Another common question that people have is how to figure out what to use for n when figuring out the credibility factor Z with Bühlmann (or more often either empirical or Bühlmann-Straub questions as that is where it gets trickier). There are a couple of things that you can keep in mind. One is that the value for n that you use in finding Z is the same as the denominator that you use in finding X-bar for the insured whose credibility you care about. I mention that since typically people are okay in finding X-bar (or X_A – bar in the empirical case) and it is the same logic as for n.
The other thing that you can look for is that n is the number of exposure units, or if you prefer, units of data, that you have. For example, in D.2 # 17 we are told that the number of claims per month for one insured has a Poisson distribution, so one unit of data is what happens to one person in one month. In the first month of our data, we have 100 insureds, so that is 100 data points for that month, etc., and that is why our n is 100 + 150 + 200.
For comparison, in D.2 #14, we are given that the claim size X has a certain distribution, and want the expected value of the next claim. That means we are looking at one unit of data = one claim, and our n = # of claims.
In order for n to be the number of months in #17, we would need to either have our data be for one single insured, so one month = one month for one insured = one data point, or we would need to change the description of the number of claims to only refer to the number of claims in a month. For example, we could say “The number of claims incurred in a month by a policyholder is … ” and then have our data be “The observed data for one policyholder is … ”
One hint that you can use in the problems is that they will almost always use the phrasing Bühlmann credibility when n is the number of years (or months) of data, or the number of claims. They will usually call it Bühlmann-Straub when we are in a case like #17 in which the number of data points per month varies (usually by varying either the number of insureds, as done here, or by varying the group size), and in that case typically your n will be the sum over all the months (or years) of the number of insureds (or number of people in the group).
Taking a peak at an empirical example, in D.2 #32, we want the estimate of the claim frequency per vehicle for Insured A in year 4. Since it says “per vehicle” then one exposure unit = one vehicle for one year. We are making an estimate for Insured A, so we only care about that insured for our n, and have 2 vehicles (=2 exposures) in the first year, 2 in the second, and 1 in the third for an n of 2 + 2 + 1 = 5.
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