On a strongly consistent estimator of the squared L_2-norm of a function

Roman Różański

Applicationes Mathematicae (1995)

  • Volume: 23, Issue: 3, page 279-284
  • ISSN: 1233-7234

Abstract

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A kernel estimator of the squared L 2 -norm of the intensity function of a Poisson random field is defined. It is proved that the estimator is asymptotically unbiased and strongly consistent. The problem of estimating the squared L 2 -norm of a function disturbed by a Wiener random field is also considered.

How to cite

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Różański, Roman. "On a strongly consistent estimator of the squared L_2-norm of a function." Applicationes Mathematicae 23.3 (1995): 279-284. <http://eudml.org/doc/219131>.

@article{Różański1995,
abstract = {A kernel estimator of the squared $L_2$-norm of the intensity function of a Poisson random field is defined. It is proved that the estimator is asymptotically unbiased and strongly consistent. The problem of estimating the squared $L_2$-norm of a function disturbed by a Wiener random field is also considered.},
author = {Różański, Roman},
journal = {Applicationes Mathematicae},
keywords = {strong consistency; stochastic integral with respect to a p-parameter martingale; Poisson random field; Wiener random field; asymptotic unbiasedness; kernel estimator; stochastic integral with respect to a -parameter martingale; intensity function},
language = {eng},
number = {3},
pages = {279-284},
title = {On a strongly consistent estimator of the squared L\_2-norm of a function},
url = {http://eudml.org/doc/219131},
volume = {23},
year = {1995},
}

TY - JOUR
AU - Różański, Roman
TI - On a strongly consistent estimator of the squared L_2-norm of a function
JO - Applicationes Mathematicae
PY - 1995
VL - 23
IS - 3
SP - 279
EP - 284
AB - A kernel estimator of the squared $L_2$-norm of the intensity function of a Poisson random field is defined. It is proved that the estimator is asymptotically unbiased and strongly consistent. The problem of estimating the squared $L_2$-norm of a function disturbed by a Wiener random field is also considered.
LA - eng
KW - strong consistency; stochastic integral with respect to a p-parameter martingale; Poisson random field; Wiener random field; asymptotic unbiasedness; kernel estimator; stochastic integral with respect to a -parameter martingale; intensity function
UR - http://eudml.org/doc/219131
ER -

References

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  1. R. Cairoli and J. B. Walsh (1975), Stochastic integrals in the plane, Acta Math. 134, 111-183. Zbl0334.60026
  2. C. W. Gardiner (1984), Handbook for Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer Series in Synergetics, Berlin. 
  3. J. Koronacki and W. Wertz (1987), A global stopping rule for recursive density estimators, Statist. Planning Inference 20, 23-39. Zbl0850.62356
  4. H. Ramlau-Hansen (1983), Smoothing counting process intensities by means of kernel functions, Ann. Statist. 12, 453-466. Zbl0514.62050
  5. P. Reveš (1968), Laws of Large Numbers, Academic Press, New York. 
  6. R. Różański (1992), Recursive estimation of intensity function of a Poisson random field, J. Statist. Planning Inference 33, 165-174. Zbl0770.62084
  7. E. F. Schuster (1974), On the rate of convergence of an estimate of a probability density, Scand. Actuar. J., 103-107. Zbl0285.62016
  8.  

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