Development of the Spatial Credibility Model - Cont.

The spatial credibility formula determines the best estimate of the projected loss cost for county t in future year "0" using a linear combination of the known observations. Using Formula 4.1 from Chapter 7 of the "Foundations of Casualty Actuarial Science" text, the objective is to determine the coefficients which minimize:

E[Xt0 – (a0 + ?i,u aiu Xiu)]2

where t is the county being evaluated. Here, Xiu represents the loss costs in county i (from 1 to N) in year u (from 1 to n). It will be assumed that the loss costs can be decomposed into three components:

The first component, m, represents the mean loss cost over all counties. Ri represents the variation of the individual county loss costs from the overall mean. These are selected such that the average deviation over all counties is zero, E(Ri) = 0. Individual year random fluctuations are represented by Qiu, with the average of the Q's over all years and counties being zero, E(Qiu) = 0. The R's are assumed to be independent of the Q's. These assumptions imply that the overall expected value E(Xiu) is m and that the expected value for a particular county i is E(Xiu|Ri) = m + Ri.