digplanet beta 1: Athena
Share digplanet:

Agriculture

Applied sciences

Arts

Belief

Chronology

Culture

Education

Environment

Geography

Health

History

Humanities

Language

Law

Life

Mathematics

Nature

People

Politics

Science

Society

Technology

The theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics.[1][2] The theory covers approaches to statistical-decision problems and to statistical inference, and the actions and deductions that satisfy the basic principles stated for these different approaches. Within a given approach, statistical theory gives ways of comparing statistical procedures; it can find a best possible procedure within a given context for given statistical problems, or can provide guidance on the choice between alternative procedures.[2][3]

Apart from philosophical considerations about how to make statistical inferences and decisions, much of statistical theory consists of mathematical statistics, and is closely linked to probability theory, to utility theory, and to optimization.

## Scope

Statistical theory provides an underlying rationale and provides a consistent basis for the choice of methodology used in applied statistics.

### Modelling

Statistical models describe the sources of data and can have different types of formulation corresponding to these sources and to the problem being studied. Such problems can be of various kinds:

Statistical models, once specified, can be tested to see whether they provide useful inferences for new data sets.[4] Testing a hypothesis using the data that was used to specify the model is a fallacy, according to the natural science of Bacon and the scientific method of Peirce.

### Data collection

Statistical theory provides a guide to comparing methods of data collection, where the problem is to generate informative data using optimization and randomization while measuring and controlling for observational error.[5][6][7] Optimization of data collection reduces the cost of data while satisfying statistical goals,[8][9] while randomization allows reliable inferences. Statistical theory provides a basis for good data collection and the structuring of investigations in the topics of:

### Summarising data

The task of summarising statistical data in conventional forms (also known as descriptive statistics) is considered in theoretical statistics as a problem of defining what aspects of statistical samples need to be described and how well they can be described from a typically limited sample of data. Thus the problems theoretical statistics considers include:

• Choosing summary statistics to describe a sample
• Summarising probability distributions of sample data while making limited assumptions about the form of distribution that may be met
• Summarising the relationships between different quantities measured on the same items with a sample

### Interpreting data

Besides the philosophy underlying statistical inference, statistical theory has the task of considering the types of questions that data analysts might want to ask about the problems they are studying and of providing data analytic techniques for answering them. Some of these tasks are:

• Summarising populations in the form of a fitted distribution or probability density function
• Summarising the relationship between variables using some type of regression analysis
• Providing ways of predicting the outcome of a random quantity given other related variables
• Examining the possibility of reducing the number of variables being considered within a problem (the task of Dimension reduction)

When a statistical procedure has been specified in the study protocol, then statistical theory provides well-defined probability statements for the method when applied to all populations that could have arisen from the randomization used to generate the data. This provides an objective way of estimating parameters, estimating confidence intervals, testing hypotheses, and selecting the best. Even for observational data, statistical theory provides a way of calculating a value that can be used to interpret a sample of data from a population, it can provide a means of indicating how well that value is determined by the sample, and thus a means of saying corresponding values derived for different populations are as different as they might seem; however, the reliability of inferences from post-hoc observational data is often worse than for planned randomized generation of data.

### Applied statistical inference

Statistical theory provides the basis for a number of data analytic methods that are common across scientific and social research. Some of these are: Interpreting data is an important objective of statistical research:

Many of the standard methods for these tasks rely on certain statistical assumptions (made in the derivation of the methodology) actually holding in practice. Statistical theory studies the consequences of departures from these assumptions. In addition it provides a range of robust statistical techniques that are less dependent on assumptions, and it provides methods checking whether particular assumptions are reasonable for a give data-set.

