Mathematical statistics

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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 2.1 Statistics in departments of mathematics 3 References

[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^[1], and makes extensive use of scientific computing, analysis, and optimization; for the design of experiments, statisticians use algebra and combinatorics. Mathematicians and statisticians often work in a department of mathematical sciences (particularly at colleges and small universities).

[edit] Statistics in departments of mathematics

Statistics is often taught in departments of mathematics.

Statisticians have long complained that many mathematics departments have assigned mathematicians (without statistical competence) to teach statistics courses, effectively giving "double blind" courses.^[2] Examing data from 2000, Schaeffer and Stasny^[3] reported

By far the majority of instructors within statistics departments have at least a master’s degree in statistics or biostatistics (about 89% for doctoral departments and about 79% for master’s departments). In doctoral mathematics departments, however, only about 58% of statistics course instructors had at least a master’s degree in statistics or biostatistics as their highest degree earned. In master’s-level mathematics departments, the corresponding percentage was near 44%, and in bachelor’s-level departments only 19% of statistics course instructors had at least a master’s degree in statistics or biostatistics as their highest degree earned. As we expected, a large majority of instructors in statistics departments (83% for doctoral departments and 62% for master’s departments) held doctoral degrees in either statistics or biostatistics. The comparable percentages for instructors of statistics in mathematics departments were about 52% and 38%.

This professional misconduct violates the "Statement on Professional Ethics" of the American Association of University Professors (which has been affirmed by many colleges and universities in the USA) and the ethical codes of the International Statistical Institute and the American Statistical Association. The principle that statistics-instructors should have statistical competence has been affirmed by the guidelines of the Mathematical Association of America, which has been endorsed by the American Statistical Association.^[4]

[edit] References

1. ^ 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:
   • 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. 
2. ^ The principle that college-instructors should have qualifications and engagement with their academic discipline has long been violated in United States colleges and universities, according to generations of statisticians:
   • Harold Hotelling (Dec. 1940). "The Teaching of Statistics". The Annals of Mathematical Statistics 11 (4): pp. 457-470. http://www.jstor.org/stable/2235726. 
   • Harold Hotelling (1988). "Golden Oldies: Classic Articles from the World of Statistics and Probability: 'The Teaching of Statistics'". Statistical Science 3 (1): pp. 63-71. doi:10.1214/ss/1177013001. http://projecteuclid.org/euclid.ss/1177013001. 
   • Harold Hotelling (1988). "Golden Oldies: Classic Articles from the World of Statistics and Probability: 'The Place of Statistics in the University'". Statistical Science 3 (1): pp. 72-83. doi:10.1214/ss/1177013002. http://projecteuclid.org/euclid.ss/1177013002. 
   These reprinted articles are followed by the comments of Kenneth J. Arrow, W. Edwards Deming, Ingram Olkin, David S. Moore, James V. Sidek, Shanti S. Gupta, Robert V. Hogg, Ralph A. Bradley, and by Harold Hotelling, Jr. (an economist and son of Harold Hotelling) in the same issue of Statistical Science.
3. ^ The qualifications of instructors in statistics have been examined (using data from 2000):
   • Scheaffer, Richard L. Scheaffer and Stasny, Elizabeth A (November 2004). "The State of Undergraduate Education in Statistics: A Report from the CBMS". The American Statistician 58 (4): pp. 265--271. doi:10.1198/000313004X5770. 
4. ^ The unprofessional teaching of statistics by mathematicians (without qualifications in statistics) has been addressed in many articles:
   • Moore, David S (Jan. 1988). "Should Mathematicians Teach Statistics?". The College Mathematics Journal 19 (1): pp. 3-7. http://www.jstor.org/stable/2686686. 
   • Cobb, George W. and Moore, David S (Nov. 1997). "Mathematics, Statistics, and Teaching". The American Mathematical Monthly 104 (9): pp. 801-823. http://www.jstor.org/stable/2975286. 

• Abraham Wald (1947). Sequential Analysis. New York: John Wiley and Sons. ISBN 0471918067. "See Dover reprint: ISBN 0486439127" 
• Abraham Wald (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. 
• Lucien Le Cam (1986). Asymptotic Methods in Statistical Decision Theory. Springer-Verlag. 
• Liese, Friedrich and Miescke, Klaus-J. (2008). Statistical Decision Theory: Estimation, Testing, and Selection. Springer. 
• 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 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 Correlation Pearson product-moment correlation · Rank correlation (Spearman's rho, Kendall's tau) · Confounding variable Linear models General linear model · Generalized linear model · Analysis of variance · Analysis of covariance Regression analysis Linear · Nonlinear · Nonparametric · Semiparametric · Logistic Statistical graphics Bar chart · Biplot · Box plot · Control chart · Forest plot · Histogram · Q-Q plot · Run chart · Scatter plot · Stemplot Category · Portal · Topic outline · List of topics

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