The roles of ISE and MISE in density estimation

Jones, M. C. (1991). The roles of ISE and MISE in density estimation. Statistics and Probability Letters, 12(1) pp. 51–56.



There is disagreement in the literature concerning the roles of integrated squared error (ISE) and mean integrated squared error (MISE) in kernel density estimation. The issues are reviewed. If best estimation of the underlying density is truly considered to be the objective, conclusions are that ISE is more appropriate than MISE for assessing the performance of density estimates using data-based bandwidth choices and, relatedly, that, in choosing bandwidths initially, aiming for the ISE-optimal bandwidth is more appropriate than aiming for the MISE-optimal target. However, it turns out that practical procedures based on MISE considerations remain one particularly sensible way to go about making automatic bandwidth selections aimed at the ISE-optimal target. Moreover, it is then argued that hoping to be able to estimate a density well from every data set associated with it is unrealistic and that one can only expect to do well in some average sense. This leads back to a conceptual preference for MISE-related procedures, a viewpoint the author commends for general use.

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