ustat-var 0.3.4

Computing unbiased variance estimators

Overview

This package contains the non-parametric unbiased estimators of the variance of teacher effects described in Rose, Schellenberg, and Shem-Tov (2022). These unbiased estimators are $U$-statistics, which provide minimum-variance unbiased estimators of population parameters for arbitrary probability distributions. The $U$-statistic approach overcomes several issues experienced by Empirical Bayes (EB) techniques when estimating an agent's 'value-added'. This readme is meant to be brief overview of how to install the package and how to use it -- the complete package documentation can be found here.

Authors

• Evan K. Rose (University of Chicago)
• Jonathan T. Schellenberg (Amazon Web Services)
• Yotam Shem-Tov (UCLA)
• Jack Mulqueeney (University of Chicago)

Installation

To install:

python3 -m pip install ustat_var

Usage

The package contains two functions that compute the non-parametric estimators presented in Appendix C of Rose, Schellenberg, and Shem-Tov (2022).

The first, ustat.varcovar estimates the variance-covariance, taking in two matrices as inputs:

import ustat_var as ustat
import numpy as np

# Seed and data
np.random.seed(18341)
X,Y = ustat.generate_test_data.generate_data(n_teachers = 10, n_time = 5, n_arrays = 2, var_fixed = 1, var_noise = 1.0, cov_factor = 0.5)

# Variance-covariance
ustat.varcovar(X, X) # Var(X)
ustat.varcovar(X, Y) # Cov(X, Y)

The second, ustat_samp_covar, estimates the sampling variance/covariance of varcovar. It takes four matrices as inputs, where ustat_samp_covar(A, B, C, D) yields $\hat{Cov}\big(\hat{Cov}(A,B) - Cov(A,B), \hat{Cov}(C,D) - Cov(C,D)\big)$. For example:

 # Base implementation through ustat_samp_covar.ustat_samp_covar() functions:
ustat.ustat_samp_covar.ustat_samp_covar(X, X, X, X) # Sampling variance of Var(X)
ustat.ustat_samp_covar.ustat_samp_covar(X, Y, X, Y) # Sampling variance of Cov(X, Y)

# Faster implementation available in ustat_samp_covar.ustat_samp_covar_fast() function
ustat.ustat_samp_covar.ustat_samp_covar_fast(X, X, X, X) # Also computes sampling variance of Var(X), but faster than above
ustat.ustat_samp_covar.ustat_samp_covar_fast(X, Y, X, Y) # Also computes sampling variance of Cov(X, Y), but faster than above

You can find further details about each function in the package documentation.