There are many long-standing open problems in mathematics for which a solution has not yet been found. The open problems of statistics , in the general case, have a different character: according to John Tukey [1], difficulties in identifying problems are significantly more significant for statistics than difficulties in solving them. A list of “one or two tasks” (actually 22) was presented by David Cox [2] .
Conclusion and Testing
- How to detect and correct systematic errors , especially in those sciences where random errors are great (a case that Tukey called an inconvenient science).
- The Greybill-Deel score is often used to estimate the total average of two normal populations with unknown and possibly unequal variances. Although this estimate is unbiased in the general case, the question of its admissibility (see en: Admissible decision rule ) remains open. [3]
- Meta-analysis : Although independent p-values can be constructed using the Fisher method , methods for working with dependent p-values are still being developed.
- The Behrens-Fisher problem : Yuri Linnik in 1966 showed that there is no uniformly most powerful criterion for distinguishing two means when variances are unknown and the probabilities are unequal. That is, there is no exact criterion (it is assumed that if the means are actually equal, then the probability of rejecting the null hypothesis is exactly equal to α), which is also the most powerful for all variance values. Although there are many approximate solutions (such as the Welch t-test ), the problem continues to attract attention [4] as one of the classic statistics problems.
- Multiple comparisons : There are various ways to adjust p-values to compensate for parallel or sequential hypothesis testing. Of particular interest is how to simultaneously control the level of errors everywhere, while maintaining statistical power, as well as how to include the interaction between tests in this adjustment. These issues are especially important when the number of simultaneous tests can be very large, as in the case of data analysis from DNA microarrays .
- Bayesian statistics : A list of Bayesian statistics problems has been proposed. [five]
Experiment Planning
- Since the theory of Latin squares is the cornerstone in the design of experiments , the solution of the problems of Latin squares would have instant application in the design of an experiment.
Problems of a More Philosophical Nature
- Sunrise Problem : What is the likelihood that the Sun will rise tomorrow?
- End of the World Theorem : How weighty is the probabilistic argument, which claims to predict the future life of humanity, based only on an estimate of the total number of people born?
- The paradox of exchange : There is still an open problem among subjectivists, a consensus on which has not yet been reached. Examples are:
- The problem of two envelopes
- The paradox of ties
Notes
- ↑ Tukey, John W. Unsolved Problems of Experimental Statistics (English) // Journal of the American Statistical Association : journal. - Journal of the American Statistical Association, Vol. 49, No. 268, 1954. - Vol. 49 , no. 268 . - P. 706-731 . - DOI : 10.2307 / 2281535 .
- ↑ Cox, DR (1984) Present position and potential developments: Some personal views - Design of experiments and regression, Journal of the Royal Statistical Society , Series A , 147 (2), 306-315
- ↑ Nabendu Pal, Wooi K. Lim (1997) “A note on second-order admissibility of the Graybill-Deal estimator of a common mean of several normal populations”, Journal of Statistical Planning and Inference , 63 (1), 71-78 . DOI : 10.1016 / S0378-3758 (96) 00202-9
- ↑ Fraser, DAS; Rousseau, J. (2008) Studentization and deriving accurate p-values. Biometrika , 95 (1), 1-16. DOI : 10.1093 / biomet / asm093
- ↑ Jordan, MI (2011). “What are the open problems in Bayesian statistics?” The ISBA Bulletin , 18 (1).
Links
- Linnik, Jurii. Statistical Problems with Nuisance Parameters. - American Mathematical Society, 1968. - ISBN 0-8218-1570-9 .
- Sawilowsky, Shlomo S. (2002). “Fermat, Schubert, Einstein, and Behrens-Fisher: The Probable Difference Between Two Means When σ 1 ≠ σ 2, ” Journal of Modern Applied Statistical Methods , 1 (2).