tweedie

The tweedie package allows likelihood computations for Tweedie distributions.

Apart from special cases (the normal, Poisson, gamma, inverse Gaussian distributions), Tweedie distributions do not have closed-form density functions or distribution functions. This package uses fast numerical algorithms (infinite oscillating integrals; infinite series) to evaluate the Tweedie density functions and distribution functions.

Installation

You can install the development version of tweedie from GitHub with:

# install.packages("pak")
pak::pak("PeterKDunn/tweedie")
#> ℹ Loading metadata database
#> ✔ Loading metadata database ... done
#> 
#> 
#> → Package library at '/tmp/RtmphZvvUG/temp_libpath222b212925bc'.
#> → Will install 4 packages.
#> → Will update 1 package.
#> → All 5 packages (1.63 MB) are cached.
#> + cli               3.6.6 [bld][cmp]
#> + lifecycle         1.0.5 [bld]
#> + rlang             1.3.0 [bld][cmp]
#> + statmod           1.5.2 [bld][cmp]
#> + tweedie   3.2.2 → 3.2.2 [bld][cmp] (GitHub: aade557)
#> ℹ No downloads are needed, 5 pkgs (1.63 MB) are cached
#> ✔ Got tweedie 3.2.2 (source) (358.83 kB)
#> ✔ Installed cli 3.6.6  (31ms)
#> ✔ Installed lifecycle 1.0.5  (46ms)
#> ✔ Installed rlang 1.3.0  (60ms)
#> ✔ Installed statmod 1.5.2  (69ms)
#> ℹ Packaging tweedie 3.2.2
#> ✔ Packaged tweedie 3.2.2 (532ms)
#> ℹ Building tweedie 3.2.2
#> ✔ Built tweedie 3.2.2 (3.6s)
#> ✔ Installed tweedie 3.2.2 (github::PeterKDunn/tweedie@aade557) (1s)
#> ✔ 1 pkg + 4 deps: upd 1, added 4, dld 1 (NA B) [8.9s]

To also build and install the vignette (this takes a little longer, and installs knitr and rmarkdown if needed):

options(pkg.build_vignettes = TRUE)
pak::pak("PeterKDunn/tweedie", dependencies = TRUE)
#> 
#> → Package library at '/tmp/RtmphZvvUG/temp_libpath222b212925bc'.
#> ℹ No downloads are needed
#> ✔ 1 pkg + 4 deps: kept 5 [916ms]
vignette("tweedie", package = "tweedie")
#> Warning: vignette 'tweedie' not found

Alternatively, using remotes:

remotes::install_github("PeterKDunn/tweedie", build_vignettes = TRUE, dependencies = TRUE)
#> Using GitHub PAT from the git credential store.
#> Skipping install of 'tweedie' from a github remote, the SHA1 (aade557d) has not changed since last install.
#>   Use `force = TRUE` to force installation

install.packages("tweedie") includes the vignette automatically.

Tweedie distributions

Tweedie distributions are exponential dispersion models, with a mean \(\mu\) and a variance \(\phi \mu^\xi\), for some dispersion parameter \(\phi > 0\) and a power index \(\xi\) (sometimes called \(p\)) that uniquely defines the distribution within the Tweedie family (for all real values of \(\xi\) not between 0 and 1).

Special cases of the Tweedie distributions are:

• the normal distribution, with \(\xi = 0\) (i.e., the variance is \(\phi\) and not related to the mean);
• the Poisson distribution, with \(\xi = 1\) and \(\phi = 1\) (i.e., the variance is the same as the mean);
• the gamma distribution, with \(\xi = 2\); and
• the inverse Gaussian distribution, with \(\xi = 3\).

For all other values of \(\xi\), the probability functions and distribution functions have no closed forms.

For \(\xi < 1\), applications are limited (non-existent so far?), but have support on the entire real line and \(\mu > 0\).

For \(1 < \xi < 2\), Tweedie distributions can be represented as a Poisson sum of gamma distributions. These distributions are continuous for \(Y > 0\) but have a discrete mass at \(Y = 0\).

For \(\xi \ge 2\), the distributions have support on the positive reals.

The vignette contains examples.