Generalized Linear Models Homework 1 Oct 13 2016

1. The list below comprises a number of distributions, including in each case, the support, parameter space, and density or probability mass function. Determine whether each of the distributions belongs to the exponential dispersion family. Similarly for the two-parameter exponential family of distributions. In both cases, justify your answers. (a) Double exponential (or Laplace) distribution. A distribution belongs to the exponential dispersion family (EDF) if its density or probability mass function can be written as f (y | θ, φ) = exp yθ − b(θ) a(φ) + c(y, φ) , (1) where θ ∈ Θ ⊆ R, φ > 0, a(φ) > 0, and the support of Y does not depend on θ and φ. A distribution belongs to the two-parameter exponential family (EF) if its density or probability mass function can be written as f (y | θ, σ) = h(y) exp 2 i=1 Q i (θ, σ)T i (y) − A(θ, σ) , (2) and the support of Y does not depend on θ and σ. The absolute value term |y − θ| of the Laplace distribution cannot give us yθ in (1) and 2 i=1 Q i (θ, σ)T i (y) for appropriate Q i (θ, σ) and T i (y) in (2). Therefore, the Laplace distribution does not belong to EDF or two parameter EF. If θ is a known and fixed constant, the Laplace distribution is in one-parameter EF where T 1 (y) = |y − θ|, Q 1 (σ) = −1/σ, A(σ) = log(2σ) and h(y) = I(y ∈ R). The support of Y depends on θ and σ, so this uniform distribution is not in the EDF and two-parameter EF.