Toward a unified approach to fitting loss models

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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

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Handout/slides

• E-mail me
• Stuart.klugman_at_drake.edu

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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

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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

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The general idea

• Limited number of distributions
• Standard way to present data
• Retain flexibility on testing and selection

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Distributions

• Should be
• Familiar
• Few
• Flexible

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A few familiar distributions

• Exponential
• Only one parameter
• Gamma
• Two parameters, a mode if agt1.
• Lognormal
• Two parameters, a mode
• Pareto
• Two parameters, a heavy right tail

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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.

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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).

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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

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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.

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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).

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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.

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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.

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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.

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Empirical cdf
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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

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Example model

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

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Distribution function plot
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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).

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CDF plot with bounds
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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.

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CDF difference plot
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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.

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Histogram plot
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Hypothesis tests

• Null-model fits
• Alternative-it doesnt
• Three tests
• Kolmogorov-Smirnov
• Anderson-Darling
• Chi-square

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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.

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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.

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Chi-square test

• You have seen this one before.
• It is the only one with an adjustment for
  estimating parameters.

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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.

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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.

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Several models
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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.

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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.