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Hereditarity of potential matrices and positive affine prediction of nonnegative risks from mixture models

ABSTRACT Nonnegative linear filtering of nonnegative risk variables necessitates positivity properties on the variance–covariance matrices of random effects, if experience rating is derived from mixture models. A variance–covariance matrix is a potential if it is nonsingular and if its inverse is diagonally dominant, with off-diagonal entries that are all nonpositive. We consider risk models with stationary random effects whose variance–covariance matrices are potentials. Positive credibility predictors of nonnegative risks are obtained from these mixture models. The set of variance–covariance matrices that are potentials is closed under extraction of principal submatrices. This strong hereditary property maintains the positivity of the affine predictor if the horizon is greater than one and if the history is lacunary. The specifications of the dynamic random effects presented in this paper fulfill the required positivity properties, and encompass the three possible levels for the length of memory in the mixing distribution. A case study discusses the possible strategies in the prediction of the pure premium from dynamic random effects.