credibility-weighted estimate

Suppose you want one final number for next year's expected claim, and you have two candidate numbers: what this risk's own recent experience suggests, and what a broad benchmark (the class average, the manual rate, the prior belief) suggests. The credibility-weighted estimate simply blends them, putting weight Z on your own data and weight (1 minus Z) on the benchmark. It is a weighted average where the weight Z, called the credibility factor, lives between 0 and 1.

The formula is short and is the heart of all credibility theory: estimate = Z times (own observation) plus (1 minus Z) times (prior or benchmark). If Z is 0.7, you take 70 percent of what your own data says and 30 percent of the benchmark. When you have a lot of relevant own-data, Z climbs toward 1 and the estimate listens mostly to you. When you have little, Z falls toward 0 and the estimate leans on the benchmark. A worked case: if own-data says the expected claim is 1,200, the benchmark says 1,000, and Z is 0.6, the credibility-weighted estimate is 0.6 times 1,200 plus 0.4 times 1,000 = 1,120.

This single line is what an actuary actually writes down when experience-rating a policy, blending a new region's data with a countrywide rate, or updating a thin mortality assumption. The whole subtlety of credibility theory is not the blending formula — that part is trivial — but choosing Z honestly. Different theories (limited-fluctuation, Buhlmann, Bayesian) are really just different principled recipes for the single number Z.