Generalized M-Estimation

Delicatessen

The one-stop sandwich (variance) shop in Python. delicatessen is a Python 3.8+ library for the generalized calculus of M-estimation.

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

M-Estimation and Estimating Equations

Here, we provide a brief overview of M-estimation theory. For more detailed introductions to M-estimation, see Ross et al. (2024), Stefanski & Boos (2002), or Chapter 7 of Boos & Stefanski (2013). M-estimation is a generalization of likelihood-based methods. M-estimators are solutions to estimating equations. To apply the M-estimator, we solve the estimating equations using observed data. This is similar to other approaches, but the key advantage of M-Estimators is variance estimation via the empirical sandwich variance estimator.

While M-Estimation is a powerful tool, the derivatives and matrix algebra can quickly become unwieldy. This is where delicatessen comes in. delicatessen takes estimating functions and data, and solves for the parameter estimates, computes the derivatives, and performs the matrix algebra calculations. Therefore, M-estimators can be more easily adopted without having to perform by-hand calculations. In other words, we can let the computer do the math for us.

To further ease use, delicatessen also comes with a variety of built-in estimating equations. See the delicatessen website for details on the available estimating equations, how to use them, and practical examples.

Installation

Installing:

You can install via python -m pip install delicatessen

Dependencies:

The dependencies are: numpy, scipy

To replicate the tests located in tests/, you will additionally need to install: panda, statsmodels, and pytest

While versions of delicatessen prior to v1.0 were compatible with older versions of Python 3 and NumPy and SciPy, the v1.0+ releases are only available for Python 3.8+ with NumPy v1.18.5+ and SciPy v1.9.0. This change was made to use a better numerical approximation procedure for the derivative. If you want to use with older versions of those packages or older versions of Python, install v0.6 instead.

Getting started

Below is a simple demonstration of calculating the mean with delicatessen

import numpy as np
y = np.array([1, 2, 3, 1, 4, 1, 3, -2, 0, 2])


Loading the M-estimator functionality, building the corresponding estimating equation for the mean, and printing the results to the console

from delicatessen import MEstimator

def psi(theta):
return y - theta[0]

estr = MEstimator(psi, init=[0, ])
estr.estimate()

print(estr.theta)     # Estimate of the mean
print(estr.variance)  # Variance estimate


For further details on using delicatessen, see the full documentation and worked examples available at delicatessen website.

References

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

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

Ross RK, Zivich PN, Stringer JS, & Cole SR. (2024). M-estimation for common epidemiological measures: introduction and applied examples. International Journal of Epidemiology, 53(2).

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

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