A NOTE ON RECURSIVE ESTIMATOR OF THE DENSITY FUNCTION WHICH IS NOT NECESSARY CONTINUOUS

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Published: Sep 1, 1991
DOI: https://doi.org/10.5556/j.tkjm.22.1991.4605
Keywords:
density estimation MISE mean integrated square error recursive kernel estimator

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M. M. EL-FAHHAM
Dept. of Mathematics, Faculty of Science, Cairo University, Cairo, Egypt.

Abstract




Consider a sequence $X_1$, $X_2, \cdots , X_n$ of independent, identically distributed random variables with unkown density function $f$, which with its first derivative are not necessarily continuous, and let


\[f_n^*=(n^2h_n)^{-1/2}\sum_{j=1}^{n}h_j^{-1/2}K\left(\frac{x-X_j}{h_j}\right)\]





be the recursive kemel estimator of $f$. It will be shown, under certain additional regularity conditions on $K$ and $|h(n)|$, that, $MISE\ [f^*_n(x)] =o (n^{-4/5})$ if $f$ and $f'$ are continuous, whereas $MISE\ [f^*_{n_1}(x)] =o (n^{-3/4})$ if $f$ is continuous and $f'$ is not and $MISE\ [f^*_{n_2}(x)] =o (n^{-1/2})$ if $f$ and $f'$ ara not continuous.


 







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How to Cite
EL-FAHHAM, M. M. (1991). A NOTE ON RECURSIVE ESTIMATOR OF THE DENSITY FUNCTION WHICH IS NOT NECESSARY CONTINUOUS. Tamkang Journal of Mathematics, 22(3), 213–221. https://doi.org/10.5556/j.tkjm.22.1991.4605
Issue
Vol. 22 No. 3 (1991)
Section
Papers
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References

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Van Eeden C., "Mean integrated squared error of Kernel estimators when the density and its derivative arc not necessarily continuous". Roppart de Recherches no. 82, Departement de mathematiques et de Statistique, Universite de Montreal. 1984.