Table 1.

Summary of Tweedie exponential dispersion models (10)

Distributions	p Domain on R	Support	α Domain on R	θ Domain	μ Domain
Extreme stable	p < 0	R	1 < α < 2	R0	R+
Normal	p = 0	R	α = 2	R	R
(do not exist)	0 < p < 1	nil	α > 2	R0	R+
Poisson	p = 1	N0	α = −∞	R	R+
Compound Poisson	1 < p < 2	R0	α < 0	R−	R+
Gamma	p = 2	R+	α = 0	R−	R+
Positive stable	2 < p < 3	R+	0 < α < ½	(−∞,0]	R+
Inverse Gaussian	p = 3	R+	α = ½	(−∞,0]	R+
Positive stable	p > 3	R+	½ < α < 1	(−∞,0]	R+

R ≡ real numbers, R₀ ≡ positive real numbers with zero, R₊ ≡ positive real numbers, R₋ ≡ negative real numbers, N₀ ≡ positive integers with zero.