The insurers interest is in estimating m for premium rating If the ob served

# The insurers interest is in estimating m for premium

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The insurer’s interest is in estimating m for premium rating. If the ob- served data is not “credible” (small sample, heterogeneous data, changes in risk characteristics) then it can be disregarded, using an exogenous estimator, a book or manual premium, M , taken from a large block of similar policies. 5

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6 CHAPTER 1. AMERICAN CREDIBILITY A more statistical estimator is ¯ X , which disregards M (i.e. full credibility ). This estimator is unbiased and has a small variance if the sample is large, with a representative risk experience. A compromising estimation approach is to use a combination of M and ¯ X ( partial credibility ), in an attempt to incorporate the qualities of both estimators into a single one and limiting the effect of their drawbacks in any given situation. We will study four basic concepts from American Credibility: 1. How to determine the criterion for Full Credibility when estimating frequencies, 2. How to determine the criterion for Full Credibility when estimating severities, 3. How to determine the criterion for Full Credibility when estimating pure premiums (loss costs), 4. How to determine the amount of partial credibility to assign when one has less data than is needed for full credibility. 1.1.1 Full Credibility If the probability of a small difference between the estimator ¯ X and the parameter it estimates, m , is “high enough”, then the insurer may find ¯ X credible as an estimator of m . Statistically, this can be defined as P {- r m ¯ X - m r m } ≥ p , (1.1) for a chosen tolerance or “confidence” level r > 0 and probability p . Equiv- alently (1.1) can be written as P braceleftbiggvextendsingle vextendsingle vextendsingle vextendsingle ¯ X - m σ/ n vextendsingle vextendsingle vextendsingle vextendsingle r m n σ bracerightbigg p . (1.2) Let y p be a two–sided p –percentile from the distribution of vextendsingle vextendsingle vextendsingle ¯ X - m σ/ n vextendsingle vextendsingle vextendsingle , that is P braceleftbiggvextendsingle vextendsingle vextendsingle vextendsingle ¯ X - m σ/ n vextendsingle vextendsingle vextendsingle vextendsingle y p bracerightbigg = p , (1.3)
1.1. LIMITED FLUCTUATIONS CREDIBILITY 7 (when equality is not possible, take a smoothed percentile; see Section 13.1 of the textbook). Then the above full credibility condition (1.2) is equivalent to y p r m n/σ , that is σ m r y p n = radicalbigg n λ 0 , (1.4) where λ 0 = ( y p /r ) 2 . Remark 1.1. : (i) Note that (1.4) means that the coefficient of variation σ/m , of X j , must be at most radicalbig n/λ 0 for full credibility. (ii) An equivalent form of (1.4) is V ( ¯ X ) = σ 2 n m 2 λ 0 , (1.5) showing that the variance of ¯ X must be sufficiently small to achieve full credibility.

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