Saddlepoint density for the Tweedie distributions

1 | ```
dtweedie.saddle(y, xi=power, mu, phi, eps=1/6, power=NULL)
``` |

`y` |
the vector of responses |

`xi` |
the value of |

`power` |
a synonym for |

`mu` |
the mean |

`phi` |
the dispersion |

`eps` |
the offset in computing the variance function.
The default is |

The Tweedie family of distributions belong to the class
of exponential dispersion models (EDMs),
famous for their role in generalized linear models.
The Tweedie distributions are the EDMs with a variance of the form
*var(Y) = phi * mu^power*
where *power* is greater than or equal to one, or less than or equal to zero.
**This function only evaluates for power
greater than or equal to one.**
Special cases include the
normal (

`power`

,
the distributions are still defined but cannot be written in closed form,
and hence evaluation is very difficult.
When *1 < power < 2*,
the distribution are continuous for *Y* greater than zero,
with a positive mass at *Y=0*.
For *power > 2*,
the distributions are continuous for *Y* greater than zero.

This function approximates the density using the saddlepoint approximation defined by Nelder and Pregibon (1987).

saddlepoint (approximate) density
for the given Tweedie distribution with parameters
`mu`

,
`phi`

and
`power`

.

Peter Dunn (pdunn2@usc.edu.au)

Daniels, H. E. (1954).
Saddlepoint approximations in statistics.
*Annals of Mathematical Statistics*,
**25**(4), 631–650.

Daniels, H. E. (1980).
Exact saddlepoint approximations.
*Biometrika*,
**67**, 59–63.

Dunn, P. K. and Smyth, G. K. (2008).
Evaluation of Tweedie exponential dispersion model densities by Fourier inversion.
*Statistics and Computing*,
**18**, 73–86.

Dunn, Peter K and Smyth, Gordon K (2001).
Tweedie family densities: methods of evaluation.
*Proceedings of the 16th International Workshop on Statistical Modelling*,
Odense, Denmark, 2–6 July

Dunn, Peter K and Smyth, Gordon K (2005).
Series evaluation of Tweedie exponential dispersion model densities
*Statistics and Computing*,
**15**(4). 267–280.

Jorgensen, B. (1987).
Exponential dispersion models.
*Journal of the Royal Statistical Society*, B,
**49**, 127-162.

Jorgensen, B. (1997).
*Theory of Dispersion Models*,
Chapman and Hall, London.

Nelder, J. A. and Pregibon, D. (1987).
An extended quasi-likelihood function.
*Biometrika*,
**74**(2), 221–232.

Tweedie, M. C. K. (1984).
An index which distinguishes between some important exponential families.
*Statistics: Applications and New Directions.
Proceedings of the Indian Statistical Institute Golden Jubilee International Conference*
(Eds. J. K. Ghosh and J. Roy), pp. 579-604. Calcutta: Indian Statistical Institute.

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