The use of the Tweedie distribution in statistical modelling

Abstract

This paper discusses the estimation of the parameters of the so-called Tweedie distribution, T _P(μ, σ²). Two special cases are considered, namely the Compound Poisson (1 < p < 2) and the Stable form (p > 2). The former is appropriate for data with a non-zero probability of zero observations and the latter is appropriate for data with a large dispersion. Our models will assume that we have data Y _i, i = 1,…, N, with differing means μ_i, with common p and σ². The T _p (μ_i, σ²) distribution can be characterised by Var(Y_i) = σ²μ ^p_i , i = 1,…, N. In general, we shall model the μ_i in terms of explanatory variates x _ij, i = 1,…, N, j= 1,…, m. We discuss how it is straightforward to construct the maximum likelihood estimates of p, μ_i, and σ ² in a GLM oriented computer package. The Tweedie distribution is used to model the alcohol consumption of British 16 and 17 year olds and randomised quantile residuals are used to validate the modelling.