ABSTRACT
As we have already mentioned at the beginning of Lecture 3, we are developing our theory in the following way: the duration of life has a distribution function F , which is used to determine all the probabilities in which we are interested. This distribution function, however, is not known; so we build statistical estimates of F based on the available data, such as the empirical distribution function F̂n, or other functions to be discussed later. For example, for the two ages x < x ′ the difference F(x ′)−F(x) is the probability P{x≤ T < x ′},
P{x≤ T < x ′}= F(x ′)−F(x), while the difference F̂n(x ′)− F̂n(x) is only its statistical estimate. Then we study the properties of our estimate: how accurate it is, how quickly it converges to F , the probability of deviations from F of a certain magnitude and so on. We may also take a somewhat different approach: Based on general considerations we can choose a hypothetical distribution function F ; and then, comparing it to our estimate, we can decide how well our hypothesised function F corresponds to the empirical data.
