16.3 Graphical comparison of the density and distribution functions

The most direct way to see how well the model and data match is to plot the respective density and distribution functions.

EXAMPLE 16.1

Consider Data Sets B and C as given in Tables 16.1 and 16.2. For this example and all that follow, in Data Set B replace the value at 15,743 with 3,476 (to allow the graphs to fit comfortably on a page). Truncate Data Set B at 50 and Data Set C at 7,500. Estimate the parameter of an exponential model for each data set. Plot the appropriate functions and comment on the quality of the fit of the model. Repeat this for Data Set B censored at 1,000 (without any truncation).

Table 16.1 Data Set B with highest value changed.

Table 16.2 Data Set C.

Payment range Number of payments
0–7,500 99
7,500–17,500 42
17,500–32,500 29
32,500–67,500 28
67,500–125,000 17
125,000–300,000 9
Over 300,000 3

For Data Set B, there are 19 observations (the first observation is removed due to truncation). A typical contribution to the likelihood function is f(82)/[1 − F(50)]. The maximum likelihood estimate of the exponential parameter is = 802.32. The empirical distribution function starts ...

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