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.