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Download Fitting a Loss Distribution: Household Theft Sample Data
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Transcript
Example: Fitting a Loss Distribution to Claims Data1 The following example lists the claims amounts (in dollars) for 120 theft claims made to a household insurance portfolio. The second table gives the sample statistics for the claims amounts. Table 1: Amounts of 120 Theft Claims Made in a Household Insurance Portfolio 3 130 233 412 716 877 1223 1436 11 138 237 423 734 942 1283 1470 27 139 254 436 743 942 1288 1512 36 140 257 456 756 945 1296 1607 47 143 259 473 784 998 1310 1699 49 153 265 475 786 1029 1320 1720 54 193 273 503 819 1066 1367 1772 77 195 275 510 826 1101 1369 1780 78 205 278 534 841 1128 1373 1858 85 207 281 565 842 1167 1382 1922 104 216 396 656 853 1194 1383 2042 121 224 405 656 860 1209 1395 2247 Table 2: Sample Statistics, Theft Data Mean 2020.291667 Median 868.50 Standard Deviation 3949.85736 Minimum 3.00 Maximum 32043.00 Count 120 2348 2377 2418 2795 2964 3156 3858 3872 4084 4620 4901 5021 A histogram of the data is shown below (with the largest three data values excluded). Frequency Histogram 50 45 40 35 30 25 20 15 10 5 0 Bin 5331 5771 6240 6385 7089 7482 8059 8079 8316 11453 22274 32043 We may attempt to fit the data to each of several conjectured loss distributions, and determine which type of loss distribution best fits the data, using the Kolmogorov distance or the Cramervon Mises statistic. If we want to fit an exponential loss distribution, the MLE for the mean of the distribution is πΜ = π₯Μ = $2020.291667. The Kolmogorov-Smirnov test of fit to an exponential loss distribution yields D = 0.2013, with p-value = 0.0001192. If we want to fit a Pareto II loss distribution, the likelihood equations are (with n = 120): ππ ππΌ π ππ = πΌ + πππ(π½) β βππ=1 ππ(π½ + π₯π ) β‘ 0, and ππ½ = ππΌ π½ 1 β (πΌ + 1) βππ=1 π½+π₯ β‘ 0. π These equations may be solved by numeric methods, yielding πΌΜ = 1.88047 and π½Μ = 1872.131756. For these parameter values, the graph of the Pareto II p.d.f. is shown below for loss values between $0.00 and $8316.00 (the 4th largest value in the data set). The Kolmogorov-Smirnov test of fit to a Pareto II distribution yields D = 0.0561, p-value = 0.8443. The Pareto II distribution provides a better fit than the exponential distribution. 1 P. J. Boland, Statistical and Probabilistic Methods in Actuarial Science, Chapman & Hall/CRC, 2007.
