ENH/Example: GLM-QMLE parameterized link and variance functions and margins

[josef-pkt]

mainly adding a reference to an application that brings several parts together

• parameterized link function - Box-Cox transformation ENH: parameterized link functions for GLM, discrete,  #624
• parameterized variance function - either power or quadratic (similar to Tweedie and Neg Bin) GLM with "foreign" variance function #1777
• based on estimating equations, not an objective function, EE joint for mean and variance function
  everything GMM, GLMGMM #1790 (GLMGMM) ENH: GLM family for estimating equations EE #2860 (GLM based on EE)
• margins and mean/expectation modelling as main objective

also related in motivation and discussion:

• use transformed endog versus GLM-like index model ENH: predict, get_prediction with log or transformed endog #2791 and shows up in several other issues

bonus: They describe how to compute start_params

Basu, Anirban, and Paul J. Rathouz. 2005. “Estimating Marginal and Incremental Effects on Health Outcomes Using Flexible Link and Variance Function Models.” Biostatistics 6 (1): 93–109. https://doi.org/10.1093/biostatistics/kxh020.

two comment on their methods:

• computational: They use Box-Cox link directly which does not enforce the domain constraints. They need additional constraints on the estimation, linpred, params, to get a non-negative or numerically valid mean response prediction. (exception in box-cox with negative linpred ?)
• theoretical: Their standard errors for margins have an extra term for sampling variance (going from sample to population effect). AFAIK we and Stata don't use this, AFAIU, and it might be related to survey sampling inference. Our margins evaluate at sample/empirical distribution of exog and only take parameter uncertainty into account.

aside:
Manning, Basu, and Mullahy. 2005 have an article on using Generalized Gamma in GLM style analysis, I haven't looked at it, but this could be another target when implementing Generalized Gamma (with log-link) for accelerated failure time AFT model.
i.e. more (general) models for continuous, positive or non-negative response variables with or without censoring. This might be a bit like a parametric analogue of the above QMLE.

Manning, Willard G., Anirban Basu, and John Mullahy. 2005. “Generalized Modeling Approaches to Risk Adjustment of Skewed Outcomes Data.” Journal of Health Economics 24 (3): 465–88. https://doi.org/10.1016/j.jhealeco.2004.09.011.