Support size estimation and the related problem of unseen species estimation have wide applications in ecology and database analysis. Perhaps the most used support size estimator is the Chao estimator. Despite its widespread use, little is known about its theoretical properties. We analyze the Chao estimator and show that its worst case mean squared error (MSE) is smaller than the MSE of the plug-in estimator by a factor of O k/n2. Our main technical contribution is a new method to analyze rational estimators for discrete distribution properties, which may be of independent interest. © 2019 IEEE.