Scientific Journal of KubSAU

Some estimators of the probability density function in spaces of arbitrary nature are used for various tasks in statistics of non-numerical data. Systematic exposition of the theory of such estimators had a start in our work [2]. This article is a direct continuation of the article [2]. We will regularly use references to conditions and theorems of the article [2], in which we introduced several types of nonparametric estimators of the probability density. We studied more linear estimators. In this article we consider particular cases - kernel density estimates in spaces of arbitrary nature. When estimating the density of the one-dimensional random variable, kernel estimators become the Parzen-Rosenblatt estimators. Asymptotic behavior of kernel density estimators in the general case of an arbitrary nature spaces are devoted to Theorem 1 - 8. Under different conditions we prove the consistency and asymptotic normality of kernel density estimators. We have studied uniform convergence. We have introduced the concept of "preferred rate differences" and studied nuclear density estimators based on it. We have also introduced and studied natural affinity measures which are used in the analysis of the asymptotic behavior of kernel density estimators. We have found the asymptotic behavior of dispersions of kernel density estimators and considered the examples including kernel density estimators in finite-dimensional spaces and in the space of square-integrable functions

The article is devoted to the nonparametric point and interval estimation of the characteristics of the probabilistic distribution (the expectation, median, variance, standard deviation, variation coefficient) of the sample results. Sample values are regarded as the implementation of independent and identically distributed random variables with an arbitrary distribution function having the desired number of moments. Nonparametric analysis procedures are compared with the parametric procedures, based on the assumption that the sample values have a normal distribution. Point estimators are constructed in the obvious way - using sample analogs of the theoretical characteristics. Interval estimators are based on asymptotic normality of sample moments and functions from them. Nonparametric asymptotic confidence intervals are obtained through the use of special output technology of the asymptotic relations of Applied Statistics. In the first step this technology uses the multidimensional central limit theorem, applied to the sums of vectors whose coordinates are the degrees of initial random variables. The second step is the conversion limit multivariate normal vector to obtain the interest of researcher vector. At the same considerations we have used linearization and discarded infinitesimal quantities. The third step - a rigorous justification of the results on the asymptotic standard for mathematical and statistical reasoning level. It is usually necessary to use the necessary and sufficient conditions for the inheritance of convergence. This article contains 10 numerical examples. Initial data - information about an operating time of 50 cutting tools to the limit state. Using the methods developed on the assumption of normal distribution, it can lead to noticeably distorted conclusions in a situation where the normality hypothesis failed. Practical recommendations are: for the analysis of real data we should use nonparametric confidence limits

In the statistical hypothesis testing, critical values often point to a priori fixed (nominal) significance levels. As such, typically researcher uses the values of three numbers 0.01, 0.05, 0.1, to which may be added a few levels: 0.001, 0.005, 0.02, and others. However, for the statistics with discrete distribution functions, which, in particular, include all nonparametric statistical tests, the real significance levels may be different from the nominal, differ at times. Under the real significance level we refer to the highest possible significance level of discrete statistics, not exceeding a given nominal significance level (ie, the transition to the next highest possible value corresponding discrete statistical significance level is greater than a predetermined nominal). In the article, we have discussed the difference between nominal and real significance levels on the example of nonparametric tests for the homogeneity of two independent samples. We have also studied two-sample Wilcoxon test, the criterion of van der Waerden, Smirnov two-sample two-sided test, sign test, runs test (Wolfowitz) and calculated the real significance levels of the criteria for nominal significance level of 0.05. The study of the power of these statistical tests is accomplished by means of Monte Carlo method. The main conclusion: the use of nominal significance levels instead of real significance levels for discrete statistics is inadmissible for small sample sizes

We have proposed the method for testing of independence of two alternative variables on the basis of statistics of non-numeric data. The method is aimed at application in problems of statistical quality control. Testing of independence is based on set of small samples, i.e., in the Kolmogorov’s asymptotics, when the number of unknown parameters of the distribution increases in proportion to the data size

