Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis. The term "mathematical statistics" is closely related to "statistical theory" but also embraces modelling for actuarial science and non-statistical probability theory, particularly in Scandinavia.
Statistics deals with gaining information from data. In practice, data often contain some randomness or uncertainty. Statistics handles such data using methods of probability theory.
Introduction
Statistics is divided into:
- descriptive statistics - the part of statistics that describes data, i.e. summarises the data and their typical properties
- inferential statistics - the part of statistics that draws conclusions from data, i.e. checks whether the data fulfill some condition and gives guarantees on the involved uncertainty.
Mathematical statistics has been inspired by and has extended many procedures in applied statistics.
Statistics, mathematics, and mathematical statistics
Mathematical statistics has substantial overlap with the discipline of statistics. Statistical theorists study and improve statistical procedures with mathematics, and statistical research often raises mathematical questions. Statistical theory relies on probability and decision theory. Mathematians and statisticians like Gauss, Laplace, and C. S. Peirce used decision theory with probability distributions and loss functions (or utility functions). The decision-theoretic approach to statistical inference was reinvigorated by Abraham Wald and his successors.[1][2][3][4][5][6][7], and makes extensive use of scientific computing, analysis, and optimization; for the design of experiments, statisticians use algebra and combinatorics.
References
- ^ * Wald, Abraham (1947). Sequential Analysis. New York: John Wiley and Sons. ISBN 0471918067. "See Dover reprint: ISBN 0486439127"
- ^ * Wald, Abraham (1950). Statistical Decision Functions. John Wiley and Sons, New York.
- ^ * Lehmann, Erich (1959). Testing Statistical Hypotheses.
- ^ * Lehmann, Erich (1983). Theory of Point Estimation.
- ^ * Bickel, Peter J. and Doksum, Kjell A. (2001). Mathematical Statistics: Basic and Selected Topics. I (Second (updated printing 2007) ed.). Pearson Prentice-Hall.
- ^ * Le Cam, Lucien (1986). Asymptotic Methods in Statistical Decision Theory. Springer-Verlag.
- ^ * Liese, Friedrich and Miescke, Klaus-J. (2008). Statistical Decision Theory: Estimation, Testing, and Selection. Springer.
Additional reading
- Borovkov, A. A. (1999). Mathematical Statistics. Taylor & Francis.
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