Measurement Techniques

, Volume 62, Issue 9, pp 769–775

# Nonparametric Estimation of the Quadratic Functional of a Multimodal Probability Density

Article

A nonparametric method for estimating the mean square functional of a multimodal probability density of a one-dimensional random variable is examined. The proposed method is based on using the Sturgis and Heinhold–Gaede formulas and an optimum sampling procedure for sampling a range of values of random quantities. This method is compared with the traditional approach based on choosing a spread coefficient using the condition for the maximum of the likelihood function. The conditions for competence of this method are determined.

## Keywords

kernel estimate of probability density estimation of a functional of the probability density multimodal probability density choice of spread coefficient Sturges rule Heinhold–Gaede rule maximum likelihood criterion

## References

1. 1.
A. V. Lapko and V. A. Lapko, “Comparison of empirical and proposed distribution functions for a random variable based on a nonparametric classifier,” Avtometriya, 48, No. 1, 44–50 (2012), DOI: .
2. 2.
A. V. Lapko and V. A. Lapko, “Construction of confidence limits for the probability density based on a regression estimate of it,” Metrologiya, No. 12, 3–9 (2013).Google Scholar
3. 3.
A. V. Lapko and V. A. Lapko, “Estimation of the parameters of a formula for optimum sampling of a region of values of a two-dimensional random variable,” Izmer. Tekhn., No. 5, 9–13 (2018).Google Scholar
4. 4.
H. A. Sturges, “The choice of a class interval,” J. Am. Stat. Assoc., 21, 65–66 (1926).
5. 5.
M. P. Wand, “Data-based choice of histogram bin width,” Am. Statistician, 51, No. 1, 59–64 (1997).Google Scholar
6. 6.
H. Shimazaki and S. Shinomoto, “A method for selecting the bin size of a time histogram,” Neural Comp., 19, No. 6, 1503–1527 (2007).
7. 7.
A. Hacine-Gharbi, P. Ravier, R. Harba, and T. Mohamadi, “Low bias histogram-based estimation of mutual information for feature selection,” Pattern Recogn. Let., 33, No. 10, 1302–1308 (2012).
8. 8.
D. W. Scott, Multivariate Density Estimation: Theory, Practice and Visualization, J. Wiley & Sons, NJ (2015).Google Scholar
9. 9.
A. V. Lapko and V. A. Lapko, “Optimum choice of the number of sampling intervals for a region of variation of a one-dimensional random variable during estimation of the probability density,” Izmer. Tekhn., No. 7, 24–27 (2013).Google Scholar
10. 10.
A. V. Lapko and V. A. Lapko, “Method for sampling a region of values of a multidimensional random variable,” Izmer. Tekhn., No. 1, 16–20 (2019), DOI: 10.32446/0368-1025it.2019-1-16-20.Google Scholar
11. 11.
E. Parzen, “On estimation of a probability density function and mode,” Ann. Math. Stat., 33, No. 3, 1065–1076 (1962).
12. 12.
A. V. Lapko and V. A. Lapko, “Analysis of methods for optimization of a nonparametric estimate of the probability density in terms of the spread coefficient of kernel functions,” Izmer. Tekhn., No. 6, 3–8 (2017).Google Scholar
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