Statistical methods for incomplete or indirectly observable data

This lecture is meant for students in the fourth or fifth year of their mathematics studies and Ph.D. students in their first two years, with basic knowledge of statistical estimation theory. We hope to give those students a glimpse of a very interesting research direction. Of course, we cannot discuss this research direction in great detail, but hope that the examples covered will motivate the students for further study. In section 2, we will briefly review the definition of a maximum likelihood estimator in specific models. As far as parametric models are concerned, this will all be standard and known. For a nonparametric model, where we observe a sample from a distribution of which we do not want to assume anything in advance (like having a specific form as the exponential or normal distribution), we will argue that the empirical distribution function is the maximum likelihood estimator. What is important in this section is that the data that are available can be considered as a sample from the distribution one wants to estimate. Section 3 deals with models where this is not the case: a censoring mechanism is active. We will see that the specific way in which the data are censored, suggests a method to compute the maximum likelihood estimator based on these censored observations. This method, the Expectation Maximization (EM) algorithm, is based on a distinction between a so called hidden space, and an observation space. The EM algorithm will be illustrated in the examples. In fact, the examples in section 3 relate to the examples in section 2 in the sense that in section 2 we are allowed to see the sample in the space we call hidden space in section 3.