In dynamic mode, positron emission tomography (PET) can be used to track the evolution of injected radio-labelled molecules in living tissue. This is a powerful diagnostic imaging technique that provides a unique opportunity to probe the status of healthy and pathological tissue by examining how it processes substrates. The spatial aspect of PET is well established in the computational statistics literature. This article focuses on its temporal aspect. The interpretation of PET time-course data is complicated because the measured signal is a combination of vascular delivery and tissue retention effects. If the arterial time-course is known, the tissue time-course can typically be expressed in terms of a linear convolution between the arterial time-course and the tissue residue. In statistical terms, the residue function is essentially a survival function - a familiar life-time data construct. Kinetic analysis of PET data is concerned with estimation of the residue and associated functionals such as flow, flux, volume of distribution and transit time summaries. This review emphasises a nonparametric approach to the estimation of the residue based on a piecewise linear form. Rapid implementation of this by quadratic programming is described. The approach provides a reference for statistical assessment of widely used one- and two-compartmental model forms. We illustrate the method with data from two of the most well-established PET radiotracers, (15)O-H(2)O and (18)F-fluorodeoxyglucose, used for assessment of blood perfusion and glucose metabolism respectively. The presentation illustrates the use of two open-source tools, AMIDE and R, for PET scan manipulation and model inference.