Mathematical statistics is a science that develops mathematical methods to systematize and use statistical data for scientific and practical conclusions.
In many of its sections, mathematical statistics are based on probability theory , which makes it possible to assess the reliability and accuracy of conclusions made on the basis of limited statistical material (for example, to estimate the necessary sample size to obtain the results of the required accuracy in a sample survey).
Content
Subject and Methods
Mathematical statistics is a branch of mathematics that develops methods for recording, describing, and analyzing observational and experimental data in order to construct probabilistic models of mass random phenomena [1] . Depending on the mathematical nature of the specific observation results, mathematical statistics are divided into statistics of numbers, multivariate statistical analysis, analysis of functions (processes) and time series, statistics of objects of non-numerical nature.
Descriptive statistics , estimation theory and hypothesis testing theory are distinguished . Descriptive statistics are a set of empirical methods used to visualize and interpret data (calculation of sample characteristics, tables, charts, graphs, etc.), usually not requiring assumptions about the probabilistic nature of the data. Some methods of descriptive statistics involve using the capabilities of modern computers . These include, in particular, cluster analysis aimed at identifying groups of objects similar to each other, and multidimensional scaling , which allows you to visualize objects on the plane.
Methods for evaluating and testing hypotheses are based on probabilistic models of data origin. These models are divided into parametric and nonparametric. In parametric models, it is assumed that the characteristics of the studied objects are described by means of distributions depending on (one or more) numerical parameters. Nonparametric models are not related to the specification of a parametric family for the distribution of the studied characteristics. In mathematical statistics, parameters and functions from them are estimated, which represent important characteristics of distributions (for example, mathematical expectation, median, standard deviation, quantiles, etc.), densities and distribution functions, etc. Use point and interval estimates .
A large section of modern mathematical statistics is statistical sequential analysis , a fundamental contribution to the creation and development of which was made by A. Wald during the Second World War . Unlike traditional (inconsistent) methods of statistical analysis based on random sampling of a fixed volume, a sequential analysis allows the formation of an array of observations one by one (or, more generally, by groups), while the decision to conduct the next observation (group of observations) is made on based on an already accumulated array of observations. In view of this, the theory of sequential statistical analysis is closely related to the theory of optimal stopping .
In mathematical statistics there is a general theory of hypothesis testing and a large number of methods devoted to testing specific hypotheses. We consider hypotheses about the values of parameters and characteristics, about checking homogeneity (that is, about the coincidence of characteristics or distribution functions in two samples), about the agreement of the empirical distribution function with a given distribution function or with a parametric family of such functions, about the symmetry of the distribution, etc.
Of great importance is the section of mathematical statistics related to conducting sample surveys , with the properties of various sampling schemes and the construction of adequate methods for estimating and testing hypotheses.
The problems of dependency recovery have been actively studied for more than 200 years since the development of the least squares method by K. Gauss in 1794.
The development of methods for approximating data and reducing the dimension of a description began more than 100 years ago when Karl Pearson created the method of principal components . Later, factor analysis [2] and numerous nonlinear generalizations [3] were developed.
Various methods of constructing (cluster analysis), analysis and use (discriminant analysis) of classifications (typologies) are also called image recognition methods (with and without a teacher), automatic classification , etc.
Currently, computers play a large role in mathematical statistics. They are used both for calculations and for simulation (in particular, in methods of multiplying samples and in studying the suitability of asymptotic results).
See also
- Applied Statistics
- Statistical Methods
- Statistical evaluation
- Probability theory
- Decision theory
- Factor analysis
- Econometrics
Notes
- ↑ Probabilistic sections of mathematics / Ed. Yu. D. Maximov. - St. Petersburg: "Ivan Fedorov", 2001. - S. 400. - 592 p. - ISBN 5-81940-050-X .
- ↑ Harman G. , Modern factor analysis. - M .: Statistics, 1972. - 486 p.
- ↑ Gorban AN, Kegl B., Wunsch D., Zinovyev AY (Eds.), Principal Manifolds for Data Visualization and Dimension Reduction , Series: Lecture Notes in Computational Science and Engineering 58, Springer, Berlin - Heidelberg - New York, 2007, XXIV, 340 p. 82 illus. ISBN 978-3-540-73749-0 (as well as online ).
Literature
- Probability and mathematical statistics. Encyclopedia / Ch. ed. Yu. V. Prokhorov. - M .: Publishing house "Big Russian Encyclopedia", 1999.
- Wald A. Sequential Analysis, Per. from English.- M .: Fizmatgiz, 1960.
- Mathematical statistics / Yu. V. Prokhorov // Big Russian Encyclopedia : [in 35 vols.] / Ch. ed. Yu.S. Osipov . - M .: Great Russian Encyclopedia, 2004—2017.
- Nathan A.A. , Gorbachev O.G., Guz S.A. Mathematical statistics. : textbook. allowance. M .: MZ Press - MIPT, 2004. ISBN 5-94073-087-6 .
- Shiryaev V.D. Statistical sequential analysis. Optimal stopping rules - M .: Nauka, 1976
- Ostapenko R. I. Mathematical foundations of psychology : a teaching aid for students and graduate students of psychological and pedagogical specialties of universities. - Voronezh: Voronezh State Pedagogical University, 2010 .-- 76 p.: Ill. - ISBN 978-5-88519-680-2