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The estimate for the Bernoulli regression function is constructed using the Bernstein polynomial for group observations. The question of its consistency and asymptotic normality is studied. A testing hypothesis is constructed on the form of the Bernoulli regression function. The consistency of the constructed tests is investigated.

In the paper, the limiting distribution is established for an integral square deviation of estimates of Bernoulli regression functions based on two group samples. Based on these results, the new test is constructed for the hypothesis testing on the equality of two Bernoulli regression functions. The question of consistency of the constructed test is studied, and the asymptotic of the test power is investigated for some close alternatives.

The Kalman–Bucy continuous model of partially observable stochastic processes is considered. The problem of optimal stopping of a stochastic process with incomplete data is reduced to the problem of optimal stopping with complete data. The convergence of payoffs is proved when ε1→0,ε2→0, where ε1,and ε2are small perturbation parameters of the non observable and observable processes respectively.

The test of homogeneity is constructed by using kernel-type estimators of a distribution density. The limit power of the constructed test is found for close Pitman-type alternatives. The constructed test is compared with Pearson’s Xu-square test.

We develop the method of maximal likelihood for infinite-dimensional Hilbert spaces and prove several theorems about consistency and asymptotic normality.