To give a feeling of what this book is about, it is perhaps best to take a look at some real-life examples. Real-life examples have the disadvantage of giving rise to a lot of discussion on the interpretation of the data, as the authors have experienced when they started a lecture with a real-life example. This often distracted the audience from the main message of the lecture. But they have the advantage of “sticking in the mind,” which might be more important than the temporary distraction they might cause. Therefore, the first four sections of this chapter are about real data. Section 1.1 is concerned with the estimation of the expected duration of ice (in days) at Lake Mendota in Wisconsin, assuming these expected durations decrease in time. In Section 1.2, a data set on time-till-onset of a nonlethal lung tumor for mice is studied. There are two groups of mice, one living in a conventional environment and the other in a germ-free environment. The main question then is whether the distribution of the time-till-onset of the tumor is affected by the choice of environment. The complication is that the times of onset are not precisely observed, but subject to censoring. The third example, in Section 1.3, concerns the estimation of a relatively complicated quantity, the transmission potential of a disease, also based on censored data on hepatitis A in Bulgaria. Section 1.4 introduces the Bangkok Metropolitan Administration injecting drug users cohort study, which is further analyzed in Chapter 12, using methods that were developed for competing risk models.

In Section 1.5, a particular shape constrained estimation problem is considered. It is argued that this problem (and many of the other problems to be considered in this book) can also be viewed from another perspective; for example, as inverse problem, mixture model, or censoring problem. As will be seen later in this book, these points of view immediately suggest methods one could use for estimating shape constrained functions and methods one could use to compute these.