You are not currently logged in.
Access your personal account or get JSTOR access through your library or other institution:
If You Use a Screen ReaderThis content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Relative Efficiencies of Kernel and Local Likelihood Density Estimators
Peter Hall and Terence Tao
Journal of the Royal Statistical Society. Series B (Statistical Methodology)
Vol. 64, No. 3 (2002), pp. 537-547
Stable URL: http://www.jstor.org/stable/3088787
Page Count: 11
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Preview not available
Local likelihood methods enjoy advantageous properties, such as good performance in the presence of edge effects, that are rarely found in other approaches to nonparametric density estimation. However, as we argue in this paper, standard kernel methods can have distinct advantages when edge effects are not present. We show that, whereas the integrated variances of the two methods are virtually identical, the integrated squared bias of a conventional kernel estimator is less than that of a local log-linear estimator by as much as a factor of 4. Moreover, the greatest bias improvements offered by kernel methods occur when they are needed most-i.e. when the effect of bias is particularly high. Similar comparisons can also be made when high degree local log-polynomial fits are assessed against high order kernel methods. For example, although (as is well known) high degree local polynomial fits offer potentially infinite efficiency gains relative to their kernel competitors, the converse is also true. Indeed, the asymptotic value of the integrated squared bias of a local log-quadratic estimator can exceed any given constant multiple of that for the competing kernel method. In all cases the densities that suffer problems in the context of local log-likelihood methods can be chosen to be symmetric, either unimodal or bimodal, either infinitely or compactly supported, and to have arbitrarily many derivatives as functions on the real line. They are not pathological. However, our results reveal quantitative differences between global performances of local log-polynomial estimators applied to unimodal or multimodal distributions.
Journal of the Royal Statistical Society. Series B (Statistical Methodology) © 2002 Royal Statistical Society