Suscripción a Biblioteca: Guest
SJR: 0.275 SNIP: 0.59 CiteScore™: 0.8

ISSN Imprimir: 1064-2315
ISSN En Línea: 2163-9337

# Journal of Automation and Information Sciences

DOI: 10.1615/JAutomatInfScien.v51.i12.20
pages 7-17

## Estimation of Experimental Distribution Function on the Basis of Finite Samples of Random Variable

Andrey B. Lozynsky
G.V. Karpenko Physico-Mechanical Institute of National Academy of Sciences of Ukraine, Lvov
Igor M. Romanyshyn
G.V. Karpenko Physico-Mechanical Institute of National Academy of Sciences of Ukraine, Lvov
Bogdan P. Rusyn
G.V. Karpenko Physico-Mechanical Institute of National Academy of Sciences of Ukraine, Lvov

### SINOPSIS

A new approach to estimating the distribution function of a random variable based on finite (including small) samples is proposed. The approach is based on determining the estimates of positions of points of the distribution function. The theoretical substantiation of the approach and the algorithm for estimating the distribution function are presented. The equations (the integral) for determining the points on the required distribution function are exact. Therefore, with the knowledge of the distribution of any order statistics the points of the required distribution function are determined exactly. In the simplest case the approach can be reduced to determining the median of statistics. Usually its exact value is unknown. Therefore the sample median is used as the median of the statistics distribution. The consistency, bias and effectiveness of estimates of the points position on the required distribution function are considered. It is shown that the proposed estimates are consistent but biased, and the bias depends on the form of the required distribution and decreases with the increasing size of the subsample. Numerical experiments for the most important practical cases indicate the increased effectiveness of the proposed approach compared to the classical one. The simplified algorithms for the approximate determination of distribution parameters are proposed and the error estimates are given which indicate the acceptability of the proposed simplified algorithms. The paper presents the results of numerical modeling which illustrate the correspondence of the constructed distribution function with the model one and the advantages of this approach. It is noted that the estimation of the distribution function based on the proposed approach in the form of the interpolated curve allows one to perform the numerical differentiation, filtering and other signal processing operations. At the same time the classical distribution function in the form of the step curve has a number of known limitations at the application of these processing operations. The method for filtering a signal with uncorrelated pulse interference based on the proposed approach is described. The Appendix gives the determination of abscissas and ordinates of the required distribution function of a random variable by solving the integral equations in the analytical form for the case of uniform distribution based on subsamples of two elements. The obtained solution is fully consistent with simplified algorithms.

### REFERENCIAS

1. Aliev T., Musaeva N., Suleymanova M., Gazizade B., Technology for calculating the parameters of the density function of the normal distribution of the useful component in a noisy process, Journal of Automation and Information Sciences, 2016, 48, No. 4,39-55, DOI: 10.1615/ JAutomatInfScien.v48.i4.50. .

2. Rosenblatt M., Remarks on some non-parametric estimates of a density function, Annals Math. Statist., 1956, 27, 832-837. .

3. ParzenE., On estimation of a probability density function and mode, Ann. Math. Statist., 1962, 33, No. 3, 1065-1076. .

4. Eddy W.F., Optimum kernel estimators of the mode, Ann Statist., 1980, 8, 870-882. .

5. Eddy W.F., The asymptotic distributions of kernel estimators of the mode, Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete, 1982, 59, 279-290. .

6. Nadaraya E.A., Some new estimates of distribution functions, Teor. Veroyatnost. i Primenen., 1964, 9, No. 3, 550-554. .

7. Nadaraya E.A., BabiluaP., Sokhadze G., The estimation of a destribution function by indirect sample, Ukrainskiy matematychnyi zhurnal, 2010, 62, No. 12, 1642-1658, DOI: 10.1007/ s11253-011-0479-y. .

8. Kuleshov E.L., Interval estimate of distribution function of probabilities, Avtometriya, 2015, 51, No. 2, 23-26, DOI: 10.15372/AUT20160104. .

9. Lvovskiy E.N., Statistical methods of empirical formulas construction, Uchebnoye posobie dlya vuzov. 2-e izdanie, pererabot. i dopolnen. [in Russian], Vysshaya shkola, Moscow, 1988. .

10. Glivenko V., Sulla determinazione empirica delle leggi di probabilita, Giornale dell'Instituto Italiano degli Attuari., 1933, 4, 92-99. .

11. CantelliF.G., Sulla determinazione empirica delle leggi di probabilita, Giornale dell'Instituto Italiano degli Attuari., 1933, 4, 421-424. .

12. Kolmogorov A.N., Sulla determinazione empirica di una legge di distribuzione, Giornale dell'Instituto Italiano degli Attuari., 1933, 4, 83-91. .

13. SmirnovN.V., Approximation of laws of random variables distribution by empirical data, Uspekhi matematicheskih nauk, 1944, 10, 179-206. .

14. Lozynsky A., Romanyshyn I., RusynB., Minialo V., Robust approach to estimation of the intensity of noisy signal with additive uncorrelated impulse interference, IEEE Second International Conference on Data Stream Mining & Processing (DSMP), August 21-25, 2018, Ukraine, Lviv, 2018, 251-254, DOI: 10.1109/DSMP.2018.8478625. .

15. Lozynsky A., Romanyshyn I., RusynB., Intensity estimation of noise-like signal in presence of uncorrelated pulse interferences, Radioelectronics and Communications, 2019, 62, No. 5, 214-222, DOI: 10.3103/S0735272719050030. .

16. RusynB.P., Lutsyk A.A., KosarevychR.Ya., Modified architecture of lossless image compression based on FPGA for on-board devices with linear CCD, Journal of Automation and Information Sciences, 2019, 51, No. 2, 41-49, DOI: 10.1615/JAutomatInfScien.v51.i2.50. .

### Articles with similar content:

USE OF HIGH-ORDER STATISTICS IN NON-GAUSSIAN PROCESS RECOGNITION FROM LINEAR PREDICTION MODELS
Telecommunications and Radio Engineering, Vol.74, 2015, issue 5
K. V. Netrebenko, V. A. Tikhonov, I.O. Fil, V. M. Bezruk
Method of Generating Realizations of Random Sequence with the Specified Characteristics Based on Nonlinear Canonical Decomposition
Journal of Automation and Information Sciences, Vol.48, 2016, issue 10
Igor P. Atamanyuk, Yuriy P. Kondratenko
Construction of Multiple Filters for Linear Algebraic Systems
Journal of Automation and Information Sciences, Vol.33, 2001, issue 1
Vladimir T. Matvienko, Nikolay Fedorovich Kirichenko
Non-Gaussian Signals Recognition against a Background of Additive Noises
Telecommunications and Radio Engineering, Vol.66, 2007, issue 18
K. V. Netrebenko, V. A. Tikhonov
Estimate of Time Series Similarity Based on Models
Journal of Automation and Information Sciences, Vol.51, 2019, issue 8
Tatyana V. Knignitskaya