Loss Models: From Data to Decisions, 4th Edition

11.2 The empirical distribution for complete, individual data

As noted in Definition 11.3, the empirical distribution assigns probability 1/n to each-data point. That definition works well when the value of each data point is recorded. An alternative definition follows.

Definition 11.5 The empirical distribution function is

where n is the total number of observations.

EXAMPLE 11.1

Provide the empirical probability functions for the data in Data Sets A and B. For Data Set A also provide the empirical distribution function. For Data Set A assume all seven drivers who had five or more accidents had exactly five accidents.

For notation, a subscript of the sample size (or of n if the sample size is not known) is used to indicate an empirical function. Without the subscript, the function represents the true function for the underlying random variable. For Data Set A, the empirical probability function is

where the values add to 0.999999 due to rounding. The empirical distribution function is a step function with jumps at each data point.

For Data Set B,

As noted in the example, the empirical ...