New statistical kernel-projection estimator in the Monte Carlo method

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.