R: Log Tweedie density evaluation

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.

### Usage

    ldTweedie(y,mu=y,p=1.5,phi=1)

### Arguments

`y` | values at which to evaluate density.
`mu` | corresponding means (either of same length as `y` or a single value).
`p` | the variance of `y` is proportional to its mean to the power `p`. `p` must be between 1 and 2. 1 is Poisson like (exactly Poisson if `phi=1`), 2 is gamma.
`phi` | The scale parameter. Variance of `y` is `phi*mu^p`.

### 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.

### Value

A matrix with 3 columns. 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`.

### Author(s)

Simon N. Wood simon.wood@r-project.org modified from Venables and Ripley's `negative.binomial` family.

### 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)
      }

[Package mgcv version 1.7-18 Index]