Generalized Minimum Bias Models

1. Generalized Minimum Bias Models By 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 upXi,jtoXi, and Yj,ito 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

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

32. 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

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