bracketrightBigg which only requires knowledge of the coefficient of variation

Bracketrightbigg which only requires knowledge of the

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bracketrightBigg , which only requires knowledge of the coefficient of variation for the severity distribution σ Y /m Y . triangle 1.1.2 Partial Credibility Clearly the applicability of the above full credibility formulas is limited by the need to estimate the coefficient of variation. In addition if the conclusion is that full credibility does not hold for a particular portfolio, then we need an alternate estimator to ¯ X . A weighted average Z ¯ X + (1 - Z ) M , where M is some known ( manual ) premium, as proposed by the actuaries that developed American credibility, seems like a simple and natural suggestion. The question here is how to esti- mate the credibility factor Z ? Intuitively it should be an increasing function of n and reach 1 asymptotically. Of the many proposals in the actuarial literature, the one that turned out to be a better choice is Z = n n + k , (1.8) where k > 0 is a parameter to be determined. All other proposals, although intuitively appealing, were not statistically sound. Even the choice in (1.8) was originally justified by flawed arguments. These, however, were subse- quently replaced by the rigorous derivation of B¨uhlmann that will be studied in the next chapter.

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10 CHAPTER 1. AMERICAN CREDIBILITY Here is one such flawed derivation. First assume that in a weighted aver- age credibility premium Z ¯ X + (1 - Z ) M we determine Z as to minimize its variance: V bracketleftbig Z ¯ X + (1 - Z ) M bracketrightbig = Z 2 V ( ¯ X ) = Z 2 σ 2 n , where Z and M are assumed here to be constants. Now, in (1.5) we see that in order to have full credibility the variance of ¯ X must be at most m 2 0 . If we set the variance of the credibility premium, that is V ( Z ¯ X +(1 - Z ) M ) = Z 2 V ( ¯ X ) = Z 2 σ 2 /n , to the same standard (for 0 Z 1), then Z 2 σ 2 n = m 2 λ 0 , which implies what is called the square root formula : Z = min braceleftbigg 1 , m σ radicalbigg n λ 0 bracerightbigg , (1.9) that is, the ratio of the full credibility standard, radicalbig n/λ 0 in (1.4), to the coefficient of variation. In the subsequent chapters we will derive k in (1.8) statistically, showing that k = s 2 a , where s 2 and a are components of the variance of ¯ X , and that it produces the correct Z = n n + k = n n + s 2 a = a n a n + s 2 . (1.10) Such a statistical approach to find Z is also called greatest accuracy credibility . 1.2 Greatest Accuracy Credibility As seen above, the estimator ¯ X can fluctuate from the parameter m that it is supposed to estimate, sometimes substantially. Such fluctuations can be due to the natural variability of a random variable around its mean, even if ¯ X is computed for homogeneous observations within a risk class. On the other hand, large fluctuations could be due to a certain heterogeneity left within a risk class.
1.2. GREATEST ACCURACY CREDIBILITY 11 One way to model this, for greater accuracy, is to assume that the risk level of each policyholder in a risk class is characterized by a risk parameter θ .

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