1. Generalized Minimum Bias ModelsBy
Luyang Fu, Ph. D.
Cheng-sheng Peter Wu, FCAS, ASA, MAAA

2. Agenda History and Overview of Minimum Bias Method
Generalized Minimum Bias Models
Conclusions
Mildenhall’s Discussion and Our Responses
Q&A

3. 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

4. History on Minimum Bias
Various bias functions proposed in the past for minimization
Examples of multiplicative bias functions proposed in the past:

5. 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

6. History on Minimum Bias
For example, using the balanced bias functions for the multiplicative model:

7. History on Minimum Bias
Past minimum bias models with the iterative method:

8. 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

9. Other Issues with Minimum Bias
What is the statistical meaning behind these models?
More models to try?
Which models to choose?

10. 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

11. 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

12. Minimum Bias and Statistical Models
Past minimum bias models and their corresponding statistical models

13. 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

14. 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?

15. 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

16. 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)

17. 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).

18. Possible Weighting Functions
Another option: using the square of relativity-adjusted exposure
This is the normal model by Brown (1988).

19. Possible Weighting Functions
Another option: using relativity-square-adjusted exposure
This is the least-square model by Brown (1988).

20. 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 relativity: WpXq 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

21. 2-parameter GMBM
2-parameter GMBM with exposure and relativity adjusted weighting function are:

22. 2-parameter GMBM vs. GLM p q GLM 1 -1 Inverse Gaussian Gamma Poisson 2
Gamma
Poisson 2
Normal

23. 2-parameter GMBM and GLM
GMBM with p=1 is the same as GLM model with the variance function of
Additional special models:
0<q<1, the distribution is Tweedie, for pure premium models
1<q<2, not exponential family
-1<q<0, 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”.

24. 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)

25. 3-parameter GMBM

26. 3-parameter GMBM
Propose 3-parameter GMBM by using the power link function f(x)=xk

27. 3-parameter GMBM When k=2, p=1 and q=1
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

28. 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.

29. 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

30. 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

31. Model Evaluation Criteria
A Severity Case Study
Model Evaluation Criteria
Weighted Absolute Bias (Bailey and Simon 1960)
Weighted Absolute Percentage Bias

32. Model Evaluation Criteria
A Severity Case Study
Model Evaluation Criteria
Pearson Chi-square Statistic (Bailey and Simon 1960)
Combine Absolute Bias and Pearson Chi-square

33. A Severity Case Study Best Fits p q k Criterion wab 2 3 wapb3
wapb
Chi-square 1
combined
-0.5
2.5

34. 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.

35. Mildenhall’s Discussion
Statistical models are always better than non-statistical models
GMBM don’t 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 (1<q<2), 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

36. Our Responses to Mildenhall’s 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.

37. Our Responses to Mildenhall’s 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

38. Our Responses to Mildenhall’s 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.

39. Q & A