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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. 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

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
the problem
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
the general idea
The general idea
  • Limited number of distributions
  • Standard way to present data
  • Retain flexibility on testing and selection
  • Should be
    • Familiar
    • Few
    • Flexible
a few familiar distributions
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
  • Add by allowing mixtures
  • That is,


and all

  • Some restrictions:
    • Only the exponential can be used more than once.
    • Cannot use both the gamma and lognormal.
why mixtures
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).
estimating parameters
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
representing the data
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.
what is the issue
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).
issue grouped data
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.
  • 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.
distribution function plot
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
example model
Example model
  • All plots and tests that follow are for a mixture of a lognormal and exponential distribution. The parameters are
confidence bands
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).
other cdf pictures
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.
histogram plot
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.
hypothesis tests
Hypothesis tests
  • Null-model fits
  • Alternative-it doesn’t
  • Three tests
    • Kolmogorov-Smirnov
    • Anderson-Darling
    • Chi-square
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.
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.
chi square test
Chi-square 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.
comparing models
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.
which is the winner
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.
can this be automated
Can this be automated?
  • We are working on software
  • Test version can be downloaded at
  • MLEs are good. Pictures and test statistics are not quite right.
  • May crash.
  • Here is a quick demo.