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Two‐moment approximations to some null distributions in order‐restricted inference: Unequal sample sizes
Author(s) -
Singh Bahadur,
Wright F. T.
Publication year - 1988
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3314733
Subject(s) - approximations of π , mathematics , null hypothesis , moment (physics) , homogeneity (statistics) , inference , simple (philosophy)
Bartholomew's statistics for testing homogeneity of normal means with ordered alternatives have null distributions which are mixtures of chi‐squared or beta distributions according as the variances are known or not. If the sample sizes are not equal, the mixing coefficients can be difficult to compute. For a simple order and a simple tree ordering, approximations to the significance levels of these tests have been developed which are based on patterns in the weight sets. However, for a moderate or large number of means, these approximations can be tedious to implement. Employing the same approach that was used in the development of these approximations, two‐moment chisquared and beta approximations are derived for these significance levels. Approximations are also developed for the testing situation in which the order restriction is the null hypothesis. Numerical studies show that in each of the cases the two‐moment approximation is quite satisfactory for most practical purposes.