An experiment with n observations X1,..., Xn is planned, with each Xi distributed as N(, 1). However,

An experiment with n observations X1,..., Xn is planned, with each Xi distributed as N(θ, 1). However, some of the observations do not materialize (for example, some of the subjects die, move away, or turn out to be unsuitable). Let Ij = 1 or 0 as X j is observed or not, and suppose the Ij are independent of the X’s and of each other and that P(Ij = 1) = p for all j.
(i) If p is known, the effective sample size M =  Ij is ancillary.
(ii) If p is unknown, there exists a UMP unbiased level-α test of H : θ ≤ 0 versus K : θ > 0. Its conditional level (given M = m) is αm = α for all m = 0,..., n.