Table 1.

Density functions (f), first and second moments (μ₁, μ₂), Laplace transforms (ϕ) and cross-ratio (C) for the distributions: gamma, inverse Gaussian, positive stable and discrete.

Gamma

f(t) = θ–1/θt(1–θ)/θ exp(–t/θ)/γ(1/θ), θ > 0

μ1 = 1, μ2 = θ + 1

ϕ(s) = (1 + θs)–1/θ

C(t1, t2) = θ + 1

Inverse Gaussian

f(t) = (πθ)–1/2 exp(2/θ)t–3/2 exp{–t/θ – 1/(tθ)}, θ ≥ 0

μ1 = 1, μ2 = θ/2

$\varphi \left(s\right)=\text{exp}\left[2\{\frac{1}{\theta }-{(\frac{1}{{\theta }^2}+\frac{s}{\theta })}^{1∕2}\}\right]$

$C(t_1,t_2)=1+\frac{\theta }{2-\theta \text{log}{S}^m(t_1,t_2)}$

Sm(t1, t2) = P(T1 ≥ t1, T2 ≥ t2) = ϕ(H1(t1) + H2(t2)) where Hi(ti) = Λ0(ti) exp(βTZi)

Positive Stable

$f\left(t\right)=-{\left(\pi t\right)}^{-1}{\Sigma }_{k=1}^{\infty }\gamma (k\theta +1){(k!)}^{-1}{(-t^{-\theta })}^k\text{sin}\left(\theta \pi k\right),,0<\theta <1$

μ1, μ2 does not exist for θ < 1

ϕ(s) = exp(–sθ)

$C(t_1,t_2)=1+\frac{1-\theta }{-\theta \text{log}{S}^m(t_1,t_2)}$

Discrete

Pr(ω = 1 + θ) = 0.5, and Pr(ω = 1 – θ) = 0.5, –1 ≤ θ < 1

μ1 = 1, μ2 = 1 + θ2

ϕ(s) = 0.5 exp {–s(1 – θ)} + 0.5 exp {–s(1 + θ)}

C(t1, t2) = 1 + 4θ2[(1 + θ){G(t1, t2)}–θ + (1 – θ){G(t1, t2)}θ]–2

G(t1, t2) = exp{H1(t1) + H2(t2)}