List of unsolved problems in statistics
There are many longstanding unsolved problems in mathematics for which a solution has still not yet been found. The notable unsolved problems in statistics are generally of a different flavor; according to John Tukey,[1] "difficulties in identifying problems have delayed statistics far more than difficulties in solving problems." A list of "one or two open problems" (in fact 22 of them) was given by David Cox.[2]
Inference and testing
- How to detect and correct for random errors are large (a situation Tukey termed uncomfortable science).
- The Graybill–Deal estimator is often used to estimate the common mean of two normal populations with unknown and possibly unequal variances. Though this estimator is generally unbiased, its admissibility remains to be shown.[3]
- Meta-analysis: Though independent p-values can be combined using Fisher's method, techniques are still being developed to handle the case of dependent p-values.
- probability exactly α) that is also the most powerful for all values of the variances (which are thus nuisance parameters). Though there are many approximate solutions (such as Welch's t-test), the problem continues to attract attention[4]as one of the classic problems in statistics.
- sequential testing of hypotheses. Of particular interest is how to simultaneously control the overall error rate, preserve statistical power, and incorporate the dependence between tests into the adjustment. These issues are especially relevant when the number of simultaneous tests can be very large, as is increasingly the case in the analysis of data from DNA microarrays.[citation needed]
- Bayesian statistics: A list of open problems in Bayesian statistics has been proposed.[5]
Experimental design
- As the theory of Latin squares is a cornerstone in the design of experiments, solving the problems in Latin squares could have immediate applicability to experimental design.[citation needed]
Problems of a more philosophical nature
- Sampling of species problem: How is a probability updated when there is unanticipated new data?[6]
- human racegiven only an estimate of the total number of humans born so far?
- Exchange paradox: Issues arise within the subjectivistic interpretation of probability theory; more specifically within Bayesian decision theory.[citation needed] This is still an open problem among the subjectivists as no consensus has been reached yet. Examples include:
- The two envelopes problem
- The necktie paradox
- Sunrise problem: What is the probability that the sun will rise tomorrow? Very different answers arise depending on the methods used and assumptions made.
Notes
- JSTOR 2281535.
- JSTOR 2981685.
- .
- .
- ^ Jordan, M. I. (2011). "What are the open problems in Bayesian statistics?" (PDF). The ISBA Bulletin. 18 (1): 1–5.
- S2CID 9416747.
References
- Linnik, Jurii (1968). Statistical Problems with Nuisance Parameters. American Mathematical Society. ISBN 0-8218-1570-9.
- Sawilowsky, Shlomo S. (2002). "Fermat, Schubert, Einstein, and Behrens–Fisher: The Probable Difference Between Two Means When σ1 ≠ σ2". Journal of Modern Applied Statistical Methods. 1 (2). .