Ioannis Kosmidis(author, maintainer)
Nicola Lunardon(author)


MIT License

Package description

GEEBRA is a Julia package that implements $M$-estimation for statistical models, either by solving estimating equations or by maximizing inference objectives, like likelihoods and composite likelihoods (see, Varin et al, 2011, for a review), using user-specified templates of the estimating function or the objective functions contributions.

A key feature is the use of only those templates and forward mode automatic differentiation (as implemented in ForwardDiff) to provide methods for reduced-bias $M$-estimation (RB$M$-estimation; see, Kosmidis & Lunardon, 2020). RB$M$-estimation takes place either through the adjustment of the estimating equations or the penalization of the objectives, or the subtraction of an estimate of the bias of the $M$-estimator from the $M$-estimates.

See the examples for a showcase of the functionaly GEEBRA provides.

See for changes, bug fixes and enhancements.

GEEBRA templates

GEEBRA has been designed so that the only requirements from the user are to:

  1. implement a Julia composite type for the data;
  2. implement a function for computing the number of observations from the data object;
  3. implement a function for calculating the contribution to the estimating function or to the objective function from a single observation that has arguments the parameter vector, the data object, and the observation index;
  4. specify a GEEBRA template (using estimating_function_template for estimating functions and objective_function_template for objective function) that has fields the functions for computing the contributions to the estimating functions or to the objective, and the number of observations.

GEEBRA, then, can estimate the unknown parameters by either $M$-estimation or RB$M$-estimation.





  • Varin, C., N. Reid, and D. Firth (2011). An overview of composite likelihood methods. Statistica Sinica 21(1), 5–42. Link
  • Kosmidis, I., N. Lunardon (2020). Empirical bias-reducing adjustments to estimating functions. ArXiv:2001.03786. Link