Barnett & Zehnwirth
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Extended link ratio family
 Assumes heteroscedastic normality
 Standard link ratio model carry assumptions not usually satisfied by data
 E(yx) = bx
 Diagnostic test  wtd stadardized residuals vs fitted value should appear rdm
 Downward trend indicated that large values are overpredicted and small values are underpredicted

Statistical modeling framework
 Possibility of parameters in:
 Acc yr  determines level from yr to yr
 Dvpmt yr  orthogonal to acc yr
 Pmt yr  trend from pay yr to pay yr
 median = exp(α + Σγ_{k})
 mean = median * exp(0.5σ²)

Statistical Modeling: Assumptions about future
 If pmt / CY trend has been stable in the more recent yrs, assumptin relatively straightforward
 If unstable, assumptions about future will depend on explanation for the instability

Statistical Modeling: prediction intervals
 In an insr writes more than one longtail line and aims for a 100(1  α)% security level on all the lines combined, then the risk margin per line decreases the more lines the company writes.
 Let se = standard error and rm = risk margin:
 se(Total) = √[se²(1) + ... + se²(n)]
 if rm(all lines) = k*se(Total) then
 rm(j) = k*se(j)/Σse(i) < k*se(j)