Bandwidth selection for kernel distribution function estimation

Abstract

Leave-one-out cross-validation is a popular and readily implemented heuristic for bandwidth selection in nonparametric smoothing problems. In this note we elucidate the role of leave-one-out selection criteria by discussing a criterion introduced by Sarda (1993) for bandwidth selection for kernel distribution function estimators (KDFEs). We show that for this problem, use of the leave-one-out KDFE in the selection procedure is asymptotically equivalent to leaving none out. This contrasts with kernel density estimation, where use of the leave-one-out density estimator in the selection procedure is critical.Unfortunately, simulations show that neither method works in practice, even for samples of size as large as 1000. In fact, we show that for any fixed bandwidth, the expected value of the derivative of the leave-none-out criterion is asymptotically positive. This result and our simulations suggest that the criteria are increasing and that for sufficiently large samples (e.g., n = 100), the smallest available bandwidth will always be selected, thus contradicting the optimality result of Sarda for this estimator.As an alternative to minimizing a selection criterion, we propose a plug-in estimator of the asymptotically optimal bandwidth. Simulations suggest that the plug-in is a good esti-*Supported by Hatch Grant 151410 NYF tsupported by NSERC (Canada) and FCAR (Quebec) 1 mator of the asymptotically optimal bandwidth even for samples as small as 10 observations and is not too far from the finite sample optimal bandwidth.

Nonparametric Kernel Distribution Function Estimation withkerdiest: AnRPackage for Bandwidth Choice and Applications

“…The plug-in bandwidth choice was studied, both theoretically and by simulation studies, by Altman and Leger (1995) and Polanski and Baker (2000). The least-squares cross-validation method was analyzed in Sarda (1993), but, as revealed in Altman and Leger (1995), it basically requires very large sample sizes to ensure good results. Hence, only the second approach, namely the modified cross-validation proposed in Bowman, Hall, and Prvan (1998), is of interest for implementation in a language programming and for application to real data sets.…”

Nonparametric Estimation of Value-at-Risk

“…That is the reason why we believe that none of them has been successful in practice. Altman and Léger (1995) discussed a plug-in method based on an expression of A-MISE which includes an additional term in the Taylor expansion of F X (x), but this leads to the need to estimate more than one functional expression and the results are still not straightforward. We will return to this problem in the next section, where a double transformation will guide the bandwidth choice.…”

Analysis of high level ozone concentrations using nonparametric methods

“…whereĈ is a preliminary nonparametric approximation of the unknown values (Altman and Leger, 1995;Polanski and Baker, 2000).…”

“…The simulation study in Bowman et al (1998) compares the cross-validation technique with the plug-in method proposed in Altman and Leger (1995), showing that better results can be obtained with the cross-validation criterion. A disadvantage of the cross-validation method when it is compared with plug-in methods is its worse performance, in terms of computing time.…”