# dtweedie.saddle

##### Tweedie Distributions (saddlepoint approximation)

Saddlepoint density for the Tweedie distributions

- Keywords
- models

##### Usage

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

##### Arguments

- y
the vector of responses

- xi
the value of \(\xi\) such that the variance is \(\mbox{var}[Y]=\phi\mu^{\xi}\)

- power
a synonym for \(\xi\)

- mu
the mean

- phi
the dispersion

- eps
the offset in computing the variance function. The default is

`eps=1/6`

(as suggested by Nelder and Pregibon, 1987).

##### Details

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
\(\mbox{var}[Y]=\phi\mu^p\)
where \(p\) is greater than or equal to one, or less than or equal to zero.
**This function only evaluates for \(p\)
greater than or equal to one.**
Special cases include the
normal (\(p=0\)),
Poisson (\(p=1\) with \(\phi=1\)),
gamma (\(p=2\))
and
inverse Gaussian (\(p=3\))
distributions.
For other values of `power`

,
the distributions are still defined but cannot be written in closed form,
and hence evaluation is very difficult.

When \(1<p<2\), the distribution are continuous for \(Y\) greater than zero, with a positive mass at \(Y=0\). For \(p>2\), the distributions are continuous for \(Y\) greater than zero.

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

##### Value

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

,
`phi`

and
`power`

.

##### References

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.
10.1093/biomet/67.1.59

Dunn, P. K. and Smyth, G. K. (2008).
Evaluation of Tweedie exponential dispersion model densities by Fourier inversion.
*Statistics and Computing*,
**18**, 73--86.
10.1007/s11222-007-9039-6

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.
10.1007/s11222-005-4070-y

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.
10.1093/biomet/74.2.221

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.

##### See Also

##### Examples

```
# NOT RUN {
p <- 2.5
mu <- 1
phi <- 1
y <- seq(0, 10, length=100)
fy <- dtweedie( y=y, power=p, mu=mu, phi=phi)
plot(y, fy, type="l")
# Compare to the saddlepoint density
f.saddle <- dtweedie.saddle( y=y, power=p, mu=mu, phi=phi)
lines( y, f.saddle, col=2 )
# }
```

*Documentation reproduced from package tweedie, version 2.3.2, License: GPL (>= 2)*