Generalized Minimum Bias Models

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Generalized Minimum Bias Models

• By
• Luyang Fu, Ph. D.
• Cheng-sheng Peter Wu, FCAS, ASA, MAAA

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Agenda

• History and Overview of Minimum Bias Method
• Generalized Minimum Bias Models
• Conclusions
• Mildenhalls Discussion and Our Responses
• QA

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History on Minimum Bias

• A technique with long history for actuaries
• Bailey and Simon (1960)
• Bailey (1963)
• Brown (1988)
• Feldblum and Brosius (2002)
• In the Exam 9.
• Concepts
• Derive multivariate class plan parameters by
  minimizing a specified bias function
• Use an iterative method in finding the
  parameters

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History on Minimum Bias

• Various bias functions proposed in the past for
  minimization
• Examples of multiplicative bias functions
  proposed in the past

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History on Minimum Bias

• Then, how to determine the class plan parameters
  by minimizing the bias function?
• One simple way is the commonly used iterative
  method for root finding
• Start with a random guess for the values of xi
  and yj
• Calculate the next set of values for xi and yj
  using the root finding formula for the bias
  function
• Repeat the steps until the values converge
• Easy to understand and can program in almost any
  tools

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History on Minimum Bias

• For example, using the balanced bias functions
  for the multiplicative model

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History on Minimum Bias

• Past minimum bias models with the iterative
  method

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Issues with the Iterative Method

• Two questions regarding the iterative method
• How do we know that it will converge?
• How fast/efficient that it will converge?
• Answers
• Numerical Analysis or Optimization textbooks
• Mildenhall (1999)
• Efficiency is a less important issue due to the
  modern computation power

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Other Issues with Minimum Bias

• What is the statistical meaning behind these
  models?
• More models to try?
• Which models to choose?

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Summary on Minimum Bias

• A non-statistical approach
• Best answers when bias functions are minimized
• Use of iterative method for root finding in
  determining parameters
• Easy to understand and can program in many tools

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Minimum Bias and Statistical Models

• Brown (1988)
• Show that some minimum bias functions can be
  derived by maximizing the likelihood functions of
  corresponding distributions
• Propose several more minimum bias models
• Mildenhall (1999)
• Prove that minimum bias models with linear bias
  functions are essentially the same as those from
  Generalized Linear Models (GLM)
• Propose two more minimum bias models

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Minimum Bias and Statistical Models

• Past minimum bias models and their corresponding
  statistical models

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Statistical Models - GLM

• Advantages include
• Commercial softwares and built-in procedures
  available
• Characteristics well determined, such as
  confidence level
• Computation efficiency compared to the iterative
  procedure
• Issues include
• Required more advanced knowledge for statistics
  for GLM models
• Lack of flexibility
• Rely on the commercial softwares or built-in
  procedures
• Assume the distribution of exponential families.
• Limited distribution selections in popular
  statistical software.
• Difficult to program yourself

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Motivations for Generalized Minimum Bias Models

• Can we unify all the past minimum bias models?
• Can we completely represent the wide range of GLM
  and statistical models using Minimum Bias Models?
  
• Can we expand the model selection options that go
  beyond all the currently used GLM and minimum
  bias models?
• Can we improve the efficiency of the iterative
  method?

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Generalized Minimum Bias Models

• Starting with the basic multiplicative formula
• The alternative estimates of x and y
• The next question is how to roll up Xi,j to Xi,
  and Yj,i to Yj

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Possible Weighting Functions

• First and the obvious option - straight average
  to roll up
• Using the straight average results in the
  Exponential model by Brown (1988)

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Possible Weighting Functions

• Another option is to use the relativity-adjusted
  exposure as weight function
• This is Bailey (1963) model, or Poisson model by
  Brown (1988).

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Possible Weighting Functions

• Another option using the square of
  relativity-adjusted exposure
• This is the normal model by Brown (1988).

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Possible Weighting Functions

• Another option using relativity-square-adjusted
  exposure
• This is the least-square model by Brown (1988).

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Generalized Minimum Bias Models

• So, the key for generalization is to apply
  different weighting functions to roll up Xi,j
  to Xi and Yj,i to Yj
• Propose a general weighting function of two
  factors, exposure and relativityWpXq and WpYq
• Almost all published to date minimum bias models
  are special cases of GMBM(p,q)
• Also, there are more modeling options to choose
  since there is no limitation, in theory, on (p,q)
  values to try in fitting data comprehensive and
  flexible

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2-parameter GMBM

• 2-parameter GMBM with exposure and relativity
  adjusted weighting function are

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2-parameter GMBM vs. GLM
p q GLM
1 -1 Inverse Gaussian
1 0 Gamma
1 1 Poisson
1 2 Normal
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2-parameter GMBM and GLM

• GMBM with p1 is the same as GLM model with the
  variance function of
• Additional special models
• 0ltqlt1, the distribution is Tweedie, for pure
  premium models
• 1ltqlt2, not exponential family
• -1ltqlt0, the distribution is between gamma and
  inverse Gaussian
• After years of technical development in GLM and
  minimum bias, at the end of day, all of these
  models are connected through the game of
  weighted average.

