Partially projected gradient algorithms for computing nonparametric maximum likelihood estimates of mixing distributions

There exist primarily three different types of algorithms for computing nonparametric maximum likelihood estimates (NPMLEs) of mixing distributions in the literature, which are the EM-type algorithms, the vertex direction algorithms such as VDM and VEM, and the algorithms based on general constrained optimization techniques such as the projected gradient method. It is known that the projected gradient algorithm may run into stagnation during iterations. When a stagnation occurs, VDM steps need to be added. We argue that the abrupt switch to VDM steps can significantly reduce the efficiency of the projected gradient algorithm, and is usually unnecessary. In this paper, we define a group of partially projected directions, which can be regarded as hybrids of ordinary projected gradient directions and VDM directions. Based on these directions, four new algorithms are proposed for computing NPMLEs of mixing distributions. The properties of the algorithms are discussed and their convergence is proved. Extensive numerical simulations show that the new algorithms outperform the existing methods, especially when a NPMLE has a large number of support points or when high accuracy is required.