How did it get this way? Disentangling the sources of teacher quality gaps through agent-based modeling.

We develop a novel simulation methodology to study the extent to which three interrelated processes-teacher attrition from the state teaching workforce, teacher mobility between teaching positions, and teacher hiring for open positions-contribute to "teacher quality gaps" (TQGs) between students of color and other students in K-12 public schools. We apply this methodology to data from Washington State to provide estimates that eliminating inequities in teacher mobility and hiring across different schools would close TQGs within 5 years, while just eliminating inequities in teacher hiring would close gaps within 10 years. On the other hand, eliminating inequities in teacher attrition without addressing mobility and hiring does little to close gaps.

A law of the iterated logarithm for Grenander's estimator.

In this note we prove the following law of the iterated logarithm for the Grenander estimator of a monotone decreasing density: If f(t0) > 0, f'(t0) < 0, and f' is continuous in a neighborhood of t0, then [Formula: see text]almost surely where [Formula: see text]here [Formula: see text] is the two-sided Strassen limit set on [Formula: see text]. The proof relies on laws of the iterated logarithm for local empirical processes, Groeneboom's switching relation, and properties of Strassen's limit set analogous to distributional properties of Brownian motion.