In previous work we have shown that non-coherent direction of arrival (DOA) estimation by an array of sensors may be implemented from magnitude-only measurements. Here we show that magnitude-only measurements collected by a sensor array may be modeled by a non-central chi-square distribution with mean and variance independent of the phase errors. We also develop a closed-form expression for the Cramer-Rao bound (CRB) of non-coherent DOA estimation for the difference in spatial frequency between two sources. Finally, we obtain closed-form expressions for the maximum likelihood estimates (MLEs) of various unknown parameters. These theoretical expressions are used to analyze the performance of non-coherent DOA estimation.