GOODNESS-OF-FIT TEST USING PAIRED χ^2-VALUES

A defect inherent in the traditional goodness-of-fit χ^2-test is pointed out, see Miyatake (1985), in which the number of classes cannot be fixed a priori. Let a population be divided into l classes and χ_<l-1>^2 be the χ^2-value obtained from a hypothesis H_0 and observed frequency in each class. We assume that l is even, and put together these classes two by two, getting a coarse division of k classes (k=l/2). Let χ_<k-1>^2 be the χ^2-value obtained from this coarse division under the same H_0 and observed frequencies. Then we get the decomposition χ_<l-1>^2=χ_<k-1>^2+χ_<l-k>^2, where χ_<l-k>^2 is a χ^2-variable of degree of freedom l-k. It will be shown that the use of the pair (χ_<k-1>^2, χ_<l-k>^2) is more effective in the goodness-of-fit test, in comparison with the traditional method using the single χ_<l-1>^2.