Delicatessen

delicatessen is a one-stop shop for all your sandwich (variance) needs. This Python 3.8+ library supports M-estimation, which is a general statistical framework for estimating unknown parameters. If you are an R user, I recommend looking into geex (Saul & Hudgens (2020)). delicatessen supports a variety of pre-built estimating equations as well as custom, user built estimating equations.

Here, we provide a brief overview of M-Estimation. For a more detailed, please refer to Stefanski & Boos (2002) or Boos & Stefanski (2013). M-estimation was developed to study the large sample properties of robust statistics. However, many common large-sample statistics can be expressed with estimating equations, so M-estimation provides a unified structure and a streamlined approach to estimation. Let the parameter of interest be the vector \(\theta = (\theta_1, \theta_2, ..., \theta_v)\) and data is observed for \(n\) independent units \(O_1, O_2, …, O_n\). An M-estimator, \(\hat{\theta}\), is the solution to the estimating equation \(\sum_{i=1}^{n} \psi(O_i, \hat{\theta}) = 0\) where \(\psi\) is a known \(v \times 1\)-dimension estimating function. M-estimators further provides a convenient and automatic method of calculating large-sample variance estimators via the empirical sandwich variance estimator:

\[V_n(O,\hat{\theta}) = B_n(O,\hat{\theta})^{-1} F_n(O,\hat{\theta}) \left(B_n(O,\hat{\theta})^{-1}\right)^T\]

where the ‘bread’ is

\[B_n(O,\hat{\theta}) = n^{-1} \sum_{i=1}^n - \psi'(O_i, \hat{\theta})\]

where the \(\psi'\) indicates the partial derivative, and the ‘filling’ is

\[F_n(O, \hat{\theta}) = n^{-1} \sum_{i=1}^n \psi(O_i, \hat{\theta}) \psi(O_i, \hat{\theta})^T\]

While M-Estimation is a general approach, widespread application is hindered by the corresponding derivative and matrix calculations. To circumvent these barriers, delicatessen automates M-estimators using numerical approximation methods.

The following description is a high-level overview. The user provides a \(v \times n\) array of estimating function(s) to the MEstimator class object. Next, the MEstimator object solves for \(\hat{\theta}\) using a root-finding algorithm. After successful completion of the root-finding, the bread is computed by numerically approximating the partial derivatives and the filling is calculated. Finally, the empirical sandwich variance is computed.

Installation:

To install delicatessen, use the following command in terminal or command prompt

python -m pip install delicatessen

Only two dependencies for delicatessen are: NumPy, SciPy.

While pandas is not necessary, several examples are demonstrated with pandas for ease of data processing. To replicate the tests in tests/ you will need to install pandas, statsmodels and pytest (but these are not necessary for use of the package).

Citation:

Please use the following citation for delicatessen: Zivich PN, Klose M, Cole SR, Edwards JK, & Shook-Sa BE. (2022). Delicatessen: M-Estimation in Python. arXiv preprint arXiv:2203.11300. URL

@article{zivich2022,
  title={Delicatessen: M-estimation in Python},
  author={Zivich, Paul N and Klose, Mark and Cole, Stephen R and Edwards, Jessie K and Shook-Sa, Bonnie E},
  journal={arXiv preprint arXiv:2203.11300},
  year={2022}
}

Contents:

Code and Issue Tracker

Please report bugs, issues, or feature requests on GitHub at pzivich/Delicatessen.

Otherwise, you may contact me via email (gmail: zivich.5).

References

Zivich PN, Klose M, Cole SR, Edwards JK, & Shook-Sa BE. (2022). Delicatessen: M-Estimation in Python. arXiv preprint arXiv:2203.11300.

Stefanski LA, & Boos DD. (2002). The calculus of M-estimation. The American Statistician, 56(1), 29-38.

Boos DD, & Stefanski LA. (2013). M-estimation (estimating equations). In Essential Statistical Inference (pp. 297-337). Springer, New York, NY.

Saul BC, & Hudgens MG. (2020). The Calculus of M-Estimation in R with geex. Journal of Statistical Software, 92(2).