The roles of ISE and MISE in density estimation

• Published in: Statistics & probability letters Vol. 12; no. 1; pp. 51 - 56
• Main Author: Jones, M.C.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.07.1991; Elsevier
• Series: Statistics & Probability Letters
• Subjects: Adaptive procedure; AMS 1980 Subject Classification: Primary 62G05; AMS 1980 Subject Classification: Primary 62G05 Secondary 62C99 Adaptive procedure bandwidth selection kernel estimation risk function squared error; bandwidth selection; Exact sciences and technology; kernel estimation; Mathematics; Probability and statistics; risk function; Sciences and techniques of general use; Secondary 62C99; squared error; Statistics; Adaptive procedure; squared error; Secondary 62C99; bandwidth selection; risk function; AMS 1980 Subject Classification: Primary 62G05; kernel estimation; Risk theory; Relative density; Estimation; Selection; Bandwidth; Kernel method
• ISSN: 0167-7152; 1879-2103
• DOI: 10.1016/0167-7152(91)90163-L

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.