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Bandwidth selection procedures based on the asymptotic Approximation of the Mean Squared Error (AMSE) criterion are investigated. The authors present the simultaneous selection method of two distinct bandwidths among a class that encompass the existing methods as special cases. They show that the asymptotic behavior of the AMSE reveals dichotomous characteristics de-pending on the sign of the product of the second derivatives for the underlying functions and that the optimal bandwidths that minimize the AMSE does not exist when the sign of the product is positive. They introduce the Modified version of the AMSE (MMSE) that exploits the higher-order bias terms as the non-vanishing penalty.
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