Geometric Properties of Non-parametric MLE and Numerical Solutions

In Chap. 6, our attention turns to numerical solutions for maximum likelihood estimation within the context of nonparametric mixture models. The challenge arises from the infinite dimension of the nonparametric space, making the resulting optimization problem seem insurmountable. However, the nonparametric maximum likelihood estimator is known to have finite support, meaning it assigns positive probabilities to a finite set of mixing parameter values. Moreover, there exists a well-behaved directional derivative function that is non-positive at the nonparametric maximum likelihood estimate. This characteristic motivates the use of iterative procedures and provides the conceptual tools necessary to verify the maximization of the likelihood function. Throughout this chapter, we present several numerical procedures and engage in discussions regarding their algorithmic convergence. We caution that a successful implementation often demands a higher level of specialized expertise.