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Operational Risk Modeling Analytics phần 9

(Nếu bạn không có ý kiến như vậy, có lẽ sự khôn ngoan của người được giao nhiệm vụ này cho bạn nên được đặt thành vấn đề.) Khó khăn là dịch kiến thức này vào một phân bố xác suất. Một cuộc thảo luận tuyệt vời về phân phối trước và nền tảng của phân tích Bayesian có thể được tìm thấy trong Lindley [76], | GRAPHICAL COMPARISON OF THE DENSITY AND DISTRIBUTION FUNCTIONS 351 Table 12.1 Data Set B with highest value changed 27 82 115 126 155 161 243 294 340 384 457 680 855 877 974 1193 1340 1884 2558 3476 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 will be used for individual data and the histogram will be used for grouped data. In order to compare the model to truncated data we begin by noting that the empirical distribution begins at the truncation point and represents conditional values that is they are the distribution and density function given that the observation exceeds the truncation point . In order to make a comparison to the empirical values the model must also be truncated. Let the truncation point in the data set be t. The modified functions are 0 F x - F t l-F i 0 ft l-F t X t X t. 12.3 GRAPHICAL COMPARISON OF THE DENSITY AND DISTRIBUTION FUNCTIONS The most direct way to see how well the model and data match up is to plot the respective density and distribution functions. Example 12.1 Consider Data Sets B and c. However for this example and all that follow in Data Set B we replace the value at 15 743 by 3 475 this is to allow the graphs to fit comfortably on a page . These data sets are reproduced here in Tables 12.1 and 12.2. 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 . For Data Set B there are 19 observations the first observation is removed due to truncation . A typical contribution to the likelihood function is 82 l F 50 . The maximum likelihood estimate of the exponential 352 MODEL SELECTION Table 12.2 Data Set c Payment range Number of payments 0-87500 99 .