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