Utente:Grasso Luigi/sandbox3/Teoria statistica
La teoria della statistica fornisce una base per l'intera gamma di tecniche, sia nella progettazione che nell'analisi dei dati, che vengono utilizzate nelle applicazioni della statistica.[1][2] La teoria comprende gli approcci ai problemi di decisione statistica e all'inferenza statistica, e le azioni e deduzioni che soddisfano i principi di base enunciati per questi diversi approcci. All'interno di un dato approccio, la teoria statistica offre modi per confrontare le procedure statistiche; può trovare la migliore procedura possibile in un certo contesto per determinati problemi statistici o può fornire indicazioni sulla scelta tra procedure alternative.[2][3]
A parte le considerazioni filosofiche su come prendere decisioni e inferenze statistiche, gran parte della teoria statistica consiste in statistiche matematiche ed è strettamente collegata alla teoria della probabilità, alla teoria dell'utilità e all'ottimizzazione.
Obiettivo modifica
La teoria statistica fornisce una logica sottostante e fornisce una base coerente per la scelta della metodologia utilizzata nelle statistiche applicate.
Modellazione modifica
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:
- Sampling from a finite population
- Measuring observational error and refining procedures
- Studying statistical relations
Statistical models, once specified, can be tested to see whether they provide useful inferences for new data sets.[4]
Raccolta dati modifica
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:
- Design of experiments to estimate treatment effects, to test hypotheses, and to optimize responses.[8][10][11]
- Survey sampling to describe populations[12][13][14]
Riepilogo dei dati modifica
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 modifica
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 modifica
Statistical theory provides the basis for a number of data-analytic approaches that are common across scientific and social research. Interpreting data is done with one of the following approaches:
- Estimating parameters
- Providing a range of values instead of a point estimate
- Testing statistical hypotheses
Many of the standard methods for those approaches 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 given data set.
See also modifica
Note modifica
- Citazioni
- ^ Cox, D.R., Hinkley, D.V. (1974) Theoretical Statistics, Chapman & Hall. ISBN 0-412-12420-3
- ^ a b (EN) Arthanari T. S.; Dodge Yadolah; Rao C. R., Foreword, in Mathematical Programming in Statistics, New York (USA), John Wiley & Sons, 1981, pp. vii–viii, ISBN 0-471-08073-X, MR 607328.
- ^ (EN) Lehmann E. L.; Romano J. P., Testing Statistical Hypotheses, 3ª ed., Springer Nature, 2008, ISBN 978-0387988641.
- ^ Freedman (2009)
- ^ Charles Sanders Peirce and Joseph Jastrow, On Small Differences in Sensation, in Memoirs of the National Academy of Sciences, vol. 3, 1885, pp. 73–83. http://psychclassics.yorku.ca/Peirce/small-diffs.htm
- ^ Ian Hacking, Telepathy: Origins of Randomization in Experimental Design, in Isis, vol. 79, n. 3, September 1988, pp. 427–451, DOI:10.1086/354775.
- ^ Stephen M. Stigler, A Historical View of Statistical Concepts in Psychology and Educational Research, in American Journal of Education, vol. 101, n. 1, November 1992, pp. 60–70, DOI:10.1086/444032.
- ^ a b Atkinson et al. (2007)
- ^ Jack Carl Kiefer, Jack Carl Kiefer: Collected papers III—Design of experiments, Springer-Verlag and the Institute of Mathematical Statistics, 1985, pp. 718+xxv, ISBN 0-387-96004-X.
- ^ Hinkelmann & Kempthorne (2008)
- ^ Bailey (2008).
- ^ Kish (1965)
- ^ Cochran (1977)
- ^ Särndal et al. (1992)
- Riviste
Bibliografia modifica
- (1876), "Note on the Theory of the Economy of Research" in Coast Survey Report, pp. 197–201 (Appendice No. 14), NOAA PDF Eprint.
- Ristampa 1958 in Collected Papers of Charles Sanders Peirce 7, paragrafi 139–157
- Ristampa 1967 in Operations Research 15 (4): pp. 643–648, Abstract da JSTOR.
- (1967) Note on the Theory of the Economy of Research, in Operations research, vol. 15, n. 4, 1967, p. 643, DOI:10.1287/opre.15.4.643.
- (1877–1878), "Illustrations of the Logic of Science"
- (1883), "A Theory of Probable Inference"
- and Jastrow, Joseph (1885), "On Small Differences in Sensation" in Memoirs of the National Academy of Sciences 3: pp. 73–83. Eprint.
- (EN) Bickel, Peter J.; Doksum, Kjell A., Mathematical Statistics: Basic and Selected Topics, I, 2 (ristampa 2007), Pearson Prentice-Hall, 2001, ISBN 0-13-850363-X.
- (EN) Davison, A.C. (2003) Statistical Models. Cambridge University Press. ISBN 0-521-77339-3
- (EN) Lehmann, Erich, Theory of Point Estimation, 2ª ed., Springer, 1998, ISBN 978-0387985022.
- (EN) Liese, Friedrich; Miescke, Klaus-J., Statistical Decision Theory: Estimation, Testing, and Selection, Springer, 2008, ISBN 0-387-73193-8.
External links modifica
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