The \(i\)th term of the distribution can be written \(q_i/Z\) where \(q_i=\lambda^i / (i!)^\nu\) and \(Z=\sum_{(i=0)}^\infty q_i\), for \(\lambda=\lambda(\mu)\) implied by its inverse relationship, the expectation formula \(\mu=\mu(\lambda)=\sum_{(i=0)}^\infty i q_i(\lambda)/Z\). The case nu=0 is the geometric distribution with parameter \(\lambda\); nu=1 is the Poisson distribution with mean \(\lambda\); and the limit as nu -> \(\infty\) is the Bernoulli distribution with expectation \(\lambda/(1+\lambda)\).
From this definition, this is an exponential family model with canonical parameters \(log(\lambda)\) and \(\nu\). When the linear predictor \(\eta\) specifies \(log(\lambda(\mu))\), the canonical link is used (e.g., Sellers & Shmueli 2010). It is here nicknamed "loglambda" and does not have a known expression in terms of elementary functions. To obtain \(\mu\) as the link inverse of the linear predictor \(\eta\), one then first computes \(\lambda=e^\eta\) and then \(\mu(\lambda)\) by the expectation formula. For other links (Huang 2017), one directly computes \(\mu\) by the link inverse (e.g., \(\mu=e^\eta\) for link "log"), and then one may solve for \(\lambda= \lambda(\mu)\) to obtain other features of the distribution.
The relationships between \(\lambda\) and \(\mu\) or other moments of the distribution involve infinite summations. These sums can be easily approximated by a finite number of terms for large nu but not when nu approaches zero. For this reason, the code may fail to fit distributions with nu approaching 0 (strong residual over-dispersion). The case nu=0 (the geometric distribution) is fitted by an ad hoc algorithm devoid of such problems. Otherwise, spaMM truncates the sum, and uses numerical integrals to approximate missing terms (which slows down the fitting operation). In addition, it applies an ad hoc continuity correction to ensure continuity of the result in nu=1 (Poisson case). These corrections affect numerical results for the case of residual overdispersion but are negligible for the case of residual underdispersion. Alternatively, spaMM uses Gaunt et al.'s (2017) approximations when the condition defined by spaMM.getOption("CMP_asympto_cond") is satisfied. All approximations reduces the accuracy of computations, in a way that can impede the extended Levenberg-Marquardt algorithm sometimes needed by spaMM.
The name COMP_nu should be used to set initial values or bounds on nu in control arguments of the fitting functions (e.g., fitme(.,init=list(COMP_nu=1))). Fixed values should be set by the family argument (COMPoisson(nu=.)).