The unbiasedness of hodges and lehman's test for a composite hypothesis of the man of a distribution with unknown variance

In 1954 Hodges and Lehmann considered the following problem: given is an i.i. normally distributed random sample with variance unknown. Under the null-hypothesis the mean is contained in a prescribed interval. Hodges and Lehmann constructed a test similar on the interval. This test is superior in power to the usual auxiliary procedure applied to this problem. Numerical calculations by Hodges and Lehmann indicated that the test is unbaised, however an analytical proof could not be given. In a recent paper the author proved unbiasedness for levels not too large, the magnitude depending on the sample size. Here the Proof is completed by establishing unbiasedness for all levels.