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Relative efficiencies of kernel and local likelihood density estimators


  • Peter Hall
  • Terence Tao


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. Copyright 2002 Royal Statistical Society.

Suggested Citation

  • Peter Hall & Terence Tao, 2002. "Relative efficiencies of kernel and local likelihood density estimators," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(3), pages 537-547.
  • Handle: RePEc:bla:jorssb:v:64:y:2002:i:3:p:537-547

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    References listed on IDEAS

    1. Rebecca A. Betensky & Jane C. Lindsey & Louise M. Ryan & M. P. Wand, 1999. "Local EM Estimation of the Hazard Function for Interval-Censored Data," Biometrics, The International Biometric Society, vol. 55(1), pages 238-245, March.
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    Cited by:

    1. Rodrigues, Clarissa Guimarães & Rios-Neto, Eduardo Luiz Gonçalves & de Xavier Pinto, Cristine Campos, 2013. "Changes in test scores distribution for students of the fourth grade in Brazil: A relative distribution analysis for the years 1997–2005," Economics of Education Review, Elsevier, vol. 34(C), pages 227-242.
    2. Rodrigues, Clarissa G. & Rios-Neto, Eduardo L. G. & Pinto, Cristine Campos de Xavier, 2012. "Changes in test scores distribution for students of the fourth grade in Brazil: a relative distribution analysis for the years 1997 to 2005," Textos para discussão 282, FGV/EESP - Escola de Economia de São Paulo, Getulio Vargas Foundation (Brazil).
    3. Otneim, Håkon & Karlsen, Hans Arnfinn & Tjøstheim, Dag, 2013. "Bias and bandwidth for local likelihood density estimation," Statistics & Probability Letters, Elsevier, vol. 83(5), pages 1382-1387.

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