Certain linear estimation procedures for randomized experimental designs are evaluated relative to the criteria of bias, variance and mean square error. For the designs considered, treatment combinations are randomly allocated to experimental units, the randomness being subject only to a wide symmetry condition. Statistical properties refer to the discrete probabilities induced by the randomization hypothesis. Section 2 defines the basic statistical model and discusses the question of conditional inference relative to this model. Certain vectorial notation and terminology is introduced in Section 3. Although the theory of the paper applies directly to $k$-factor designs with general $k,$ the notation is set up in Section 3 for a three factor design, and the three factor notation is used throughout, except for Section 5 which discusses an even simpler example. Two general classes of linear unbiased estimators are defined in Section 4 and illustrated in Section 5. In Section 6 it is shown that estimators of the types defined in Section 4 have optimum properties in a wide class of linear estimators. Finally, the theory for the basic model is generalized in Section 7 to cover the case of observations with error. Formal proofs of stated theorems are to be found at the ends of Sections 4.2, 4.3 and 6.
"Random Allocation Designs I: On General Classes of Estimation Methods." Ann. Math. Statist. 31 (4) 885 - 905, December, 1960. https://doi.org/10.1214/aoms/1177705665