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Generalized M-Estimation

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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 a detailed introduction to M-estimation, see 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 estimation of the variance via the sandwich variance.

While M-Estimation is a powerful tool, the derivatives and matrix algebra can quickly become unwieldy. This is where delicatessen comes in. delicatessen takes an array estimating equations and data, and solves for the parameter estimates, numerically approximates the derivatives, and does the matrix calculations. Therefore, M-estimators can be more widely adopted without by-hand calculations. We can let the computer do all the math for us.

delicatessen also comes with a variety of built-in estimating equations. See the delicatessen website for the full set of available estimating equations and how to use them.



You can install via python -m pip install delicatessen


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 estimating equation, and printing the results to the console

from delicatessen import MEstimator

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

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

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

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


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.

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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