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

Abstract

AbstractIn 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].