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Asymptotic bias and variance for a general class of varying bandwidth density estimators
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  • Published: June 1996

Asymptotic bias and variance for a general class of varying bandwidth density estimators

  • Ola Hössjer1 

Probability Theory and Related Fields volume 105, pages 159–192 (1996)Cite this article

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Summary

We consider a general class of varying bandwidth estimators of a probability density function. The class includes the Abramson estimator, transformation kernel density estimator (TKDE), Jones transformation kernel density estimator (JTKDE), nearest neighbour type estimator (NN), Jones-Linton-Nielsen estimator (JLN), Taylor series approximations of TKDE (TTKDE) and Simpson's formula approximations of TKDE (STKDE). Each of these estimators needs a pilot estimator. Starting with an ordinary kernel estimator\(\hat f_1\), it is possible to iterate and compute a sequence of estimates\(\hat f_2 ,...,\hat f_t\), using each estimate as a pilot estimator in the next step. The first main result is a formula for the bias order. If the bandwidths used in different steps have a common orderh=h(n), the bias of\(\hat f_k\) is of orderh 2k∧m,k=1, ...,t. Hereh m is the bias order of the ideal estimator (defined by using the unknownf as pilot). The second main result is a recursive formula for the leading bias and stochastic terms in an asymptotic expansion of the density estimates. Ifm<∞, it is possible to make\(\hat f_t\) asymptotically equivalent to the ideal estimator.

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Authors and Affiliations

  1. Department of Mathematical Statistics, Lund University, Box 118, S-221 00, Lund, Sweden

    Ola Hössjer

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  1. Ola Hössjer
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Hössjer, O. Asymptotic bias and variance for a general class of varying bandwidth density estimators. Probab. Th. Rel. Fields 105, 159–192 (1996). https://doi.org/10.1007/BF01203834

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  • Received: 21 September 1994

  • Revised: 07 November 1995

  • Issue Date: June 1996

  • DOI: https://doi.org/10.1007/BF01203834

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Mathematics Subject Classifications (1991)

  • 62G07
  • 62G20
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