Development of the Spatial Credibility Model - Cont.

  • Equation 2 requires that the covariances be evaluated. This can be done by considering the following identity:

    (3) Cov(Xiu, Xjv) = E[Cov(Xiu, Xjv|Ri,Rj)] + Cov[E(Xiu|Ri,Rj), E(Xjv|Ri,Rj)]

  • But,

    E(Xiu|Ri,Rj) = m + Ri

    and:

    E(Xjv|Ri,Rj) = m + Rj

  • This permits the second term on the right side of equation 3 to be simplified:

    Cov[E(Xiu|Ri,Rj), E(Xjv|Ri,Rj)] = Cov(m + Ri, m + Rj) = Cov(Ri, Rj)

  • It will be assumed that:

    Cov(Ri, Rj) = f(d(i,j))

  • That is, the spatial covariance is a function of the distance d between counties i and j. If another pair of counties are separated by the same distance, the value of f will be assumed to be identical. For simplicity in notation, f(d(i,j)) will be replaced by f(i,j).

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