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#1




GLMM & Credibility
Hi All,
Apologies if this is the wrong section. So I am a L&H actuary and I have recently introduced GLMs into our mortality assumption setting process. As part of the implementation process I had to fit these models into the existing limited fluctuation credibility framework we use. Using LFC + factors from a GLM is fine enough for the time being. Backtesting helped my approvers gain enough comfort to let this fly. Naturally I did some research and ended up coming across the relationship between GLMMs and Buhlmann credibility. There was a section in ASM manual for exam C that covered roughly this I think, that was a long time ago. I also read some CAS papers by Klinker. Those were particularly helpful in my understanding. My understanding of GLMMs is that the variable modeled as a random effect will have its level's factors shrunk back to the mean. For certain variables in my mortality models this makes a lot of sense but there is a catch with some other variables. Let me sketch out my example below. Let's say my model is specified as such. Since I am using an expected count offset I am really modeling A/E ratios. Poission family and log link. Also let's assume my aggregate A/E is 100% to simply further. ActualCount ~ RiskClass + IssueAge + offset(log(ExpectedCount)) So the catch arises from the fact I have an expected basis (call it VBT15). I model IssueAge as a random effect and because my underlying expected mortality basis varies by IssueAge I am content with factors being shrunk to 100%. (ok so here's actually my question) Let's say I used some relative risk tool and came up with an expected A/E by risk class; How would / could / should I adjust my model to account for the the relative risk tool and get my GLMM risk class factors to shrink to the risk class factors recommended my relative risk tool? Is this even statistically sound? Is this too oddly specific to be easily answered? So essentially of shrinking towards the mean, shrink towards predefined values? Does this make sense? The ugly work around I see is to model IssueAge as random effect, let GLMM shrink those parameters then model Risk Class as a fixed effect and independently credibility weight my GLMM fixed effect risk class factors to the relative risk tool's predictions. Unsure of how statistically unsound this is but I bet if I backtest this work around it would work well. Aiming to do better though. Thanks for the help y'all. 
#3




Quote:
We are setting a mortality assumption for a life product. We have emerging experience. We are using the GLM to model experience. We aren’t directly modeling qx’s, we are modeling A/Es. We are looking to adjust our expected basis for emerging experience. To be specific a coworker saw higher A/Es on permanent products than term. Turned out to be an unequal weighting of Super Preferred and Standard risks between the two. 
#4




I can't understand your post. A lot of times people can leave out background information and I can deduce what their model is doing/how it works, but in this case, I can't. If you can provide a little more background on what it is you are modeling, you might get better answers and someone might chime in with a succinct reason showing that your results are spurious.

#5




It's a frequency model. I don't think it's particularly egregious.
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#6




If you want to use predefined risk relativities as your complement of credibility, you're better off using a Bayesian glm. This way, instead of shrinking towards the grand mean, you shrink towards the relativities given by your risk tool.
As far as I am aware, there aren't any papers that discuss applying Bayesian GLMs in insurance contexts, but there are plenty of references on the model outside of insurance. 
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