# MEstimation.jl

## Authors

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

## Package description

MEstimation is a Julia package that implements M-estimation for statistical models (see, e.g. Stefanski and Boos, 2002, for an accessible review) 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 just

1. the estimating function or the objective functions contributions
2. a function to compute the number of independent contributions in a given data set

A key feature is the use of those templates along with forward mode automatic differentiation (as implemented in ForwardDiff) to provide methods for reduced-bias M-estimation (RBM-estimation; see, Kosmidis & Lunardon, 2020).

See the documentation for more information, and the examples for a showcase of the functionality MEstimation provides.

See NEWS.md for changes, bug fixes and enhancements.

## MEstimation templates

MEstimation 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 MEstimation 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.

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

## References

• Varin C, Reid N, and Firth D (2011). An overview of composite likelihood methods. Statistica Sinica 21(1), 5-42. Link
• Kosmidis I, Lunardon N (2020). Empirical bias-reducing adjustments to estimating functions. ArXiv:2001.03786. Link
• Stefanski L A and Boos D D (2002). The calculus of M-estimation. The American Statistician(56), 29-38. Link