Vol 220, No 6 (2017)
- Year: 2017
- Articles: 14
- URL: https://journals.rcsi.science/1072-3374/issue/view/14805
Article
Kernel Estimation of a Characteristic Function
Abstract
This article deals with the study of asymptotic properties of a nonparametric kernel estimator of a characteristic function. Uniformly strong consistency of kernel estimator with fixed and expanding interval is established. The law of iterated logarithm type is also proved.
On Nonparametric Estimation of the Mathematical Expectation of a Function of Random Variables with Identical Distributions
Abstract
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.
Limit Distributions of a Random Term of a Variational Series
Abstract
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 Behavior of Chi-Square Statistics Using Unbiased Estimators in the Case of One-Parameter Distribution from Exponential Family
Abstract
We propose two new variants of chi-square-type statistics for testing the hypothesis on the shape of a distribution belonging to the one-parameter exponential family. Each variant assumes the use of unbiased estimators of the probabilities for the random variable to hit the atoms (intervals) of decomposition of its range. We establish the convergence of each statistics’ distribution to the chi-square distribution with (r − 1) degrees of freedom, where r is the number of decomposition atoms.
The Rate of Convergence of the Distributions of Regular Statistics Constructed from Samples with Negatively Binomially Distributed Random Sizes to the Student Distribution
Abstract
New estimates are obtained for the rate of convergence of the negative binomial distribution with parameters (r, p) to the gamma-distribution with parameters (r, r) when p → 0. The main result is a considerable improvement of convergence rate estimates for regular statistics constructed from samples with negatively binomially distributed random sizes to the Student distribution.
On Stability of Sequential Wald Test to a Violation of the Assumption About Observations’ Independence
Abstract
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.
Some Properties of Two-Sample Kolmogorov–Smirnov Test in the Case of Contamination of One of the Samples
Abstract
We discuss some properties of the two-sample Kolmogorov–Smirnov test for trimmed samples, one of which is drawn from the standard normal distribution and another is a mixture of two normal distributions. Using mathematical simulation, we show the insensibility of the test to small portions of contamination. Some properties of the test for moderate rates of contamination are demonstrated.
An Analog of the Chi-Square Distribution for Normalized Sums with Small Number of Summands
Abstract
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.
On the Estimates of Average Characteristics of Some Birth and Death Processes
Abstract
Inhomogeneous birth and death processes with intensities close to periodic are studied. Limit mean and double mean of such processes are analyzed, their estimates are obtained, and the method of their approximate calculation is developed. Also some examples from queuing systems theory are considered.
On Convergence Rate in CLT for Smooth Distributions
Abstract
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.
Estimation of Distributions Under Dose-Effect Dependence with Fixed Experiment Plan
Abstract
In the present paper we find the limit distributions of the Nadaraya–Watson estimators and asymptotically unbiased estimators of a distribution function under dose-effect dependence with fixed experiment plans. We prove the asymptotic normality of integrated square errors of the distribution function’s estimators.
Asymptotics of the Optimal Confidence Region for Shift and Scale, Based on Two Order Statistics
Abstract
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.