### Nonparametric estimate of quadrate of the probability densities and its properties under the large sample

#### T. K. Yuldashev^{1}

Annotation | Efficiency of nonparametric data processing algorithms, based on kernel estimators of square of probability density and blur factor, largely determined by the volume of statistical data. It is considered in this article the nonparametric estimate of quadrate of the probability densities of Rosenblatt-Parzen type. It is studied the asymptotic properties of the nonparametric estimate of quadrate of the probability densities. It is tested the convergence of nonparametric estimates of the probability densities with increasing amount of experimental data to the desired probability density function. It is obtained the formula of asymptotically unbiased of the desired estimates. It is proved the mean-square convergence and the solvency of quadrate density. It is produced the computer experiment to determine the dependence of mean square error of nonparametric estimate approximation of the probability densities quadrate from the volume of statistical data. The nonparametric estimate of quadrate of the probability densities, obtained in this article can be used for constructing decision algorithms when the primary source of information is statistical data. |
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Keywords | quadrate of probability density, nonparametric estimate, asymptotic properties, Rosenblatt-Parzen type estimation, kernel function |

^{1}Associate professor of Higher Mathematics Chair, M. F. Reshetnev Siberian State Aerospace University, Krasnoyarsk, tursunbay@rambler.ru

**Citation**: T. K. Yuldashev, "[Nonparametric estimate of quadrate of the probability densities and its properties under the large sample]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,17:4 (2015) 60–69 (In Russian)