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#53




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#54




This thread got a bit distracted. Here's what I've gathered from user response and other readings I have done.
Per the "Practitioners Guide to GLMS (I'd link it, but it links to a direct pdf): Page 3: log linked Poisson GLMs are equivalent to multiplicative balance principles of minimum bias estimations (minimum bias estimation goes all the way back to the 1960s and were used when computing power was limited) Page 19: Loglinked Poisson is commonly used for Frequencies because the log link makes it a multiplicative model (much easier to implement and compare factors) and because it is invariant to measures of time: modeling frequencies per year will yield the same results as per month. Page 20: Loglinked Gamma is commonly used for Severities because the log link makes it multiplicative and Gamma is invariant to measures of currency. Measuring severity in dollars and cents will yield same results. The log linked Tweedie distribution w/ p in (1,2) is considered a compound PoissonGamma distribution. The closer to 1, it acts more like Poisson, and closer to 2, it acts more like Gamma. Common values are 1.51..65. It also makes the assumption that frequency and severity are highly correlated. I'm not too familiar with the Tweedie distribution likelihood function, but due to its complexity, it's a bit harder to grab some metrics. It really depends on your data if it is appropriate. When you select a family, you're choosing the meanvariance relationship. For Poisson GLMs, the meanvariance relationship is the identity. Despite the warnings that most statistical software gives you, it's completely reasonable to model a relationship in continuous data in which the relationship between two variables is linear on the log scale, and the variance increases in accordance with the mean. If you look at the residuals, you can determine whether the Poisson meanvariance relationship is accurate. If not, may be better to use Gamma whose meanvariance is x^2. Back to my original question: Is there any major disadvantages of using Poisson over Tweedie? No, but it's worth also checking Gamma. 
#56




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For a true poisson distribution, the error is proportional to the square root of the mean. So if you measured value is 110, and your model thinks the true value is 100, then you are like 1 sigma off. And if your measured value is 1032 and your model thinks the true value is 1000, then that is also 1 sigma off. It weights the two errors appropriately relative to each other. For gamma, the error is proportional to the mean. So if 110 vs 100 is equivalent to 1100 vs 1000. The variances add in your pure premium: the variance from the poisson process that is probably driving your claims, plus some severity distribution's variance. So the poisson model probably doesn't assume enough variance, and your model will give to much weight to the data point at 1000. Just how much does that matter? You mileage will vary. The other problem is that the poisson will get our standard errors wrong as well. The over dispersed poisson model helps with that part. 
#57




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#58




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For standard Poisson, dispersion is not even estimated, as it's assumed to be 1. In practice, dispersion is higher than 1, and way higher than 1 if you're using Poisson to model pure premium. As long as you're using Pearson estimate of dispersion, your standard errors should be more or less in the same ballpark regardless of distribution you pick, but if you use something like a deviance estimate, then you may be off by a mile if your distribution is off by a mile. 
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glm, poisson, tweedie 
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