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.

Contents 1 Introduction 2 Statistics, mathematics, and mathematical statistics 3 References 4 Additional reading

[edit] 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.

[edit] 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.

[edit] References

1. ^ * Wald, Abraham (1947). Sequential Analysis. New York: John Wiley and Sons. ISBN 0471918067. "See Dover reprint: ISBN 0486439127" 
2. ^ * Wald, Abraham (1950). Statistical Decision Functions. John Wiley and Sons, New York. 
3. ^ * Lehmann, Erich (1959). Testing Statistical Hypotheses. 
4. ^ * Lehmann, Erich (1983). Theory of Point Estimation. 
5. ^ * Bickel, Peter J. and Doksum, Kjell A. (2001). Mathematical Statistics: Basic and Selected Topics. I (Second (updated printing 2007) ed.). Pearson Prentice-Hall. 
6. ^ * Le Cam, Lucien (1986). Asymptotic Methods in Statistical Decision Theory. Springer-Verlag. 
7. ^ * Liese, Friedrich and Miescke, Klaus-J. (2008). Statistical Decision Theory: Estimation, Testing, and Selection. Springer. 

[edit] Additional reading

• Borovkov, A. A. (1999). Mathematical Statistics. Taylor & Francis.

v • d • e Statistics Descriptive statistics Continuous data Location Mean (Arithmetic, Geometric, Harmonic) · Median · Mode Dispersion Range · Standard deviation · Coefficient of variation · Percentile Moments Variance · Semivariance · Skewness · Kurtosis Categorical data Frequency · Contingency table Inferential statistics and hypothesis testing Inference Confidence interval (Frequentist inference) · Credible interval (Bayesian inference) · Significance · Meta-analysis Design of experiments Population · Sampling · Stratified sampling · Replication · Blocking · Sensitivity and specificity  · Optimal design Sample size estimation Statistical power · Effect size · Standard error General estimation Bayesian estimator · Maximum likelihood · Method of moments · Minimum distance · Maximum spacing Specific tests Z-test (normal) · Student's t-test · F-test · Chi-square test · Pearson's chi-square test · Wald test · Mann–Whitney U · Wilcoxon signed-rank test Survival analysis Survival function · Kaplan–Meier · Logrank test · Failure rate · Proportional hazards models Correlation and regression analysis Correlation Pearson product-moment correlation · Rank correlation (Spearman's rho, Kendall's tau) · Confounding variable Linear regression Simple linear regression · Ordinary least squares  · General linear model · Analysis of variance · Analysis of covariance Non-standard Nonlinear regression · Nonparametric  · Semiparametric · Robust Non-normal errors Generalized linear model · Binomial · Poisson · Logistic Statistical graphics Bar chart · Biplot · Box plot · Control chart · Correlogram · Forest plot · Histogram · Q-Q plot · Run chart · Scatter plot · Stemplot Category · Portal · Outline · Index

v • d • e Major fields of mathematics Logic · Set theory · Category theory · Algebra (elementary – linear – abstract) · Number theory · Analysis/Calculus · Geometry · Topology · Dynamical systems · Combinatorics · Game theory · Information theory · Numerical analysis · Optimization · Computation · Probability · Statistics · Trigonometry

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