In recent work by O'Reilly, a necessary and sufficient condition for the existence of an unbiased estimate of the distribution function of a "future" observation was given. The result was obtained under the condition that the model had full rank. Here, the result is generalized to any number of future observations and the full rank condition is relaxed. The corresponding uniformly minimum variance unbiased estimator is identified from the density estimates given by Ghurye and Olkin.
"On a Criterion for Simultaneous Extrapolation in Nonfull Rank Normal Regression." Ann. Statist. 4 (3) 625 - 628, May, 1976. https://doi.org/10.1214/aos/1176343469