In many areas - the economy, quality management, medicine, the ecology, in safety of flights and others - the problems of analysis, estimation and management of risks have much in common. Therefore, we consider it necessary to develop a general theory of risk. Approaches and methods of this theory will allow in the future solving problems of uniform risk management in specific subject areas. Based on the analysis of scientific publications and industry regulations it must be noted that private risk theories tend to become isolated within themselves, create their own internal standards and systems of regulations. Separately - for banking, separately - for safety, separately - for industrial accidents, etc. In order to construct a general theory of risk we analyze use of the term "risk" in various fields, consider the variety of types of risks, give the basic definitions in the field of analysis, estimation and management of risk. We discuss planetary risks (at Earth as a whole), global risks (at the level of one or more States), financial risks, commercial risks (risks at the level of the immediate environment of the company), and production (internal, operational) risks relating to the activities of individual enterprises (organizations), personal risks. Instruments of total risk theory allow us equally solve the basic problems of analysis, estimation and management of risk for all areas

In various applications it is necessary to analyze some expert orderings, ie clustered rankings of examination objects. These areas include technical studies, ecology, management, economics, sociology, forecasting, etc. The objects may make samples of the products, technologies, mathematical models, projects, job applicants and others. We obtain clustered rankings which can be both with the help of experts and objective way, for example, by comparing the mathematical models with experimental data using a particular quality criterion. The method described in this article was developed in connection with the problems of chemical safety and environmental security of the biosphere. We propose a new method for constructing a clustered ranking which can be average (in the sense, discussed in this work) for all clustered rankings under our consideration. Then the contradictions between the individual initial rankings are contained within clusters average (coordinated) ranking. As a result, ordered clusters reflects the general opinion of the experts, more precisely, the total that is contained simultaneously in all the original rankings. Newly built clustered ranking is often called the matching (coordinated) ranking with respect to the original clustered rankings. The clusters are enclosed objects about which some of the initial rankings are contradictory. For these objects is necessary to conduct the new studies. These studies can be formal mathematics (calculation of the Kemeny median, orderings by means of the averages and medians of ranks, etc.) or these studies require involvement of new information from the relevant application area, it may be necessary conduct additional scientific research. In this article we introduce the necessary concepts and we formulate the new algorithm of construct the coordinated ranking for some cluster rankings in general terms, and its properties are discussed

We have studied the asymptotic behavior of a broad class of nonparametric statistics, which includes statistics of omega-square type and Kolmogorov-Smirnov type. Limit theorems have been proved. We have also developed the method of approximation with step functions. With the help of this method we have obtained a number of necessary and sufficient conditions

In this article we propose a general theoretical model of estimation of the feasibility of an innovation-investment project. For specifying a general model to estimate the feasibility of a project we have highlighted the stages of development of projects in the aerospace industry. Organizational-economic approaches to estimation of the feasibility of projects to create rocket and space technology are presented in terms of algorithms. They take into account the specifics of the space industry, by virtue of which such projects have both innovative and investment components

Small business is an important part of modern Russian economy. We give a wide panorama developed by us of possible approaches to the construction of economic-mathematical models that may be useful to describe the dynamics of small businesses, as well as management. As for the description of certain problems of small business can use a variety of types of economic-mathematical and econometric models, we found it useful to consider a fairly wide range of such models, which resulted in quite a short description of the specific models. In this description of the models brought to such a level that an experienced professional in the field of economic-mathematical modeling could, if necessary, to develop their own specific model to the stage of design formulas and numerical results. Particular attention is paid to the use of statistical methods of non-numeric data, the most pressing at the moment. Are considered the problems of economic-mathematical modeling in solving problems of small business marketing. We have accumulated some experience in application of the methodology of economic-mathematical modeling in solving practical problems in small business marketing, in particular in the field of consumer goods and industrial purposes, educational services, as well as in the analysis and modeling of inflation, taxation and others. In marketing models of decision making theory we apply rankings and ratings. Is considered the problem of comparing averages. We present some models of the life cycle of small businesses - flow model projects, model of capture niches, and model of niche selection. We discuss the development of research on economic-mathematical modeling of small businesses

In accordance with the new paradigm of mathematical statistics the statistics of objects of nonnumerical nature (statistics of nonnumerical objects, non-numerical data statistics, non-numeric statistics) is one of the four main areas of mathematical statistics. Statistics of objects of nonnumerical nature consists of a central core - statistics in spaces of arbitrary nature - and statistical theories of analysis of specific types of non-numeric data. To identify possibilities of application of statistics of objects of nonnumerical nature it is useful to explore the multiformity of objects of non-numeric nature. This is the subject of this article. We have considered the results of measurements at scales other than absolute; binary relations; dichotomous (binary) data; sets. We have also analyzed the objects of non-numerical nature as statistical data, and their importance in the formation of statistical or mathematical model of a real phenomenon, as a result of data analysis