## Notes

1. ^ Cox & Hinkley (1974, p.1)
2. ^ a b Rao, C. R. (1981). "Foreword". In Arthanari, T. S.; Dodge, Yadolah. Mathematical programming in statistics. Wiley Series in Probability and Mathematical Statistics (New York: Wiley). pp. vii–viii. ISBN 0-471-08073-X. MR 607328.
3. ^ Lehmann, Erich (1959). Testing Statistical Hypotheses.
4. ^ Freedman, David A. (2009). Statistical Models: Theory and Practice (Second ed.). Cambridge University Press. ISBN 978-0-521-67105-7.
5. ^ Charles Sanders Peirce and Joseph Jastrow (1885). "On Small Differences in Sensation". Memoirs of the National Academy of Sciences 3: 73–83. http://psychclassics.yorku.ca/Peirce/small-diffs.htm
6. ^ Hacking, Ian (September 1988). "Telepathy: Origins of Randomization in Experimental Design". Isis 79 (3): 427–451. doi:10.1086/354775. JSTOR 234674. MR 1013489.
7. ^ Stephen M. Stigler (November 1992). "A Historical View of Statistical Concepts in Psychology and Educational Research". American Journal of Education 101 (1): 60–70. doi:10.1086/444032.
8. ^ a b Atkinson, A. C.; Donev, A. N.; Tobias, R. D. (2007). Optimum Experimental Designs, with SAS. Oxford University Press. pp. 511+xvi. ISBN 978-0-19-929660-6.
9. ^ Kiefer, Jack Carl (1985). Brown, Lawrence D.; Olkin, Ingram; Sacks, Jerome et al., eds. Jack Carl Kiefer: Collected papers III—Design of experiments. Springer-Verlag and the Institute of Mathematical Statistics. pp. 718+xxv. ISBN 0-387-96004-X.
10. ^ Hinkelmann and Kempthorne (2008)
11. ^ Bailey (2008).
12. ^ Kish (1965)
13. ^ Cochran, William G. (1977). Sampling Techniques (Third ed.). Wiley. ISBN 0-471-16240-X.
14. ^ Särndal, Carl-Erik, Swensson, Bengt, and Wretman, Jan (1992). Model Assisted Survey Sampling. Springer-Verlag. ISBN 0-387-40620-4.

## References

• Bickel, Peter J. and Doksum, Kjell A. (2001). Mathematical Statistics: Basic and Selected Topics I (Second (updated printing 2007) ed.). Pearson Prentice-Hall. ISBN 0-13-850363-X.
• Lehmann, Erich (1959). Testing Statistical Hypotheses.
• Lehmann, Erich (1983). Theory of Point Estimation.
• Liese, Friedrich and Miescke, Klaus-J. (2008). Statistical Decision Theory: Estimation, Testing, and Selection. Springer. ISBN 0-387-73193-8.

• Davidson, A,C. (2003) Statistical Models. Cambridge University Press. ISBN 0-521-77339-3

Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Statistical_theory — Please support Wikipedia.
A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia.
 99353 videos foundNext >
 1.4 Statistical Decision Theory | 1 Introduction | Pattern Recognition Class 2012The Pattern Recognition Class 2012 by Prof. Fred Hamprecht. It took place at the HCI / University of Heidelberg during the summer term of 2012. Website: http... 03 Introduction to Statistical Decision Theory II (Classification)Course webpage: http://www.helwan.edu.eg/university/staff/Dr.WaleedYousef/PatternRecognition.html Dr. Waleed Yousef webpage: http://www.helwan.edu.eg/univers... @ Feb 4, 2013 Statistical Theory I ความทรงจำดีดี Mod-07 Lec-20 Overview of Statistical Learning Theory; Empirical Risk MinimizationPattern Recognition by Prof. P.S. Sastry, Department of Electronics & Communication Engineering, IISc Bangalore. For more details on NPTEL visit http://nptel... 02 Introduction to Statistical Decision Theory I (Regression)Course webpage: http://www.helwan.edu.eg/university/staff/Dr.WaleedYousef/PatternRecognition.html Dr. Waleed Yousef webpage: http://www.helwan.edu.eg/univers... Molecular Structure & Statistical Mechanics 131B. Lecture 04. Group Theory Applications.Molecular Structure and Statistical Mechanics (Chem 131B) is part of OpenChem: http://learn.uci.edu/openchem This video is part of a 26-lecture undergraduate... 04 Introduction to Statistical Decision Theory III (Classification)Course webpage: http://www.helwan.edu.eg/university/staff/Dr.WaleedYousef/PatternRecognition.html Dr. Waleed Yousef webpage: http://www.helwan.edu.eg/univers... Considering statistical mechanics problems mathematically, as problems in probability theoryThe Sakagawa Group in Keio University's Department of Mathematics studies problems in probability theory, motivated by topics in statistical mechanics. In pr... Almost Quantum Theory: Classical Theories with a Statistical RestrictionMaxEnt 2011 — Robert Spekkens, "Almost Quantum Theory: Classical Theories with a Statistical Restriction" Thursday 14th July 2011 www.maxent2011.org. The Theoretical Limits of Statistical High Dimensional Nearest Neighbor Algor...Google Tech Talks September 6, 2007 ABSTRACT Suppose we have \$n\$ Bernoulli(1/2) long sequences of bits. Let \$n-2m\$ sequences be completely independent, while...
 99353 videos foundNext >
 11 news items
 Testing The Statistical Theory 'Sell In May And Go Away' Seeking Alpha Tue, 11 Jun 2013 02:32:34 -0700 Testing The Statistical Theory 'Sell In May And Go Away'. Jun 11 2013, 05:26 by: Wall Street Elevator | includes: SPY. Introduction. According to the Stock Trader's Almanac, since 1950, the Dow Jones Industrial Average has had an average return of only ... Sorting the Wheat from the Chaff: How Do We Know Which Cancer Therapies ... Huffington Post Wed, 12 Jun 2013 16:10:43 -0700 The development of modern statistical theory and methods in the 20th century, enabling the collaboration between statisticians and oncologists, was stimulated especially by the pioneering work of R.A. Fisher, the famous English statistician and ... The Numbers Game: The Luck of Champions Stanley Cup of Chowder Wed, 12 Jun 2013 07:04:53 -0700 It's going to get a little heavy with the maths down below, but I had to muddle through a lot of this myself not having any schooling in such statistical theory, so maybe my novice point of view can help make this a little less clinical for those of ... The BJP's poll star Business Standard Mon, 10 Jun 2013 09:18:11 -0700 This is not merely an arcane matter of arid statistical theory. Polls in advance of the 2004 elections uniformly predicted a thumping victory for the National Democratic Alliance. We all know how accurate those polls turned out to be. As my co-author ... Industry veteran takes the helm Cogeneration & On-Site Power Production Magazine Thu, 06 Jun 2013 12:53:01 -0700 However, we could showed via statistical theory and real-life tests that having multiple smaller generators connected to the grid could provide a higher stability and security of supply than a few large power plants. CHP and on-site power production ... NorthJersey.com Klapisch: Lower expectations have befallen Yankees NorthJersey.com Sun, 02 Jun 2013 21:47:50 -0700 That's a worrisome trend line, considering Boston's percentage has surged to 76.5 percentage. True, this is all statistical theory, subject to change, but for now it's just another way of saying you can only live so long with Jayson Nix/Reid Brignac at ... Student's social media monitoring tool wins gold Collingwood Enterprise Bulletin Mon, 27 May 2013 05:44:53 -0700 The latter award is presented to an outstanding senior project in statistical theory, while the Actuarial Foundation Award is given to an outstanding project that demonstrates effective use of one or more mathematics, database manipulation and ... NCAA Lacrosse Tournament Final Four: The Look of a Champion College Crosse (blog) Mon, 20 May 2013 12:55:33 -0700 Ignoring statistical theory because I don't want your brain to explode, green means that current ranking positions are similar to that of prior champions; for metrics where wiggle room seems to exist, the cells are shaded yellow. Here's how everything ...
 Limit to books that you can completely read online Include partial books (book previews) .gsc-branding { display:block; }

Oops, we seem to be having trouble contacting Twitter