Series evaluation of Tweedie exponential dispersion model densities
By Peter K. Dunn, Gordon K. Smyth. Statistics and computing, 2005
366 citations recorded by the source corpus.
Exponential dispersion models, which are linear exponential families with a dispersion parameter, are the prototype response distributions for generalized linear models. The Tweedie family comprises those exponential dispersion models with power mean-variance relationships. The normal, Poisson, gamma and inverse Gaussian distributions belong to theTweedie family. Apart from these special cases, Tweedie distributions do not have density functions which can be written in closed form. Instead, the densities can be represented as infinite summations derived from series expansions. This article describes how the series expansions can be summed in an numerically efficient fashion. The usefulness of the approach is demonstrated, but full machine accuracy is shown not to be obtainable using the series expansion method for all parameter values. Derivatives of the density with respect to the dispersion parameter are also derived to facilitate maximum likelihood estimation. The methods are demonstrated on two data examples and compared with with Box-Cox transformations and extended quasi-likelihoood.
Subfields: Artificial Intelligence, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/ca79e85932d2c6df1bd792bc1eabe1f2020c0647
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In Statistics and Probability
- Natalya Pya
- Benjamin Säfken
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- Gareth William Peters
- Pavel V. Shevchenko
- Wagner Hugo Bonat
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- Gail MacInnis
Citations in the corpus
- Pollination by wild bees yields larger strawberries than pollination by honey bees (2019, methodology): “For the GLMs with strawberry mass as a response variable, a tweedie distribution with an inverse link function was implemented using r packages ‘tweedie’ (Dunn & Smyth, 2005) and ‘statmod’ (Giner & Smyth, 2016).”
- A simple monotone process with application to radiocarbon‐dated depth chronologies (2008, background): “Dunn and Smyth (2005) suggested that α=4 is a conservative limit and this is what we adopt.”
- Model Uncertainty in Claims Reserving within Tweedie's Compound Poisson Models (2009, methodology): “Dunn and Smyth (2005) derive the following bounds, c(y; φ, p)− c̃(y; φ, p) < WRL−1 1 − qRL−1L 1 − qL + WRU +1 1 1 − qU (3.6) with qL = exp ( ∂ log Wr ∂r )∣∣∣∣ r=RL−1 , qU = exp ( ∂ log Wr ∂r )∣∣∣∣ r=RU+1 .”
- Smoothing Parameter and Model Selection for General Smooth Models (2015, background): “The sum is interesting in that the early terms are near zero, as are the later terms, so that it has to be summed-fromthe-middle, which can be a bit involved: Dunn and Smyth (2005), give the details, but basically they show that the series maximum is around jmax = y2−p/{φ(2− p)}.”
- Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models (2011, methodology): “All computations were performed with R 2.9.2 (R Development Core Team, 2008) and R package mgcv version 1.6-1 (which includes a Tweedie family based on Dunn and Smith (2005)).”
- Flexible Tweedie regression models for continuous data (2016, methodology): “Further applications include property and causality insurance, where Jørgensen and Paes De Souza [5] and Smyth and Jørgensen [6] fit the Tweedie family to auto-mobile insurance claims data.”