On the asymptotic null distribution of the F ‐statistic for testing a partial null hypothesis in a randomized PBIB design with m associate classes under the Neyman model

In the paper [11], the authors have shown that the familiar central F ‐distribution can be considered as an asymptotically approximate distribution, in a heuristic sense, of the null distribution of the F ‐statistic for testing a partial null hypothesis in a randomized PBIB design with m associate classes under the Neyman model. This result is the most general among those given in previous related papers [6, 7, 8, 9, 11]. The term “asymptotically approximate distribution” has been used, however, in a rather heuristic and unsophisticated way and the rigorous formulation of the notion of asymptotically approximate distribution has not yet been attempted. The purposes of the present article are to give a satisfactory formulation of the asymptotically approximate distribution based on Ikeda's theory of the asymptotic equivalence of probability distributions [2, 5] and to show that the familiar centrals F ‐distribution is “an asymptotically approximate distribution”, in the sense just defined, of the F ‐statistic for testing a partial null hypothesis in the analysis of a PBIB design with m associate classes, even under the Neyman model when it is randomized. The tedious calculations of the previous paper [11] are eliminated and the rather rigid uniformity conditions are weakened in the present article. The present work throws some light on the asymptotic nature of the power function of the F ‐statistic under consideration; this will be presented in a forthcoming paper based on [4].