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# Toward a unified approach to fitting loss models - PowerPoint PPT Presentation

Toward a unified approach to fitting loss models. Jacques Rioux and Stuart Klugman, for presentation at the IAC, Feb. 9, 2004. Handout/slides. E-mail me [email protected] Overview. What problem is being addressed? The general idea The specific ideas Models to consider

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### Toward a unified approach to fitting loss models

Jacques Rioux and Stuart Klugman, for presentation at the IAC, Feb. 9, 2004

• 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

• 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

• Limited number of distributions

• Standard way to present data

• Retain flexibility on testing and selection

• Should be

• Familiar

• Few

• Flexible

• Exponential

• Only one parameter

• Gamma

• Two parameters, a mode if a>1.

• Lognormal

• Two parameters, a mode

• Pareto

• Two parameters, a heavy right tail

• 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.

• 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).

• 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

• Why do we care?

• Graphical tests require a graph of the empirical density or distribution function.

• Hypothesis tests require the functions themselves.

• 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).

• 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.

• 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.

• 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.

• 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

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

• 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).

• 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.

• 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.

• Null-model fits

• Alternative-it doesn’t

• Three tests

• Kolmogorov-Smirnov

• Anderson-Darling

• Chi-square

• 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.

• 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.

• You have seen this one before.

• It is the only one with an adjustment for estimating parameters.

• 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.

• 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.

• 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.

• 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.