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Development of the Spatial Credibility Model - Cont
The reader can confirm that this result is consistent with the classical credibility formula under the assumption of the covariance structure f(i,j) = ?ij ?2 and g(i,j) = ?ij s2.
It should also be noted that the functions f and g, or more properly, ft and gt, represent the spatial covariance and the intertemporal covariance in a neighborhood At around county t. The neighborhood needs to be large enough to reliably estimate ft and gt, but small enough to represent the covariance structure near county t. In practice, it may be appropriate to assume that fw and gw are essentially identical to ft and gt for all counties w in a neighborhood Bt of t. Further refinements in the model could be achieved by permitting the functions f and g to depend on location and direction, rather than on distance only.
A simplification of the spatial credibility result can be obtained by approximating f(i,j) and g(i,j) with discrete valued functions. Recall that these functions depend solely on the distance between counties i and j, not on their specific locations. For example, let f and g take distinct values for a series of distances such as 0 to 50 miles, 51 to 100 miles, 100 to 150 miles, and so on. This would enable the references to specific counties for the functions f(t,j), f(i,j) and g(i,j) in equation 5 to be replaced by a small number of values. If it can be assumed that the functions f and g are independent of county t, and if all counties formed a uniform pattern such as a rectangular grid, then the resulting bi coefficients would be independent of t. In this situation, the same coefficients would be applied for all counties, eliminating the need to reevaluate the coefficients for each individual county t. It may be advantageous to superimpose a rectangular grid to replace the actual county structure in order to achieve this simplification. The result would be a smoothing process analogous to the current concentric circle technique, with more weight given to nearby counties than to those that are further away.
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