ldTweedie {mgcv} | R Documentation |

## Log Tweedie density evaluation

### Description

A function to evaluate the log of the Tweedie density for variance powers between 1 and 2, inclusive.
Also evaluates first and second derivatives of log density w.r.t. its scale parameter, `phi`

, and `p`

,
or w.r.t. `rho=log(phi)`

and `theta`

where `p = (a+b*exp(theta))/(1+exp(theta))`

.

### Usage

```
ldTweedie(y,mu=y,p=1.5,phi=1,rho=NA,theta=NA,a=1.001,b=1.999,all.derivs=FALSE)
```

### Arguments

`y` |
values at which to evaluate density. |

`mu` |
corresponding means (either of same length as |

`p` |
the variance of |

`phi` |
The scale parameter. Variance of |

`rho` |
optional log scale parameter. Over-rides |

`theta` |
parameter such that |

`a` |
lower limit parameter (>1) used in definition of |

`b` |
upper limit parameter (<2) used in definition of |

`all.derivs` |
if |

### Details

A Tweedie random variable with 1<p<2 is a sum of `N`

gamma random variables
where `N`

has a Poisson distribution. The p=1 case is a generalization of a Poisson distribution and is a discrete
distribution supported on integer multiples of the scale parameter. For 1<p<2 the distribution is supported on the
positive reals with a point mass at zero. p=2 is a gamma distribution. As p gets very close to 1 the continuous
distribution begins to converge on the discretely supported limit at p=1.

`ldTweedie`

is based on the series evaluation method of Dunn and Smyth (2005). Without
the restriction on `p`

the calculation of Tweedie densities is less straightforward. If you really need this
case then the `tweedie`

package is the place to start.

The `rho`

, `theta`

parameterization is useful for optimization of `p`

and `phi`

, in order to keep `p`

bounded well away from 1 and 2, and `phi`

positive. The derivatives near `p=1`

tend to infinity.

Note that if `p`

and `phi`

(or `theta`

and `rho`

) both contain only a single unique value, then the underlying
code is able to use buffering to avoid repeated calls to expensive log gamma, di-gamma and tri-gamma functions (`mu`

can still be a vector of different values). This is much faster than is possible when these parameters are vectors with different values.

### Value

A matrix with 6 columns, or 10 if `all.derivs=TRUE`

. The first is the log density of `y`

(log probability if `p=1`

).
The second and third are the first and second derivatives of the log density w.r.t. `phi`

. 4th and 5th
columns are first and second derivative w.r.t. `p`

, final column is second derivative w.r.t. `phi`

and `p`

.

If `rho`

and `theta`

were supplied then derivatives are w.r.t. these. In this case, and if `all.derivs=TRUE`

then the 7th colmn is the derivative w.r.t. `mu`

, the 8th is the 2nd derivative w.r.t. `mu`

, the 9th is the mixed derivative w.r.t. `theta`

and`mu`

and the 10th is the mixed derivative w.r.t. `rho`

and `mu`

.

### Author(s)

Simon N. Wood simon.wood@r-project.org

### References

Dunn, P.K. and G.K. Smith (2005) Series evaluation of Tweedie exponential dispersion model densities. Statistics and Computing 15:267-280

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.

### Examples

```
library(mgcv)
## convergence to Poisson illustrated
## notice how p>1.1 is OK
y <- seq(1e-10,10,length=1000)
p <- c(1.0001,1.001,1.01,1.1,1.2,1.5,1.8,2)
phi <- .5
fy <- exp(ldTweedie(y,mu=2,p=p[1],phi=phi)[,1])
plot(y,fy,type="l",ylim=c(0,3),main="Tweedie density as p changes")
for (i in 2:length(p)) {
fy <- exp(ldTweedie(y,mu=2,p=p[i],phi=phi)[,1])
lines(y,fy,col=i)
}
```

*mgcv*version 1.9-1 Index]