A Problem of Minimax Estimation with Directional Information
Thomas S. Ferguson
This problem is in the area of minimax selection of experiments.
Nature chooses a number x in the closed interval [-1,1]. The
statistician chooses a number y in the same interval (an experiment)
and is informed whether x<y, x=y, or x>y. Based on this
information, the statistician then estimates x with squared error
loss. The minimax solution of this problem is found. The minimax
value is 1/(2e). The least favorable distribution involves the
truncated t-distribution with two degrees of freedom. The minimax
choice of experiment involves the truncated t-distribution with zero
degrees of freedom.