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]
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]
Behrens–Fisher problem:
Yuri Linnik showed in 1966 that there is no
uniformly most powerful test for the difference of two means when the variances are unknown and possibly unequal. That is, there is no
exact test (meaning that, if the means are in fact equal, one that rejects the
null hypothesis with
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.
Multiple comparisons: There are various ways to adjust p-values to compensate for the simultaneous or
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]
^Tukey, John W. (1954). "Unsolved Problems of Experimental Statistics". Journal of the American Statistical Association. 49 (268): 706–731.
doi:
10.2307/2281535.
JSTOR2281535.
^Cox, D. R. (1984). "Present Position and Potential Developments: Some Personal Views: Design of Experiments and Regression". Journal of the Royal Statistical Society. Series A (General). 147 (2): 306–315.
doi:
10.2307/2981685.
JSTOR2981685.
^Pal, Nabendu; Lim, Wooi K. (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: 71–78.
doi:
10.1016/S0378-3758(96)00202-9.