Toward a unified approach to fitting loss models

1. Toward a unified approach to fitting loss models Jacques Rioux and Stuart Klugman, for presentation at the IAC, Feb. 9, 2004

2. Handout/slides • E-mail me • Stuart.klugman@drake.edu

3. Overview • What problem is being addressed? • The general idea • The specific ideas • Models to consider • Recording the data • Representing the data • Testing a model • Selecting a model

4. The problem • Too many models • Two books – 26 distributions! • Can mix or splice to get even more • Data can be confusing • Deductibles, limits • Too many tests and plots • Chi-square, K-S, A-D, p-p, q-q, D

5. The general idea • Limited number of distributions • Standard way to present data • Retain flexibility on testing and selection

6. Distributions • Should be • Familiar • Few • Flexible

7. A few familiar distributions • Exponential • Only one parameter • Gamma • Two parameters, a mode if a>1. • Lognormal • Two parameters, a mode • Pareto • Two parameters, a heavy right tail

8. Flexible • Add by allowing mixtures • That is, where and all • Some restrictions: • Only the exponential can be used more than once. • Cannot use both the gamma and lognormal.

9. Why mixtures? • Allows different shape at beginning and end (e.g. mode from lognormal, tail from Pareto). • By using several exponentials can have most any tail weight (see Keatinge).

10. Estimating parameters • Use only maximum likelihood • Asymptotically optimal • Can be applied in all settings, regardless of the nature of the data • Likelihood value can be used to compare different models

11. Representing the data • Why do we care? • Graphical tests require a graph of the empirical density or distribution function. • Hypothesis tests require the functions themselves.

12. What is the issue? • None if, • All observations are discrete or grouped • No truncation or censoring • But if so, • For discrete data the Kaplan-Meier product-limit estimator provides the empirical distribution function (and is the nonparametric mle as well).

13. Issue – grouped data • For grouped data, • If completely grouped, the histogram represents the pdf, the ogive the cdf. • If some grouped, some not, or multiple deductibles, limits, our suggestion is to replace the observations in the interval with that many equally spaced points.

14. Review • Given a data set, we have the following: • A way to represent the data. • A limited set of models to consider. • Parameter estimates for each model. • The remaining tasks are: • Decide which models are acceptable. • Decide which model to use.

15. Example • The paper has two example, we will look only at the second one. • Data are individual payments, but the policies that produced them had different deductibles (100, 250, 500) and different maximum payments (1,000, 3,000, 5,000). • There are 100 observations.

16. Empirical cdf

17. Distribution function plot • Plot the empirical and model cdfs together. Note, because in this example the smallest deductible is 100, the empirical cdf begins there. • To be comparable, the model cdf is calculated as

18. Example model • All plots and tests that follow are for a mixture of a lognormal and exponential distribution. The parameters are

19. Distribution function plot

20. Confidence bands • It is possible to create 95% confidence bands. That is, we are 95% confident that the true distribution is completely within these bands. • Formulas adapted from Klein and Moeschberger with a modification for multiple truncation points (their formula allows only multiple censoring points).

21. CDF plot with bounds

22. Other CDF pictures • Any function of the cdf, such as the limited expected value, could be plotted. • The only one shown here is the difference plot – magnify the previous plot by plotting the difference of the two distribution functions.

23. CDF difference plot

24. Histogram plot • Plot a histogram of the data against the density function of the model. • For data that were not grouped, can use the empirical cdf to get cell probabilities.

25. Histogram plot

26. Hypothesis tests • Null-model fits • Alternative-it doesn’t • Three tests • Kolmogorov-Smirnov • Anderson-Darling • Chi-square

27. Kolmogorov-Smirnov • Test statistic is maximum difference between the empirical and model cdfs. Each difference is multiplied by a scaling factor related to the sample size at that point. • Critical values are way off when parameters estimated from data.

28. Anderson-Darling • Test statistic looks complex: • where e is empirical and m is model. • The paper shows how to turn this into a sum. • More emphasis on fit in tails than for K-S test.

29. Chi-square test • You have seen this one before. • It is the only one with an adjustment for estimating parameters.

30. Results • K-S: 0.5829 • A-D: 0.2570 • Chi-square p-value of 0.5608 • The model is clearly acceptable. Simulation study needed to get p-values for these tests. Simulation indicates that the p-values are over 0.9.

31. Comparing models • Good picture • Better test numbers • Likelihood criterion such as Schwarz Bayesian. The SBC is the loglikelihood minus (r/2)ln(n) where r is the number of parameters and n is the sample size.

32. Several models

33. Which is the winner? • Referee A – loglikelihood rules – pick gamma/exp/exp mixture • This is a world of one big model and the best is the best, simplicity is never an issue. • Referee B – SBC rules – pick exponential • Parsimony is most important, pay a penalty for extra parameters. • Me – lognormal/exp. Great pictures, better numbers than exponential, but simpler than three component mixture.

34. Can this be automated? • We are working on software • Test version can be downloaded at www.cbpa.drake.edu/mixfit. • MLEs are good. Pictures and test statistics are not quite right. • May crash. • Here is a quick demo.