Loss Models: From Data to Decisions, 4th Edition

16.2 Representations of the data and model

All the approaches to be presented compare the proposed model to the data or to another model. The proposed model is represented by either its density or distribution function or perhaps some functional of these quantities such as the limited expected value function or the mean excess loss function. The data can be represented by the empirical distribution function or a histogram. The graphs are easy to construct when there is individual, complete data. When there is grouping or observations have been truncated or censored, difficulties arise. Here, the only cases covered are those where the data are all truncated at the same value (which could be zero) and are all censored at the same value (which could be infinity). Extensions to the case of multiple truncation or censoring points are detailed in [63].³ It should be noted that the need for such representations applies only to continuous models. For discrete data, issues of censoring, truncation, and grouping rarely apply. The data can easily be represented by the relative or cumulative frequencies at each possible observation.

With regard to representing the data, the empirical distribution function is used for individual data and the histogram will be used for grouped data.

To compare the model to truncated data, we begin by noting that the empirical distribution begins at the truncation point and represents conditional values (i.e., they are the distribution and density function given ...