A note on the uniform distribution on the arcsin points

In his book Pukelsheim pointed out that designs supported at the arcsin points are very efficient for the statistical inference in a polynomial regression model. In this note we determine the canonical moments of a class of distributions which have nearly equal weights at the arcsin points. The class contains the D-optimal arcsin support design and the D1-optimal design for a polynomial regression. The results allow explicit representations of D-, and D1-efficiencies of these designs in all polynomial models with a degree less than the number of support points of the design.