Mean Estimation of Truncated Mixtures of Two Gaussians: A Gradient Based Approach

Abstract

Even though data is abundant, it is often subjected to some form of censoring or truncation which inherently creates biases. Removing such biases and performing parameter estimation is a classical challenge in Statistics. In this paper, we focus on the problem of estimating the means of a mixture of two balanced d-dimensional Gaussians when the samples are prone to truncation. A recent theoretical study on the performance of the Expectation-Maximization (EM) algorithm for the aforementioned problem showed EM almost surely converges for d=1 and exhibits local convergence for d>1 to the true means. Nevertheless, the EM algorithm for the case of truncated mixture of two Gaussians is not easy to implement as it requires solving a set of nonlinear equations at every iteration which makes the algorithm impractical. In this work, we propose a gradient based variant of the EM algorithm that has global convergence guarantees when d=1 and local convergence for d>1 to the true means. Moreover, the update rule at every iteration is easy to compute which makes the proposed method practical. We also provide numerous experiments to obtain more insights into the effect of truncation on the convergence to the true parameters in high dimensions.