## Brazilian Journal of Probability and Statistics

### A criterion for the fuzzy set estimation of the density function

Jesús A. Fajardo

#### Abstract

In this paper we propose a criterion to estimate the density function by means of a nonparametric and fuzzy set estimator, based on $n$ i.i.d random variable, obtaining a reduction of the integrated mean square error of the fuzzy set estimator regarding the integrated mean squared error of the classic kernel estimators. This reduction shows that the fuzzy set estimator has better performance than the kernel estimations. Also, the convergence rate of the optimal scaling factor is computed, which coincides with the convergence rate in classic kernel estimation. Finally, these theoretical findings are illustrated using a numerical example.

#### Article information

Source
Braz. J. Probab. Stat., Volume 28, Number 3 (2014), 301-312.

Dates
First available in Project Euclid: 17 July 2014

https://projecteuclid.org/euclid.bjps/1405603503

Digital Object Identifier
doi:10.1214/12-BJPS208

Mathematical Reviews number (MathSciNet)
MR3263049

Zentralblatt MATH identifier
1304.62075

#### Citation

Fajardo, Jesús A. A criterion for the fuzzy set estimation of the density function. Braz. J. Probab. Stat. 28 (2014), no. 3, 301--312. doi:10.1214/12-BJPS208. https://projecteuclid.org/euclid.bjps/1405603503

#### References

• Fajardo, J., Ríos, R. and Rodríguez, L. (2012). Properties of convergence of an fuzzy set estimator of the density function. Brazilian Journal of Probability and Statistics 26, 208–217.
• Falk, M. and Liese, F. (1998). Lan of thinned empirical processes with an application to fuzzy set density estimation. Extremes 1, 323–349.
• Reiss, R. D. (1993). A Course on Point Processes. Springer Series in Statistics. New York: Springer.
• Zadeh, L. A. (1965). Fuzzy sets. Information and Control 8, 338–353.