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3-parameter GMBM

• One model published to date not covered by the
  2-parameter GMBM Chi-squared model by Bailey and
  Simon (1960)
• Further generalization using a similar concept of
  link function in GLM, f(x) and f(y)
• Estimate f(x) and f(y) through the iterative
  method
• Calculate x and y by inverting f(x) and f(y)

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3-parameter GMBM
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3-parameter GMBM

• Propose 3-parameter GMBM by using the power link
  function f(x)xk

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3-parameter GMBM

• When k2, p1 and q1
• This is the Chi-Square model by Bailey and Simon
  (1960)
• The underlying assumption of Chi-Square model is
  that r2 follows a Tweedie distribution with a
  variance function

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Further Generalization of GMBM

• In theory, no limitation in selecting the
  weighting functions - another possible
  generalization is to select the weight functions
  separately and differently between x and y
• For example, suppose x factors are stable and y
  factors are volatile. We may only want to use x
  in the weight function for y, but not use y in
  the weight function for x.
• Such generalization is beyond the GLM framework.

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Numerical Methodology for the Iterative Method

• Use the mean of the response variable as the base
• Starting points
• Use the latest relativities in the iterations
• All the reported GMBMs converge within 8 steps

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A Severity Case Study

• Data the severity data for private passenger
  auto collision given in Mildenhall (1999) and
  McCullagh and Nelder (1989).
• Testing goodness of fit
• Absolute Bias
• Absolute Percentage Bias
• Pearson Chi-square Statistic
• Fit hundreds of combination for k, p and q k
  from 0.5 to 3, p from 0 to 2, and q from -2.5 to 4

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A Severity Case Study

• Model Evaluation Criteria
• Weighted Absolute Bias (Bailey and Simon 1960)
• Weighted Absolute Percentage Bias

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A Severity Case Study

• Model Evaluation Criteria
• Pearson Chi-square Statistic (Bailey and Simon
  1960)
• Combine Absolute Bias and Pearson Chi-square

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A Severity Case Study

• Best Fits

Criterion p q k
wab 2 0 3
wapb 2 0 3
Chi-square 1 1 2
combined 1 -0.5 2.5
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Conclusions

• 2 and 3 Parameter GMBM can completely represent
  GLM models with power variance functions
• All published to date minimum bias models are
  special cases of GMBM
• GMBM provide additional model options for data
  fitting
• Easy to understand and does not require advanced
  statistical knowledge
• Can program in many different tools
• Calculation efficiency is not a issue because of
  modern computer power.

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Mildenhalls Discussion

• Statistical models are always better than
  non-statistical models
• GMBM dont go beyond GLM
• - GMBM (k,p,q) can be replicated by the
  transformed GLMs
• with rk as the response variable, wp as the
  weight, and variance function as V(µ)µ2-q/k.
• - When it is not exponential family (1ltqlt2), GLM
  numerical algorithm (recursive re-weighted least
  square) can still apply
• Recursive re-weighted least square is extremely
  fast.
• In theory, agree with Mildenhall in practice,
  subject to discussion

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Our Responses to Mildenhalls Discussion

• Are statistical models always better in practice?
• Require at least intermediate level of
  statistical knowledge.
• Statistical model results can only be provided by
  statistical softwares. For example, GLM is very
  difficult to implement in Excel without
  additional software
• Popular statistical softwares provide limited
  distribution selections.

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Our Responses to Mildenhalls Discussion

• Are statistical models always better in practice?
• Few softwares provide solutions for distributions
  with other power variance functions, such as
  Tweedie and non-exponential distributions
• It requires advanced statistical and programming
  knowledge to program the above distributions
  using the recursive re-weighted least square
  algorithm
• Costs involved acquiring softwares and knowledge

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Our Responses to Mildenhalls Discussion

• Calculation Efficiency
• Recursive re-weighted least square algorithm
  converges with fewer iterations.
• GMBM also converges fast with actuarial data. It
  generally converges within 20 iterations by our
  experience.
• The cost in additional convergence is small and
  the timing difference between GMBM and GLM is
  negligible with modern powerful computers.

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Q A