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

Authors

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

DOI:

https://doi.org/10.5556/j.tkjm.22.1991.4605

Keywords:

density estimation, MISE, mean integrated square error, recursive kernel estimator

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.

 

References

Parzen, E., "On estimation of a probability density and mode". Ann. Math. Statist. 33, 1065-1076, 1962.

Rosenblatt, M., "The central limit. theorem for mixing sequences of random variables". Z. Wahrscheinl, chkeitsch., 12, 155-171, 1956.

Wegman, E. J. and J. I. Davies. "Remarks on some recursive estimators of a probability density". Ann. Math. Statist., Vol. 7, 316-327, 1979.

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.

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Published

1991-09-01

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

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Papers