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

Applicationes Mathematicae (1995)

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

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topRóżański, 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

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

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