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Necessary and sufficient consistency conditions for a recursive kernel regression estimate

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  • Greblicki, Wlodzimierz
  • Pawlak, Miroslaw

Abstract

A recursive kernel estimate [summation operator]i = 1n YiK[+45 degree rule](x - Xi)hi)[+45 degree rule][summation operator]j = 1n K((x - Xj)[+45 degree rule]hj) of a regression m(x) = E{YX = x} calculated from independent observations (X1, Y1),..., (Xn, Yn) of a pair (X, Y) of random variables is examined. ForEY1 + [delta] 0, the estimate is weakly pointwise consistent for almost all ([mu]) x [set membership, variant] Rd, [mu] is the probability measure of X, if and only if[summation operator]i-1n hid I{hi > [var epsilon] } [+45 degree rule] [summation operator]j = 1n hjd --> 0 as n --> [infinity], all [var epsilon] > 0, and[summation operator]i = 1[infinity] hid = [infinity], d is the dimension of X. For EY1 + [delta] 0, the estimate is strongly pointwise consistent for almost all ([mu]) x [set membership, variant] Rd, if and only if the same conditions hold. ForEY1 + [delta] 0, weak and strong consistency are equivalent. Similar results are given for complete convergence.

Suggested Citation

  • Greblicki, Wlodzimierz & Pawlak, Miroslaw, 1987. "Necessary and sufficient consistency conditions for a recursive kernel regression estimate," Journal of Multivariate Analysis, Elsevier, vol. 23(1), pages 67-76, October.
    • Handle: RePEc:eee:jmvana:v:23:y:1987:i:1:p:67-76
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    Citations

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    Cited by:

    1. Aboubacar Amiri, 2013. "Asymptotic normality of recursive estimators under strong mixing conditions," Statistical Inference for Stochastic Processes, Springer, vol. 16(2), pages 81-96, July.
    2. Miroslaw Pawlak, 1991. "On the almost everywhere properties of the kernel regression estimate," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 43(2), pages 311-326, June.
    3. Juan Vilar-Fernández & José Vilar-Fernández, 2000. "Recursive local polynomial regression under dependence conditions," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 9(1), pages 209-232, June.
    4. Györfi, László & Walk, Harro, 1997. "On the strong universal consistency of a recursive regression estimate by Pál Révész," Statistics & Probability Letters, Elsevier, vol. 31(3), pages 177-183, January.
    5. Harro Walk, 2001. "Strong Universal Pointwise Consistency of Recursive Regression Estimates," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(4), pages 691-707, December.
    6. Zhou, Yong & Liang, Hua, 2000. "Asymptotic Normality for L1 Norm Kernel Estimator of Conditional Median under [alpha]-Mixing Dependence," Journal of Multivariate Analysis, Elsevier, vol. 73(1), pages 136-154, April.

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