


Том 228, № 5 (2018)
- Год: 2018
- Статей: 12
- URL: https://journals.rcsi.science/1072-3374/issue/view/14881
Article
Application of Unbiased Estimators to Group Classification Risk Estimation
Аннотация
We consider the problem of construction of point and interval estimators for the Bayesian risk of a group classification decision rule provided that the elements of training samples have distributions belonging to the same one-parameter exponential family. We propose a solution to this problem using the asymptotic normality and efficiency of an unbiased risk estimator. An example of application of the obtained theoretical results is given.



On Bayes Equality and Related Issues
Аннотация
To make calculations in the Bayesian analysis, the formalism of which is based on the layering of a probability measure defined on the product of measurable spaces, it is useful to have a summary of the properties of this layering. In this paper we formulate and prove those of them that are used in calculations more often than others. Particularly, we prove Hoeffding-type inequalities using direct elementary techniques.



Comparative Analysis of the Powers of the Two-Sample Kolmogorov–Smirnov and Anderson–Darling Tests Under Various Alternatives
Аннотация
In this paper we conduct a comparative analysis of the powers of the two-sample Kolmogorov–Smirnov and Anderson–Darling tests under various alternatives using simulation. We consider two examples. In the first example the alternatives to the standard normal distribution are the distributions of the so-called contaminated normal model. We study the influence of a small contamination with a positive shift on the powers of the test. In the second example the alternatives are the logistic and the Laplace distributions, which are symmetric and differ in shape from the normal distribution having a larger kurtosis coefficient and heavier tails.



Nonparametric Method of Least Squares: Accounting for Seasonality
Аннотация
We consider the problem of restoring a nonparametric dependence described by the sum of linear trend and seasonal component, i.e., a periodic function with the known period. We obtain the asymptotic distribution of the parameter estimates and the trend component. We find the mathematical expectation of the residual sum of squares. We also develop the methods of estimation of the seasonal component and construction of the interval forecast.



Perturbation Bounds for Markov Chains with General State Space
Аннотация
The aim of this paper is to investigate the stability of Markov chains with general state space. We present new conditions for the strong stability of Markov chains after a small perturbation of their transition kernels. Also, we obtain perturbation bounds with respect to different quantities.



Efficiency of a Certain Modification of the Studentized Range of Symmetric Stable Random Variables
Аннотация
We study the properties of tests constructed by a simple modification of the studentized range of the sample from a symmetric stable population in the problem of testing the hypothesis \( \mathrm{\mathscr{H}} \)α (the stability index equals α, α ∈ (1, 2)) against the alternative \( \mathrm{\mathscr{H}} \)2. We obtain approximate formulas for the calculation of critical values and estimation of the test power and develop a method for the estimation of the accuracy of these approximations. The major part of the paper deals with the construction of the approximations to the distribution function of the normalized sum of squares of symmetric stable random variables and estimation of the accuracy of approximations.



Multiparameter Methods are the New Field in Statistics
Аннотация
During the last decade statistical problems of a new kind have appeared more and more often, in which the dimensionality of observations is large, and the sample size is small or comparable with the dimensionality of observations. These problems may be characterized as multiparameter problems. The theory of multiparameter analysis, proposed by A.N.Kolmogorov and supported by the studies on the spectral theory of random matrices of V. A.Marchenko, L. A.Pastur, V. L.Girko, and others, revealed some specific phenomena appearing in statistics with a large number of weakly dependent variables. Particularly, there are stable relations between the principal parts of parameter set functions and the set of observed variables, which may be used to improve the statistical analysis. The use of these phenomena allows one to develop a thorough, always stable, and approximately optimal (irrespective of samples) versions of mostly used statistical procedures. The multiparameter versions of the discriminant analysis are of special interest.



D-Guaranteed Discrimination of Statistical Hypotheses: a Review of Results and Unsolved Problems
Аннотация
We compare two sequential d-guaranteed tests and an optimal d-guaranteed test based on a fixed number of observations with respect to the average number of observations within the most accepted practical applications of Bayesian probabilistic models. We consider the sequential “first skipping” test, the sequential locally efficient test based on the score statistic, and the test based on a fixed number of observations that minimizes the necessary sample size. We study various characteristics of these tests connected with the number of observations within three probabilistic models, namely, the normal (ϑ, σ2) distribution of the observed random variable and the normal a priori distribution of ϑ with fixed σ2; the exponential distribution with the intensity parameter ϑ and the a priori gamma distribution of ϑ; and the Bernoulli sampling with the success probability ϑ and the a priori beta distribution of ϑ. We discuss the connection of the d-posterior approach with the compound decision problem as applied to the analysis of data provided by microchips (when the false discovery rate, or FDR, for short, is treated as the d-risk of the first kind). We present the vast data on characteristics of the mentioned tests obtained by the method of mathematical modeling in several tables. We discuss unsolved problems of the d-guaranteed discrimination of hypotheses with the minimal number of observations and approaches to their solution.



Oja Median: Center-Locating Property
Аннотация
The property of consistency with the point of symmetry is a natural requirement for an arbitrary estimate of the center (for the centrally symmetric distribution this follows from the affine equivariance of the estimate). Different notions of multidimensional symmetry and some of their properties are given in this paper. We establish that in the general case, the multidimensional simplicial Oja median identifies the point of symmetry, if it exists. By this criterion we compare the Oja median with the other best-known multidimensional medians.






A Method of Parametric Solution of Convolution Equations
Аннотация
A method of moments is developed for the parametric form the solution of convolution equations of the first kind. Two models, the gamma model and the shifted gamma model, are studied in details and a non-parametric approach based on regularization is presented.





