## Abstract

The statistical kernel estimator in the Monte Carlo method is usually optimized based on the preliminary construction of a "microgrouped" sample of values of the variable under study. Even for the two-dimensional case, such optimization is very difficult. Accordingly, we propose a combined (kernel-projection) statistical estimator of the two-dimensional distribution density: a kernel estimator is constructed for the first (main) variable, and a projection estimator, for the second variable. In this case, for each kernel interval determined by the microgrouped sample, the coefficients of a particular orthogonal decomposition of the conditional probability density are statistically estimated based on preliminary results for the "micro intervals." An important result of this work is the mean-square optimization of such an estimator under assumptions made about the convergence rate of the orthogonal decomposition in use. The constructed estimator is verified by evaluating the bidirectional distribution of a radiation flux passing through a layer of scattering and absorbing substance.

Original language | English |
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Pages (from-to) | 313-317 |

Number of pages | 5 |

Journal | Doklady Mathematics |

Volume | 102 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jul 2020 |

## Keywords

- kernel density estimator
- kernel-projection estimator
- Monte Carlo method
- projection estimator
- ANGULAR-DISTRIBUTIONS