Mathematical statistics is the application of mathematics to statistics, which was originally conceived as the science of the state — the collection and analysis of facts about a country: its economy, land, military, population, and so forth. Mathematical techniques which are used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory.
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.
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, and makes extensive use of scientific computing, analysis, and optimization; for the design of experiments, statisticians use algebra and combinatorics.
- Lakshmikantham,, ed. by D. Kannan,... V. (2002). Handbook of stochastic analysis and applications. New York: M. Dekker. ISBN 0824706609.
- Schervish, Mark J. (1995). Theory of statistics (Corr. 2nd print. ed.). New York: Springer. ISBN 0387945466.
- Freedman, D.A. (2005) Statistical Models: Theory and Practice, Cambridge University Press. ISBN 978-0-521-67105-7
- Wald, Abraham (1947). Sequential analysis. New York: John Wiley and Sons. ISBN 0-471-91806-7. "See Dover reprint, 2004: ISBN 0-486-43912-7"
- Wald, Abraham (1950). Statistical Decision Functions. John Wiley and Sons, New York.
- Lehmann, Erich (1997). Testing Statistical Hypotheses (2nd ed.). ISBN 0-387-94919-4.
- Lehmann, Erich; Cassella, George (1998). Theory of Point Estimation (2nd ed.). ISBN 0-387-98502-6.
- Bickel, Peter J.; Doksum, Kjell A. (2001). Mathematical Statistics: Basic and Selected Topics 1 (Second (updated printing 2007) ed.). Pearson Prentice-Hall.
- Le Cam, Lucien (1986). Asymptotic Methods in Statistical Decision Theory. Springer-Verlag. ISBN 0-387-96307-3.
- Liese, Friedrich and Miescke, Klaus-J. (2008). Statistical Decision Theory: Estimation, Testing, and Selection. Springer.
- Borovkov, A. A. (1999). Mathematical Statistics. CRC Press. ISBN 90-5699-018-7
- Virtual Laboratories in Probability and Statistics (Univ. of Ala.-Huntsville)
- StatiBot, interactive online expert system on statistical tests.