Vol 221, No 4 (2017)

This paper studies a nonparametric kernel estimator of a mean residual life function. We find asymptotic decompositions for the mean, variance, and mean square deviation of the estimator.

This article proposes a group of new criteria for testing hypothesis about the identity of distributions in two small size samples. Their special feature is the dependence on the sample generating distribution even under the null hypothesis. The result of this paper is a certain comprehensive methodology for answering the question about identity of distributions of two given samples, containing one to three tens of observations. The main components of this methodology are: (a) visual sample comparison by a “regression” graph of dependence of their quantiles of the same order, (b) choosing the suitable test from the given group of regression goodness-of-fit tests, using the fact that under the null hypothesis the quantile graph must be close to the line y = x, (c) a modernized method of modeling bootstrap samples for determining different quality characteristics of the chosen goodness-of-fit test. The article does not contain theoretical results that might be formulated as a mathematical theorem. Its conclusions are based on statistical analysis of a large number of sample pairs of different size and origin. Description of such analysis methodology is based on three demonstrative examples from biology.

We construct a model of one-dimensional Brownian motion in the form of a random walk as an alternative to the Wiener random process.

The goal of the present article is to establish the asymptotic normality of L₂-deviations of the kernel estimators of the distribution function F_n(x), defined as M_n = ∫ (F_n(x) − R(x))²ω(x)dx, where R(x) is a conditional average distribution function of a random variable X, ω(x) is a weight function under dose-effect dependence based on the sample U⁽ⁿ⁾ = {(W_i, Y_i), 1 ≤ i ≤ n}, W_i = I(X_i < U_i) is an indicator of the event (X_i < U_i), and Y is a random variable that depends on U and defines the measurement error in the injected random dose. These results may be used to construct goodness-of-fit and homogeneity tests under dose-effect dependence.

We study the power of the two-sample Anderson–Darling test for fixed alternatives, when one sample is drawn from the standard normal distribution and the other is a mixture of two normal distributions. Using the modeling we study the behavior of the power function depending on the proportion of contamination for different sample sizes. For the same alternatives we compare the powers of Anderson–Darling and Kolmogorov–Smirnov tests. Similar results are obtained for the trimmed samples.

The use of probabilistic models is often useful for proving combinatorial identities and for studying the properties of special numbers and functions. The proof in this case becomes more simple, intuitive and natural compared to traditional combinatorial and algebraic-analytical methods. In this article we demonstrate the advantages of a probabilistic approach for studying the asymptotic behavior of Bell numbers.