Table 1.

Natural parameterizations relating π_i to linear predictors $z_i^{\prime }\gamma$ for commonly used two-component models and link functions relating μ_i to linear predictors $x_i^{\prime }\beta$ when y^* = 0

Nondegenerate distribution G	Link function h(·) relating μi to $x_i^{\prime }\beta$	Probability gi(0) = g̃(μi)	Parameterization relating πi to $z_i^{\prime }\gamma$
Poisson with mean μi	log{μi}	$exp\{-exp\{x_i^{\prime }\beta \}\}$	$exp\{-exp\{z_i^{\prime }\gamma \}\}$
Binomial with success probability μi and mi trials	logit{μi}	${\{1+exp\{x_i^{\prime }\beta \}\}}^{-m_i}$	${\{1+exp\{z_i^{\prime }\gamma \}\}}^{-m_i}$
	Φ−1(μi)	${\{1+\mathrm{\Phi }(x_i^{\prime }\beta )\}}^{m_i}$	${\{1+\mathrm{\Phi }(z_i^{\prime }\gamma )\}}^{m_i}$
	log{−log{μi}}	${\{1-exp\{-exp\{x_i^{\prime }\beta \}\}\}}^{m_i}$	${\{1-exp\{-exp\{z_i^{\prime }\gamma \}\}\}}^{m_i}$
Negative binomial with mean μi and over dispersion parameter κ	log{μi}	${\{1+\kappa exp\{x_i^{\prime }\beta \}\}}^{-1/\kappa }$	${\{1+\kappa exp\{z_i^{\prime }\gamma \}\}}^{-1/\kappa }$

Note: θ = β for the Poisson and the binomial processes;

θ = {β, κ} for the negative binomial process.