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

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

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

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

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

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

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

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

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

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

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

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

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

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?

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

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)

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).

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

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

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

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

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

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”.

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)

3-parameter GMBM

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

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

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.

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

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

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

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

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

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.

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

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.

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

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.

Q & A