See Also Examples

Density, distribution function, quantile function and random generation for the Truncated Generalised Gamma Distribution in linear space.

1 2 3 4 5 |

`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. |

`scale` |
Vector of values for scale, which controls the transition regime between the power law slope and the exponential cut-off of the TGGD. This is analogous to the scale parameter for the standard Gamma distribution (see |

`a` |
Vector of values for a, which controls the power law slope of the TGGD. |

`b` |
Vector of values for b, which controls the exponential cutoff of the TGGD. |

`xmin` |
Vector of values for xmin, which controls the lower limit at which to trancate the TGGD. |

`res.approx` |
The resolution used to create the inverted CDF required to map probability integrals back onto quantiles. |

`log, log.p` |
Logical; if TRUE, probabilities/densities p are returned as log(p). |

`lower.tail` |
Logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]. |

This distribution function is described in detail in Murray, Robotham and Power 2016. The PDF is given by:

*f(x;a,b,s,m) = b.((x/s)^(a).exp(-(x/s)^b))/(s.gamma_inc((a+1)/b,(m/s)^b))*

where, from the argument list above, we use x=x, a=a, b=b, s=scale and m=xmin. *gamma_inc* is the upper incomplete Gamma function as defined by the gsl `gamma_inc`

function, using the same argument ordering, where `gamma_inc(a,x)==pgamma(x,a,lower=FALSE)*gamma(x)`

for a>0. `gamma_inc`

is used because it allows for the computation of upper incomplete integrals in cases where a<=0.

dtggd gives the density, ptggd gives the distribution function, qtggd gives the quantile function, and rtggd generates random deviates.

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

The length of the result is determined by n for rtggd, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

Aaron Robotham, Steven Murray

Murray, Robotham and Power (2016)

`GammaDist`

regarding the Gamma distribution.
`RNG`

about random number generation in R.
`Distributions`

for other standard distributions.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
r <- rtggd(100,a=-2)
hist(log10(r))
##Ideally the output below should equal 1, in practice it will be very close:
qtggd(ptggd(r))/r
#These should be the same:
integrate(dtggd,lower=1e10,upper=1e11,a=-1.5,b=0.7,xmin=1e10)
ptggd(1e11,a=-1.5,b=0.7,xmin=1e10)
#This should be very close to 1 (for a true PDF):
ptggd(1e18,a=-1.5,b=0.7,xmin=1e10)
#To show the link to the log10 (called log) and ln variants (and the slight inaccuracies)
#these outputs should be a sequence from 0 to 1 (by=0.1):
ptggd(10^qtggd_log(seq(0,1,by=0.1)))
ptggd(exp(qtggd_ln(seq(0,1,by=0.1))))
``` |

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