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#1




Confidence Intervals from combined GLM outputs
Hi everyone, long time lurker, first time poster.
Recently at work we have been debating how to go about creating confidence intervals from GLM output. Specifically the confidence interval around the predicted risk premium. Say for instance we had:
To get the risk premium estimate, we are combining the parameter estimates produced. To get the 95% Confidence intervals can we use the "Wald 95% Confidence Limits" from Proc GENMOD? One of my colleagues is trying to use the Compound Poisson distribution as a proof that we can combine the model parameters (implying each "segment" in a regression is it's own Poisson/Gamma variable with parameters equal to the combined model efficients  no idea on how we get shape and scale parameters this way for gamma models). I've found literature on this topic really sparse, so I would appreciate if anyone could point me in particular direction, or has had some experience with this. Also due to the CLT are the regression coefficients normally distributed? Would it be a better approach to simulate the risk premium and assume that the coefficients between all models are independent? Cheers for any help cj 
#2




What did you assume the distribution to be for Frequency & severity?
Are you assuming any correlation between perils?
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#3




Using Poisson and Gamma for frequency and severity. We have assumed perils are independent (and I think they generally are), I was wondering if there might be correlation arising in (say) the Sum Insured parameter estimates across all the models, which means using the combined 95% Wald CI understate the true range.

#4




Just so I understand better, is this how you're approaching it?
Z=N1*X1 + N2*X2 + N3*X3 + N4*X4 where each Ni~Poisson(lambda i) and Xi ~ Gamma (alpha i, beta i). You have lambda's, alpha's, and beta's estimated from your data on each peril. Do you want a confidence interval around Z given these estimates? Last edited by AMedActuary; 05122016 at 06:44 PM.. 
#6




Quote:
Exactly what I would have done myself. Riley 
#7




I agree. If you want the uncertainty in your estimated parameters (lambda, alpha, beta) to be incorporated in the confidence interval and if there's prior information on these parameters you want to use, you would have to use a Bayesian model. If not, then bootstrapping is the way to go.

#8




Quote:

#9




I believe you are talking about a confidence interval on the expected loss, and not a prediction interval around the actual losses.
In that case, if the standard errors on the frequency and severity are relatively small compared to their values, and if they are independent between frequency and severity, then you can use the following approximations: 1) Pure Premium = frequency * severity : In this case, the relative standard errors add. (error in pure premium/pure premium) = sqrt((error in frequency/frequency)^2 + (error in severity/severity)^2) 2) Pure Premium = Pure Premium of Peril 1 + Pure Premium of Peril 2 : In this case, the absolute errors add. (error in total) = sqrt((error in peril 1)^2 + (error in peril 2)^2) This is not as accurate as bootstrapping or simulation. But it can be a useful check. 
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combined, confidence interval, glm 
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