Volume 220, Nº 6 (2017)

We consider the classical problem of nonparametric estimation of the mathematical expectation of a function of independent random variables. In contrast to the traditional formulation, it is assumed that some random variables have identical distributions. For estimation, there are the samples whose number coincides with the number of unknown distributions. Traditional nonparametric estimation uses empirical distribution functions, which leads to biased estimates. The resampling approach proposes the following procedure. For each random function’s argument, an element from the corresponding sample is drawn at random without replacement, and it is taken as the argument’s value in the given realization. Then the function’s value is calculated and stored. After that all drawn elements are returned to their samples and the procedure is repeated many times. The estimator is the arithmetic mean of the obtained function’s values. This estimator is unbiased. This paper describes the process of calculation of the resampling estimator variance and the bias of the traditional nonparametric estimator.

There are a lot of papers devoted to the study of order statistics, in particular, the asymptotics of the kth term of a variational series for a sample of size n of independent identically distributed random variables with distribution F(x) with different relations between k and n. In the “central” part, where min(k, n − k) → ∞, the normal limit distribution dominates. In the present paper we study the asymptotic behavior of a random term of the variational series: its number ν is a random variable, taking values 1, 2,…, n with equal probabilities. Under mild assumptions on the density of underlying distribution, we find all possible limit distributions for the νth term of variational series.

The theory of integration along Poisson trajectories is developed. Its probabilistic, statistical, and physical applications are specified.

In the present paper we study the stability of Wald test for simple hypotheses on the parameters of binomial distribution to violation of the assumption on the observations’ independence. We obtain approximations of the test’s probabilistic characteristics, linear in distortion level. We also provide results of numerical experiments confirming theoretical conclusions.

In the present paper we obtain approximations to the distribution of a sum of squared normalized variables with small number of summands, and provide an estimate of approximation accuracy. We compare confidence intervals for the unknown parameter σ constructed with the use of the obtained approximations when α₁ is known, with the intervals constructed with the use of the classical chi-square distribution under the assumption of normality of normalized sums.

There are obtained nonuniform estimates of the first and second order for the rate of convergence in the central limit theorem for sums of independent identically distributed random variables with bounded density and finite absolute moments of order 2+δ, where 0 < δ ≤ 1. The obtained estimates represent a sum of two terms, the first one being the Lyapunov fraction of order 2 + δ with a factor depending only on δ, and the second one decaying exponentially. The values of the factor of the Lyapunov fraction are considerably less than the known. This is an updated version of the original paper.

Using two order statistics, we construct the two-dimensional optimal strong confidence region for shift and scale parameters and study its asymptotics when the sample size tends to infinity. Also, we construct simpler confidence region asymptotically equivalent to the optimal one. The constructed confidence region is compared with similar confidence regions, based on sample mean and sample mean-square deviation.