Unbiasedness on rays of the tests of locations of homogeneous means in isotonic regression

The location of a normal mean can be specified by a vector or by an ellipsoid. In isotonic regression the mean must be from an order restricted cone. For the likelihood ratio tests (LRTs) for the locations of an isotonic mean a simple condition is derived for the tests to be unbiased on a ray. Based on this result the most favorable point within the isotonic regression specification and the set that contains the least favorable points within the null hypothesis are identified. It is also shown that when the null hypothesis points to a homogeneous vector, the LRTs are unbiased.