There are many longstanding unsolved problems in mathematics for which a solution has still not yet been found. The unsolved problems in statistics are generally of a different flavor; according to John Tukey, "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.
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.
Multiple comparisons: There are various ways to adjust p-values to compensate for the simultaneous or sequential testing of hypothesis. 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.
^Tukey, John W. (1954). "Unsolved Problems of Experimental Statistics". Journal of the American Statistical Association (Journal of the American Statistical Association, Vol. 49, No. 268) 49 (268): 706–731. doi:10.2307/2281535. JSTOR2281535.
^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