Mathematical statistics

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 the term "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

Statistical science is concerned with the planning of studies, especially with the design of randomized experiments and with the planning of surveys using random sampling. The initial analysis of the data from properly randomized studies often follows the study protocol.

Of course, the data from a randomized study can be analyzed to consider secondary hypotheses or to suggest new ideas. A secondary analysis of the data from a planned study uses tools from data analysis.

Data analysis 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 (using some model for the data): For example, inferential statistics involves selecting a model for the data, checking whether the data fulfill the conditions of a particular model, and with quantifying the involved uncertainty (e.g. using confidence intervals).

While the tools of data analysis work best on data from randomized studies, they are also applied to other kinds of data --- for example, from natural experiments and observational studies, in which case the inference is dependent on the model chosen by the statistician, and so subjective.^[1]

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. Mathematicians 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,^{[2][3][4][5][6][7][8]} and makes extensive use of scientific computing, analysis, and optimization; for the design of experiments, statisticians use algebra and combinatorics.

See also

• Asymptotic theory (statistics)

References

1. ^ Freedman, D.A. (2005) Statistical Models: Theory and Practice, Cambridge University Press. ISBN 9780521671057
2. ^ Wald, Abraham (1947). Sequential analysis. New York: John Wiley and Sons. ISBN 0471918067. "See Dover reprint: ISBN 0486439127" 
3. ^ Wald, Abraham (1950). Statistical Decision Functions. John Wiley and Sons, New York. 
4. ^ Lehmann, Erich (1997). Testing Statistical Hypotheses (2nd ed.). ISBN 0387949194. 
5. ^ Lehmann, Erich; Cassella, George (1998). Theory of Point Estimation (2nd ed.). ISBN 0387985026. 
6. ^ Bickel, Peter J.; Doksum, Kjell A. (2001). Mathematical Statistics: Basic and Selected Topics. 1 (Second (updated printing 2007) ed.). Pearson Prentice-Hall. 
7. ^ Le Cam, Lucien (1986). Asymptotic Methods in Statistical Decision Theory. Springer-Verlag. ISBN 0387963073. 
8. ^ Liese, Friedrich and Miescke, Klaus-J. (2008). Statistical Decision Theory: Estimation, Testing, and Selection. Springer. 

Additional reading

• Borovkov, A. A. (1999). Mathematical Statistics. CRC Press. ISBN 90-5699-018-7
• Virtual Laboratories in Probability and Statistics (Univ. of Ala.-Huntsville)