Identifying common normal distributions

Consider a one-way layout where random samples of data are available from k populations, where the distributions of the data from each population are considered to be completely unknown. This paper discusses a methodology for investigating whether it can be concluded that the k unknown distributions, or any subsets of these distributions, can be taken to be equal to a common normal distribution, and if so it is shown how to identify these common normal distributions. This is accomplished with an exact specified error rate by constructing confidence sets for the parameters of the common normal distributions using Kolmogorov’s (G. Ist. Ital. Attuari 4:83–91, 1933) procedure. The relationship of this methodology to standard tests of normality and to standard procedures for constructing confidence sets for the parameters of a normal distribution are discussed, together with its relationship to functional data analysis and other standard test procedures for data of this kind. Some examples of the implementation of the methodology are provided.