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Development of the Spatial Credibility Model - Cont
Since all of the terms on the right hand side are known and depend only on j, the right hand side can be written more simply as cj:
This is a system of Nn equations in v and j, with Nn unknowns aiv. This can be written as the product of the transpose of NxN matrix [g(i,j)] with Nxn matrix [aiu]. Since the Nxn product matrix [cju] has cju = cj for all u, its column rank is 1. Assuming that matrix [g(i,j)] is non-singular, the column rank of matrix [aiu] must also be 1. This means that each column is a multiple of the first column, that is, aiv = kv ai1. Substituting this into the previous formula results in the conclusion that kv = 1 for all v. This conclusion can also be reached intuitively by noting that the right hand side of the equation is independent of v. This permits the symbol ai to be used in place of aiv, so that:
for each value of j. An immediate solution to this system of equations can be obtained by observing that bi = ?v aiv = ?v ai = n ai. Since the values of bi are known, aiv = ai = bi / n. The value of a0 can also be determined from a0 = m (1 - ?i,u aiu) = m (1 – ?i bi). This also shows that the coefficients depend on the county but are independent of the year.
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